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ARITHMETIC

FOR YOU7

By JAMES R. OVERMAN

FREDERICK S. BREED

CLIFFORD WOODY

Illustrated by

Miriam Story Hurford, A. F. Hurford

LYONS AND CARNAHANChicago Dallas Los Angeles Atlanta New York



Copyright 1945

by

Lyons & Carnahan

Printed in the United States of America

51J53

When requesting answers for this hook, specify numher AFY1457.



FOREWORD

Time marches on, says a familiar voice, and with it social

change. Education, too, is part and parcel of our general Bocial

structure and must be continually reshaped to fit our changing

purposes.

In the spirit of progress Arithmetic for You is now offered as the

successor of the Child-Life Arithmetics. The many good features

of the former books have been retained. Revisions have been

made to meet changed social conditions and in the light of what

has been learned from the latest developments of psychology,

from educational experiment, and from the operation of the earlier

series in thousands of classrooms. The following are special

features of the seventh and eighth grade books.

The books are highly socialized. Each arithmetical topic-

grows out of an interesting social situation and is then applied to

others. Use is made of the pupil's growing interest in the activities

of the adult world. The organization is largely social, centering

around topics, such as Using Arithmetic in the Home. This

social organization is important since the purpose is to teach the

pupil to use mathematics in life.

In these books, learning becomes an active process—the pupils

learn by doing. The need for new information arises in connec-

tion with some interesting life situation, and the pupils discover

the needed knowledge inductively and form their own generaliza-

tions. The experimental method is frequently used.

Because of the increased use of algebra and geometry in industry

and many other fields, the more useful concepts, methods, and

facts of these subjects must now be included in seventh and

eighth year arithmetics. In these books this is done in a simple,

concrete way that not only greatly increases the pupils' stock of

mathematical tools but actually makes arithmetic easier and more

interesting.

One of the major objectives of these books is the development

and maintenance of skill in the fundamentals. At the beginning

of each year each process is reviewed, and the common sources of

error are pointed out before starting practice. Practice is con-

centrated where needed by the systematic use of diagnostic



testa and remedial practice. Pupil Lnteresl is secured by frequent

self-measuremenl and the stimulation of the desire for self-im-

provement. Review practice exercises are of the mixed type,

whi.h has been shown by experiment to be superior, except on a

nrw process. Accuracy is developed by systematic checking,

reduction of score for errors, and by avoiding the use of a time

limit so shori as to force many pupils to sacrifice accuracy for

speed. Finally, the co-operation of the pupils is secured by dis-

cussing with them, in simple terms, the psychology of effective

practice.

A second important objective of these books is the develop-

ment of the ability to use arithmetical tools in the solution of

simple problems. Each new arithmetical topic grows out of a

social situation and is immediately applied to others. The tool

and the social situations in which it is used are presented together

and kept together. This constant emphasis on the meaning and

uses of the arithmetical fact or process is the primary secret of

developing the ability to apply mathematics in life. Further aid

is given the pupils by a simple presentation of the steps in problem

solving. Frequent use is also made of such devices as diagrams,

problems with extra data, problems with insufficient data,

generalized problems, and the making of problems to fit given

data.

Psychology today is emphasizing teaching for spread and trans-

fer. Experiments show that much that is learned in connection

with one situation can be and is used in other related situa-

tions, if the first teaching is made sufficiently general. Every

effort is made in these books to secure as wide a spread as possible

and to avoid teaching in watertight compartments . The pupils

are encouraged to make their own generalizations, and these are

prominently displayed for emphasis and for ease of future refer-

ence

THE AUTHORS.

IV



CONTENTS

CHAPTER PAGE

1 Things We Learned Last Year 1

2 Improving Your Work with

Fractions 39

3 Improving Your Work with

Decimals... 59

4 Learning to Work with Per Cents.. 87

5 Using Arithmetic in the Home 127

6 Helps in Problem Solving... 179

7 Practical Measurements 203

8 Picturing Numbers. Graphs... 249

9 Using Geometry 279

Practice Tests 307-340

Score Card 341

tables for reference ...342-343

Index 344



C II A P T E R

/

/

Zkings We teamed Cast year

IMPROVING YOUR WORK WITH WHOLE NUMBERS

The Indians who lived in North America before the coming

of the white men, probably used little arithmetic except simple

counting. Today, everyone has to make considerable use of

this subject.

Many of you, when you are older, expect to work in stores,

offices, or shops; some of you will be aviators, farmers, nurses,

carpenters, teachers, bookkeepers, stenographers, doctors,

dentists, lawyers, mechanics, engineers, contractors, or archi-

tects. All of you will be interested in running your own per-

sonal affairs economically. For many of these types of work

you will need to study more advanced mathematics, but for

all of them you will need arithmetic and some simple algebra

and geometry.

In this book you will learn how the Cunninghams, a typical

American family, used mathematics. You will become ac-

quainted with such subjects as accounts, budgets, banks,

interest, formulas, areas, volumes, scale drawings, statistical

tables, graphs, and many others.

There are four members of the Cunningham family, Mr. L.

D. Cunningham, Mrs. Cunningham, Dick, and Agnes.

In this chapter you will find out how accurately and rapidly

you can add, subtract, multiply, and divide whole numbers;

and will receive many suggestions that will help you to im-

prove your work. Accuracy and reasonable speed are both

important in all practical applications of arithmetic.



I.-* '* '...,..- 25 «—»'«»rf'

OUR NATIONAL PARKS

One summer the Cunninghams took an automobile trip to

Yellowstone National Park. Before starting, Dick found the

following table in a booklet Our National Parks.

State

or

Territory



Z,i z 0, 7 9 Zj/ 45

Reading Large Numbers

The method of writing numbers which we use was perfected

by the Hindus over a thousand years ago. Since it was brought

into Europe by theArabs, it is often called the Arabic system.

In reading large numbers, you must know the names and

values of each of the different places. The names of the first

ten places, starting at the right, are given on the number

written on the blackboard at the top of the page.

This number is read Two billion, one hundred twenty-eight

million, seven hundred ninety-two thousand, one hundred forty-

five. Notice that you should not use the word and between

the different parts of the number.

1. Copy this statement and complete it: Each place, as

you go to the left, has a value times as great as the pre-

ceding place.

2. Learn the names of the first ten places, in order, starting

fromthe right. Practice giving them from memory.

3. Read the following. Do not use the word and between

the different parts of the numbers.

a. 386 b. 702 c. 500 d. 7,000 e. 28,597 f. 396,085

g. 97,489,107 h. 3,148,307,209 i. 5,891,283,756

4. Read the numbers in the table on page 2.



*^y Ufc*Aft*>*>- vv /.,>*>»--..

RUNNING THE 100-YARD DASH

When Dick's school organized a track team, Dick and

several other boys decided to try the hundred-yard dash. The

first afternoon the coach had all of the boys run a hundred

yardswhile he timed them. It took Dick just 13 seconds.

After that, Dick practiced almost every afternoon and asked

the coach for suggestions. By the end of the year he could run

a hundred yards in 12 seconds. Next year, by practicing and

following the suggestions of the coach, Dick expects to improve

this record.

You can improve your record in arithmetic in the same way

that Dick improvedhis record in the hundred-yard dash, and

you can have just as much fun as Dick had. In this book you

will find a number of Improvement Tests. By taking these

tests, you can find out how rapidly and how accurately you can

work with whole numbers, fractions, and per cents. You will

take each of these tests several times this year. By keeping

your scores, you can measure the improvement which you make.



Improvement Tests 5

MEASURING YOUR ACCURACY AND SPEED

The Improvement Tests are intended to help you measure

your increase in both accuracy and speed in the use of mathe-matical tools. Success in practical life calls for a high degree of

accuracy and a reasonable degree of speed. Speed without

accuracy is worse than useless. For this reason your score on

these tests will be the number of examples you have right less

the number wrong, or omitted. This means that you must take

every precaution to avoid errors. You will make better scores

and more improvementif

you follow the suggestions givenbelow.

Direction for Taking Tests

1. Do not copy the examples unless it is necessary to do so.

Place the top edge of a sheet of paper below the first row and

write the answers, or work the examples, on the paper.

After you finish the first row, fold the paper so the answers

are hidden and place the folded edge below the second row of

examples. Do this for each row.

2. Work the examples in order. Your score is decreased by

one for each example omitted.

3. Do not hurry. Errors lower your score.

4. Work carefully. Carelessness causes errors.

5. Check each step before starting the next.

6. Keep your attention fixed on the example until you

finish it. If your attention wanders, you are liable to make a

mistake.

7. Rest between examples, if necessary.

8. Check each example before starting the next. This is

time well spent, since each example wrong reduces your score

by one.

9. Work as fast as you can with comfort. If you always do

this, you will find that your speed will gradually increase with-

out loss of accuracy.



IMPROVEMENT TEST NUMBER ONE

Graphs

PICTURING YOUR PROGRESS

Mary Andrews, a pupil in the seventh year of the Roosevelt

Junior High School, made the picture shown above to show

her own progress and that of her class on Improvement Test

Number One. Such a picture is called a graph.

1. How many times did Mary and her class take the test?

2. What was Mary's score on the first trial? What was the

class average? Answer the same questions for the third trial.

The sixth. The tenth.

3. On which trials was Mary's score below the class average?

Above?

4. On which trials did Mary make a higher score than on

the previous trial? A lower score? The same score?

5. Did Mary make more or less progress than the class as

a whole?

Measuring YourProgress

Take Improvement Test Number One. Copy the Score

Card given on page 341 in a notebook and record your score.

On the next page in your notebook start a graph like Mary's,

showing your own progress and that of your class on Improve-

ment Test Number One.



Improvement Test Number One 7

THE FOUR PROCESSES WITH WHOLE NUMBERS

Work each example on your paper.

(

'lucki lie

addition examplesby adding again in the reverse order. Check the subtraction

examples by addition. Go over each step in the multiplication

and division examples a second time before going on to the next

step. Time, 16 minutes.

1. Copy and



8 How Addition Is Used

1. Ruth saved the money which she earned and deposited

it in a savings bank. She deposited $2.37 on June 1st, 76^ on

July 12th, $1.89 on July 30th, and $2.17 on August 5th. Howmuch did she deposit altogether?

2. Last summer Joseph went on an automobile trip with

his father. They drove 231 miles the first day, 187 the second,

254 the third, 89 the fourth, and 315 the last day. How far

did they drive on the trip?

3. Louise wrote a short sentence describing the kind of

problems in which addition is used. Study her statement at

the bottom of the page. Is it correct?

4. Make up at least one good problem to be solved by addi-

tion. Then solve it.

The Vocabulary of Addition

If you cannot answer the questions below, look up the mean-ing of addends, sum, and total in the dictionary.

1. Copy and fill the blanks in the following statements.

a. In the example 8+9 = 17, the addends are and

, and the sum is .

b. Total is another name for the .

c. The sign + is read

2. Name the addends and the sum in each of the following

examples.

32

7+9 = 16 384 15

598 49

2+5+8 = 15 982 96

Addition is used to find the combined value of two or

more numbers.

Vocabulary: Addends, sum, total.



Avoiding Errors in Addition 9

If you are like most seventh grade pupils, you know how to

add but make too many mistakes. Most errors in addition

occur at the points described below. Study the explanations

carefully. If you master these troublesome points, you should

be able to increase your accuracy and speed.

1. Basic Addition Facts. On page 309 you will find the

100 basic facts needed in addition. See how rapidly you can

give the sums. Note the ones that cause you trouble and give

them extra practice.

2. Higher Decade Facts. In adding more than two

numbers, you must be able to recognize the basic facts when

you meet them in the tens, twenties, etc. In the example

at the left, starting at the top, you must know 5+7,

12+4, and 16+8. Do you recognize the last two? Al-

though these are not among the 100 basic facts, they are

— closely related to them as shown in the examples below.

2



10 \>oidiii£ Errors in Addition

3. Thinking Results Only. Agnes Cunningham was slow

in addition. In adding the numbers at the left, she thought

5 and 7 are 12 , 12 and 4 are 16 , and 16 and 8 are

24 . Her teacher suggested that she save time by looking

at the numbers and thinking 12, 16, 24 . Can you do

this? Turn to Practice Test Number 3, on page 310, and

practice. Think results only.

4. Carrying. Three types of errors are common when the

sum of a column is more than nine: (a) Failure to add the

tens of the sum to the next column to the left; (b) adding the._ wrong figure of the sum to the next column;

' ~ ,„ (c) carrying when there is nothing to carry.

— — — Explain the error in each of the examples at

61 54 68the left> Add correct iy .

5. Placing the Numbers. Mary wanted to add 324 and

15. She wrote them and added as shown at the left. Is

this correct? Why? State a rule about writing whole

—^- numbers in addition. Which column must be kept474

straight? Why? Copy and add correctly. Turn to

Practice Test Number 4, on page 311, for more practice.

6. Carrying in Adding Columns. Most pupils add down-

ward and check by adding upward. In adding downward, carry

to the top of the next column. In adding upward, carry to the

bottom. Why? The number carried should not be written but

added mentally. Why? Turn to Practice Tests Number 4 or 5,

on page 311, for further practice in carrying.

7. Compound Numbers. Dick added the numbers at the

left. Explain how he changed 5 ft. 15 in. to 6 ft. 3 in. Is his

work correct? Dick decided it was not necessary to write the

3 ft. 7 in. 5 ft. 15 in. He changed the 15 in. to 1 ft. 3 in.

2 ft. 8 in. mentally, wrote the 3 in. and added the 1 ft. to

5 ft. 15 in. the other feet. Can you do this?

or For further practice, turn to Practice Test

6 ft. 3 in. Number 48a, on page 338.



Safety First. Checking Addition LI

Even the most accurate computers occasionally make mis-

takes, so it is customary in practical work, where the result must

be right, to check all computations. You should form the habit

of checking while you arc in school.

389

756 Dick added the numbers at the left, starting at

898 the top. To check his result he added the numbers

577 a second time, starting at the bottom. Is his resull

635 correct?

3255

1. Add in each of the following examples, and check.

583 2859 136 756 41

344 6139 89 289 388

206 7581 7 175 285

347



12 Diagnostic Tests in Addition

LOCATING YOUR DIFFICULTIES

Do not ropy the examples unless directed to do so. (See page 5.)

Add, and check each example before starting the next. Work as

fast as you can without hurrying.



mump*

\



14 How Subtraction Is Used

1. Henry is treasurer of his class. He collected $13.85 in

dues and paid bills amounting to $8.37. How much was left

in the treasury?

2. Mary took an automobile ride. When she started, the

speedometer read 8154 miles. When she got back, it read

8932 miles. How far did she ride?

3. Helen's little sister is saving coupons from breakfast

food to get a doll. She has saved 37 coupons. How many

more must she save if the doll costs 65 coupons?

4. John wrote the statement at the bottom of the page

describing the three uses of subtraction. Tell which of the

first three problems illustrates each use.

5. Write a problem illustrating each of the three uses of

subtraction. Solve these problems.

The Vocabulary of Subtraction

1. Copy and fill the blanks in each statement:

a. In the example 12-7 = 5 the subtrahend is ,the

minuend is , and the difference is

b. Remainder and balance are other names for the

c. The sign — is read

2. Name the minuend, subtrahend, and difference in each

of these examples

15-9 = 6 425 3829

138 153

7 from 9 leaves 2 287 3676

Subtraction is used for three purposes:

a. To find how many are left when a smaller number

is taken from a larger number.

b. To find how much larger, or smaller, one number is

than another.

c. To find how much must be added to a smaller number

to make it equal to a given larger number.

Vocabulary: Minuend, subtrahend, remainder, difference,

balance.



Avoiding Errors in Subtraction i >

You have probably heard the old saying that Prevention

is better than cure. It is better to go to the dentist before

you have a bad tooth. The same is true in arithmetic. It is

better to avoid errors than to correct them. You can prevent

errors in subtraction by watching the following troublesome

points.

1. Basic Subtraction Facts. Turn to page 312 and see

how rapidly you can give the answers to the subtraction facts.

Watch the ones that cause you trouble and give them extra

practice.

2. Smaller Minuend Figure. If you know the facts,

subtraction is easy until you come to a figure in the minuend

that is smaller than the figure below it

in the subtrahend. Subtract 25 from 83.

One method is shown in the box. How

Tens



L6 Avoiding Errors in Subtraction

6. ( Jompoxjnd Numbers.

r> yd. 1 ft.

2 >d. 2 ft.

2 yd. 2 ft.

1 yd.1 ft.

2yd. 2 ft.

2 v<l. 2 ft.

Subtract 2 yd. 2 ft. from 5 yd. 1 ft.

Study the solution in the box. Ex-

plain how you get the 4 ft. in the

minuend. The 4 yd. Think the

chunges in the box. Do not write

them. Turn to Practice Test Number

48 b, page 388.

Safety First. Checking Subtraction

Ralph had $87.25 in the bank. He wrote a check for $2.50

and subtracted to find out how much he would have left in the

87.25 bank. As a check he added his result, $85.75, to the

2.50 subtrahend, $2.50, and got $88.25. Was his subtraction

85/75 correct? If not, copy and subtract correctly. Check.

1. Ralph asked if he should write the sum of the subtrahend

and the minuend in checking. Histeacher suggested that it would

be better to check mentally. Why?

2. Subtract and check.

1347



Diagnostic Tests in Subtraction 17


Do not copy tin* examples unless directed to do so. (Sec page 5.

Subtract, and write the remainders on your paper. Check each

example before starting the next. Work as fasl as you can without

hurrying.

1.



18 Ccnpar. , Numbers by Subtraction

W.en Dick Cunning came home tarn schoo^he

boasted, Our baseballteam

,

made 12

„^1

mns did

Mrs. Cunningham laughed nd *£ H^the other team make? It is the

fe to

An important u*»**£*£ a^you^ w

rrsSKS-«—rcompared

*?»»—* -- -8

-Which team won? By how many runs. v

, Dick is 5 ft. 4 in. tall, and his sister Agnes ts 4 ft. 9 m.

lal,- how much taUet^ ^3 -

LaSf^2 871 Their otal expenses for the year m-

^uchdtd the sales exceed the expenses?

SSS^dt received how many more

than Mr. Willkie? , , ,

tones anu phe increase .

population was 138,439,0by. rin

neoole lived in New York?people live

subtraction,

• TtmTh: rrs^Vaaiofin^ompanng

number,

^mwS of hree uses of subtraction, given on page 14,

does the name remainder come?

T From which of the three uses of subtract™ does the

.- ~~,^V

name balance come?



Mysterious Addition 19

Agnes Cunningham learned a trick in school and played it

on her father. She asked Mr. Cunningham to write two

numbers of the same number of figures. He wrote3 ' 8

? 3789 and 2495. Agnes then wrote 7504 and asked

her father to write another number. He wrote 8726.

' * Agnes wrote 1273 and told her father that she could

8tell him the sum without adding. Mr. Cunningham

12' J added the numbers and Agnes told him the sum was

23787 23,787. Was Agnes right? Check by adding.

Agnes's father tried to guess how she got the sum but finally

gave up. Agnes explained. The trick is in choosing the num-

bers I wrote. I chose the third number so that the sum of the

second and third numbers is 9999, and the fifth number so the

sum of the fourth and fifth numbers is also 9999. The sum of

the last four numbers is 9999+9999, or 19,998, to be added to

the first number. But 19,998 is just 2 less than 20,000, so

adding 19,998 is the same as adding 20,000 and subtracting 2.

I wrote the sum by writing 2 in front of the first number and

subtracting 2 from the last figure of this number.

1. Do you see how to play this trick? Try it on some of

your friends.

2. Agnes played the trick on her mother. Mrs. Cunningham

wrote two numbers. Then Agnes wroteone. Mrs. Cunning-

ham's first number was 50,761. What was the sum? Explain.

3. Agnes played the trick on her brother, Dick. This time

she had Dick write two numbers; then Agnes

3 5 8 9 2 wrote one. Dick wrote one, Agnes wrote one,

5 12 4 8 Dick wrote one, and Agnes wrote one. The

? ? ? ? ? numbers Dick wrote are shown at the left. Those

3 6 9 18 that Agnes wrote are represented by question

? ? ? ? ? marks. Can you supply the numbers Agnes

7 16 4 3 wrote? How did she find the result? Explain.

? ? ? ? ? Copy and complete. Check by adding.



\N.

«'j* * ''

How Multiplication Is Used

1. Ruth and George found the cost of 4 pounds of crackers

at 12c1

a pound. Which solution was shorter?

2. Make a short statement telling how multiplication is used.

3. Write at least one good multiplication problem. Solve.

The Vocabulary of Multiplication

1. Copy and fill the blanks in the following:

a. In the example 5X7 = 35, the multiplier is 5, the

multiplicand is ,and the product is

b. The sign X is read

2. Name the multiplier, multiplicand, and product in each

of the following examples:

7X9 = 63 84

3

252

Multiplication is used to find the total value of several

equal numbers. It is a short method of adding when the

addends are all equal.

Vocabulary: Multiplier, multiplicand, product.

20



Avoiding Errors in Multiplication 21

Most pupils can improve their work in multiplication by

watching the following common sources of trouble.

1. Basic Multiplication Facts. Turn to page 315 and see

how rapidly you can give the answers to the multiplication facts.

Watch for the ones that cause you trouble and give them extra

practice.

2. Zeros. John said that 7X0 = 7. Is this correct? Explain.

How much is 0X7? Explain. Remember that any number

times zero or zero times any number is zero.

3. Carrying. Many mistakes in multiplication are caused

by forgetting to add the tens from the preceding multiplication,

adding the wrong number of tens, or making an error in adding

the tens. Practice on Practice Test Number 12, page 316, will

help you avoid these errors.

4. Placing Partial Products. Many errors are due to

placing partial products in the wrong place. Such mistakes are

common when there are zeros in the multiplier. Remember to

always place the right-hand figure of the partial product directly

under the figure by which you are multiplying. Study the follow-

ing examples and explain. Copy and correct any that are wrong.

Turn to Practice Tests Number 13 and 14, page 817, for further

practice.

233 320

320 30

322



22 Safety First. Checking Multiplication

First Method. There are several methods of check-

837 ing multiplication. The three suggested below are

68 probably the most useful. You can increase your ac-

6695 curacy by using them regularly. One of the best ways

5022 is t0 nnd an error as soon as lt is made-Check each

,6

step by going over it a second time before starting the

next. There are three steps in the example at the left;

the multiplication by 8, then the multiplication by 6, and finally

the addition of the partial products. Check each step. Is the result

correct?

Multiplication Check SECOND METHOD. You know that it

38 27 makes no difference in the product which

27 38 number is written first when you multiply.

27X38 is the same as 38X27.

In the example at the left the product

is checked byinterchanging the multi-

1026 1026 plicand and multiplier. Go over each

multiplication. Is the result correct?

266 216

76 81

Multiplication Check Third Method. If you divide the

328 328 product by the multiplier, the result

8 8)2624 always equals the multiplicand. Thus

4X7 = 28, and 28^-4 = 7. Also 5X9 =

2624 45?anc[ 45-^5 = 9. This principle can

be used to check multiplication as shown

at the left. Go over both the multiplication and the division. Is

the result correct?

Go overeach step in multiplication a second time before

starting the next. Check the final result by either

a. Multiplying a second time, interchanging the multi-

plier and multiplicand, or

b. Dividing the product by the multiplier. The result

should equal the multiplicand.



Safety First. Checking Multiplication 23

1. Agnes said, You can often find errors in multiplication

by going over each step a second time, but there IS always the

chance that, you will make the same mistake twice. For this

reason you should use either the second or third method (page 22;

for checking your final product. Do you agree'.'

2. Multiply and check each step by going over it a second time

before starting the next, Check the final result by interchanging

the multiplier and multiplicand.

38 49 756 836 597

76 58 395 472 869

3. Check each of the multiplications given 348 46

at the side by interchanging the multiplier _7 _34

and multiplicand. Then check each by divi- 2426 184

sion. Which method of checking is shorter 128

for the first example? For the second? 1464

4. Multiply 756 by 32 and check each step by going over it

a second time. Check the final result by the easier method.

5. Multiply and check each step by going over it a second

time before starting the next step. Check the final result by the

most convenient method.

38917 49719

306 8

23842 38795

_9 79

39187 87562

7 86

Locating Your Difficulties in Multiplication

On the next page you will find several Diagnostic Tests. Use

these to locate your weaknesses in multiplication.

432



24 Diagnostic Tests in Multiplication


Do not copy the examples. (See page 5.) Multiply. Check

each step before starting the next. Note the trouble spots. Work

as fast as you can without hurrying.

9



Saving Time. Short Methods in Multiplication

Inthe lower grades you learned how to multiply a number

by 10 by annexing a zero to the number. You also learned how

to multiply a number by 100 by annexing two zeros.

10X38= 380 100X38 = 3800 10X472 = ?

OneWay

148

_?5

740

296

3700

3

325

15

6

9_

975

AnotherWay

3700

4)14800

325

3

975

Many seventh year pupils use a short

method of multiplying by 25. If they

want to multiply 148 by 25, instead of

multiplying in the usual way they annex

two zeros to 148 and then divide by 4.

Study the example at the left. Explain

the second way.

Harry wishes to find the cost of 325

sweaters at S3 each. He used the first

method at theleft. His father suggested

the second method. Which is shorter?

You can save time by remembering and using the short

methods given at the bottom of the next page.

25



26 Saving Time. Short Methods in Multiplication

1. Can you explain why annexing a zero to a number mul-

tiplies it by 10?

2. Why does annexing two zeros to a number multiply it

by 100?

3. State a short rule for multiplying a number by 1000.

4. Can you explain the rule for multiplying by 25? What

is the effect of annexing two zeros to the multiplicand? Why

must this result be divided by 4?

5. Can you state a rule for multiplying by 33J? By 16f?

By 12J? Remember that 33| = J of 100, that 16j=fc of 100,

and 12^=| of 100. If you cannot state these rules, ask your

teacher to help you.

6. 28x9. Find the result in the shortest way.

7. Find the cost of each of the following:

a. 47 yards of goods @ 25^ a yard

b. 36 yards of ribbon @ 33J ^ a yard

c. 17 ft. of braid @ 12J ^ a foot

8. Find the products for these examples in the easiest way.

How many can you find without using a pencil?

a. 10X278

h. 33JX78 o. 25x735

b. 1000X394 i. 16JX126 p. 16fxl52

c. 100X835j. 25X389 q. 100x83

d. 10X59 k. 33^X85 r. 25x247

e. 763X5 1. 12^X79 s. 10x98

f. 25X76 m. 16fx52 t. 1000x7

g. 12^X84 n. 39X7 u. 12|x48

1. To multiply any whole number by 10, annex a zero at

the right of the number.2. To multiply any whole number by 100, annex two

zeros at the right of the number.

3. To multiply any number by 25, annex two zeros and

divide the result by 4.

4. Always use the smaller number for the multiplier.



Problems. Using Multiplication 27

1. Frank Owen belongs to a 4-H Club. Last summer he

sold the following from his garden. How much did he receive?

22 dozenears of corn

@15^ a dozen

41 bushels of tomatoes @ 60^ a bushel

22 bushels of tomatoes @ 40^ a bushel

74 bushels of potatoes @ 7(ty a bushel

35 quarts of beans @ 8<t a quart

18 quarts of peas © 9£ a quart

2. Frank's total expenses for his garden were $27.45. How

much did he make on his garden project?

3. Frank's father owns a dairy farm. Last year he sold an

average of 127 gallons of milk a day. How many gallons did

he sell during the year?

4. Last year Mr. Owen raised 25 acres of wheat that aver-

aged 31 bushels to the acre. How many bushels did he raise?

5. Mr. Owen putcommercial fertilizer on 87 acres at a

cost of $4 an acre. How much did the fertilizer cost?

6. Frank's brother worked for a neighbor and received 50^

an hour. During one month he worked 8 hours a day for 18

days, 4 hours a day for 4 days, and 10 hours a day for 2 days.

How much did he earn?

7. Mr. Owen read in a farm paper that there are 6,812,350

farms in the United States and that the average size is 155

acres to the farm. How many acres of farm land are there in

the United States?

8. The paper stated that the average income for the year

was $1,434 per farm. Find the total farm income for the year.

(Problem 7)

9. Harriet Owen worksin a dry goods store. She found

the cost of each of the following without using pencil or paper.

Can you do this?

a. 24 yards of ribbon © 12^ a yard

b. 60 yards of goods @ 33^ a yard

c. 75 yards of goods @ 16f£ a yard



28 How Division Is Used

1. John's father's automobile averages 18 miles on a gallon

of gasoline. How many gallons will it take to run the car 252

miles?

2. John's father drove 156 miles in 4 hours. How far did he

drive in one hour, on the average?

3. Howard wrote a short summary of the two uses of

division. His statement is given at the bottom of the page.

Study it carefully. Which of the first two problems illustrates

each use?

4. One Saturday, John and Howard earned $3.80 by mow-

ing lawns for their neighbors. The boys divided the money

equally. How much did each earn? Which type of division does

this illustrate?

5. Write at least one good problem illustrating each of the

two uses of division. Solve.

The Vocabulary of Division

1. Copy and fill the blanks in these statements:

a. In the example 56 -=-8 = 7, the dividend is , the

divisor is , and the quotient is .

b. The sign 4- is read

2. Name the dividend, 32 24

divisor, quotient, and re- 4)128 21)508

mainder, if any, in each 42

of the following examples. 88

84

36-^4=9~4

Division is used for two purposes:

a. To find how many of one number there are in another

number.

b. To find the part of a number.

Vocabulary: Dividend, divisor, quotient, remainder.



A.voiding Errors in Division 29

Many people find division the hardest of the fundamental

processes. Master the following points and you should have

no more difficulty with this process.

L. Basic Division Facts. Turn to Practice Test Number

15, page 318, and practice on the division facts. Watch for

the ones that cause you trouble and give them extra practice.

2. Facts with Remainders. In performing the division at

the left, you must know that there are eight 7's in

85 59, with a remainder of 3. This is the most difficult

7)595 Pomt m snort division. Turn to Practice Test Number

16, page 319, lor practice.

3. Placing the Quotient Figures. In the example at the

left, think 4 into 12 is 3. The 12 is really 12 tens so

31 the 3 belongs in tens place or above the 2. Always

4)124 Place tne nrst Quotient figure above the last figure

of the dividend which you use in the first division.

Failure to do this causes many errors in dividing decimals.

4. Estimating Quotient Figures. This is the most dif-

ficult step in long division. In the example at the

3 left, to find how many 21's in 76, think how many

21)761 2's in 7.

To find how many 39's in 177, in the example at

4 the left, think how many 4's in 17. Why? Is 39

39)1777 nearer 40 or 30? For further practice, turn to Prac-

tice Test Number 17, page 319.

Place the first quotient figure above the last figure of the

dividend which you use in the first division.

When the second figure of the divisor is 0, 1, 2, 3, or 4,

use the first figure of the divisor as a trial divisor. When

the second figure of the divisor is 5, 6, 7, 8, or 9, use one

more than the first figure of the divisor as a trial divisor.



30 Avoiding Errors in Division

5. Correcting the Quotient Figure. No matter how care-

fully you estimate the quotient figure, your first estimate may

be too large or too small.

Study each of the examples below. In which is the trial quotient

too large? How can you tell? In which is it too small? How can

you tell?

7 1 7

86)5886 34)2385 29)2386

602 34 203

35

You will avoid trouble if you always watch for the two trouble

points given at the bottom of the page. Can you explain each?

6. Zeros in Quotient. Some pupils forget to place zeros in

the quotient when needed. Study the examples below and see

if you know why the zeros are placed in the quotients. Practice

on PracticeTest Number 19, page 320.

4003 209 30

4)16012 21)4389 32)987

42 96

T89 27

189

7. Compound Numbers. Study the examples below. Explain

each step in the first, In the second why is 6 in. changed to 18 in.?

In the third why is 4 oz. changed to 36 oz.? After you understand

you should think these changes instead of writing. Turn to

Practice Test Number 50a, page 340, for practice.

3 yd. 1 ft. 2 ft. 9 in . 2 lb. 12 oz.

Y\ 18 \ 36

2/6yd.2ft.2/5ft. 6 in. 3/81b. 4 oz.

If the product is larger than the number from which it is

to be subtracted, the trial quotient is too large.

If the remainder is as large, or larger, than the divisor,

the trial quotient is too small.



Safety I irst. Checking Division 31

97

76)7392 John worked the example at the Left. He went

684 over each multiplication a seaond time and checked

1^52 each subtraction by addition. Do you think this

532 is a ^ood plan? Why?

~20

Division Check

27 27 Quotient In division the dividend is al-

32)~876~ 32 Divisor ways equal to the product of the

—divisor and quotient, plus the

— remainder. This principle gives a

236 *LL convenient method of checking

224 864the gnai result in division, as

12 12 Remainder shown in the example at the left.

876 Dividend Is the work correct?

1. Divide. Check each step before starting the next. Check

the final result.

5)185 3)918 9)83756 21)5783 32)6849

2. If you need further practice on checking division, make

more examples like those given above. You should form the

habit of checking all divisions.

Locating Your Difficulties in Division

Turn to the Diagnostic Tests on the next page. They will help

you locate your difficulties in division.

Check each step in division before starting the next. Do

each multiplication twice. Check each subtraction by addi-

tion. Be sure you place a figure in the quotient each time a

figure is brought down for a new dividend. That figure maybe a zero.

Check the final result by multiplying the quotient by the

divisor, and adding the remainder (if any). The result

should equal the dividend.



32 Diagnostic Tests in Division


Do not copy the examples unless directed to do so. (See page

5.) Divide. Check each step before starting the next. Work as

fast as you can without hurrying.

1. Write the quotient.

7)56 9)63 8)72 6)54 8)64 7>49~ 9)81

2. Write quotient and remainder.

5)48 7)60 8)67~ 6)57 9)79 7)53 8)68

3. Estimate the first quotient figure.

32)928 47)878 84)589 79)686 63)472

4. Mark trial quotient C, S, or L, according as it is correct,

too small, or too large. Carry the work just far enough to makeyour decision.

9 7 7 8 7

41)398 58)473 93)637 89)685 68)499

5. Copy and divide.

79)67869 86)26475 49)28457 92)75638 38)19034

324)8679 756)52968 579)98758 859)601586 736)81597

6. 2)8 hr. 18 min. 3)2 yd. 12 in. 10 min. 8 sec. 4-4

OVERCOMING YOUR DIFFICULTIES

Study your errors carefully and note the trouble spots. Read

pages 29 and 30 again, if necessary. Then turn to the Practice

Tests (pages 307 to 340) for further practice, as suggested below.

Test 1 — 15 Test 3— 17, 15 Test 5— 19

Test 2— 16, 15 Test 4— 18 Test 6— 50,45a



Finding Averages 33

In many practical problems it is necessary to find the aver-

age of two or more numbers. Do you know how to do this ?

DickCunningham found the average weight of his high

school football team. The eleven regular members of the

team weighed 128 pounds, 119 pounds, 106 pounds, 145

pounds, 122 pounds, 118 pounds, 152 pounds, 112 pounds,

123 pounds, 137 pounds, and 135 pounds. Dick found the

total weight of the eleven players was 1397 pounds. He then

divided by 11 and found the average weight was 127 pounds.

1. State a rule for finding the average of two or more num-

bers. Compare your rule with the one at the bottom of the

page.

2. Check Dick's addition. Then check his division. Was

his work correct?

3. The team played four home games. The attendance

was 593, 725, 897, and 1049. Find the average attendance.

4. Thegate receipts for the four games were $186, $231,

$295, and $312. Find the average receipts.

5. The team played 7 games and scored a total of 56

points. Find the average number of points they scored per

game. Since the total is known, do you have to add?

6. Their opponents scored 42 points in the 7 games. Find

the average.

7. Dick earned money byworking after school and Satur-

days His earnings for 8 weeks were $4.25, $6.00, $3.80, $4.60,

$1.50, $5.25, $4.80, and $2.90. Find his average earnings for

a week. How much would he earn in 36 weeks at the same

rate?

8. Agnes Cunningham typed a manuscript for her father.

She counted the words in several lines. There were 9, 10, 8,

10, 9, 12,11, 10, 11, and 10. Find the average number of

words in'a line. About how many words were there in the

manuscript if there were 20 lines on a page and 12 pages?

To find the average of two or more numbers, add the

numbers and divide the total by the number of addends.



34 Selecting the Process to Use

/y

Many pupils have trouble in solving problems because they

do not know whether to add, subtract, multiply, or divide.

Write the numbers from 1 to 5 on paper. Read the first

problem. If you think it should be solved by addition, write

+ after 1 on your paper. Do the same for the other prob-

lems, using -, X, and -i- for subtraction, multiplica

tion, and division respectively. Do not solve

the problems.

1. Louise made doll

dresses for Christmas

presents. How many

inches of goods did

she need to make six

dresses, if each dress

took 8 inches?

2. Mary Adamsweighs herself each

week. On September 1, she weighed 79 pounds. Two weeks

later she weighed 81 pounds. How many pounds had she

gained?

3. Each shelf of Harry's bookcase is to be 3 ft. long. How

many shelves can he make out of a board that is 10 ft. long?

4. John weighs 94 pounds. He learned in school that a boy

of his age and height should weigh 102 pounds. How many

pounds must he gain to reach normal weight?

, 5. On Monday Agnes had $2.17 in her bank. She put in

25 cents on Tuesday, 10 cents on Thursday, and 18 cents on

Saturday. How much did she then have in her bank?

6. Dick helped make badges for theofficials of a school

track meet. They needed 18 badges and had 72 inches of

ribbon. How long could they make each badge?

7. Helen works in a store on Saturdays. She sold 9 yards

from a bolt of goods containing 24 yards. How many yards

were left on the bolt?



- CL

Telling How to Solve Problems

Two processes must be used to solve each problem on this

page Write the numbers from 1 to 5 on paper. If you would

first subtract and then multiply in problem 1, write - X

after 1 on your paper. Do not solve the problem.

1 When Mr. Atkins left home, his speedometer read

9864 miles. After driving for three hours it read 9978 miles.

How many miles did heaverage each hour?

2. Joe sold tickets for a class play. He sold 9 tickets to

children at 15* each and one ticket to his mother at 25*.

How much money should he receive for these tickets?

3 Richard received a score of 94 on the first arithmetic

test, a score of 89 on the second, and a score of 93 on the

third. What was his average on these three tests?

4 Mary made 6 towels. How many inches of material did

she need if each towel was 20 inches long when finished and

she had allowed 3 inches on each towel for hemming?

5 At Gordon's Grocery, Marguerite bought \ lb. of cheese

at 60* a pound and 1 lb. of crackers at 12* a pound. How

much did they both cost?

35



36 Problem Test

1. Mr. Gerard owns a house valued at $12,500 which he

rents for $720 a year. Repairs are $125 a year, taxes $160 a

year,and insurance $48 a year. What is his income from the

house after all expenses are paid?

2. Last winter a farmer fed 65 head of cattle, using 97

tons of hay and 43 tons of straw. This winter he proposes

to feed 195 head of cattle. Judging by last year, how much

hay and straw should he have?

3. A farmer weighed his corn in the fall and found that he

had 85,925 pounds. About how many bushels will he have in

the spring if the corn loses £ of its weight through drying?

One bushel of corn weighs 70 pounds.

4. A man paid $1200 for an automobile, drove it 24,500

miles, and at the end of four years sold it for $475. During

this time he spent $331 for gasoline, $98 for oil and grease,

$95 for tires, $129 for repairs, $30 a year for insurance, $6

a year for license, and $4 a month garage rent. How much

per mile did it cost him to run the car?

5. Last year Mr. Bates had ten acres in tomatoes. His

expenses were: planting, cultivating, etc., 300 hours man

labor; 250 hours horse labor; plants, $100; fertilizer, $20;

picking and hauling crops, $440. He marketed 102 tons of

tomatoes at $14 a ton. Estimating the cost of man labor at

40 i an hour and horse labor at 20^ an hour, find his profit

on the crop. On each acre.

6. A fruit grower shipped 600 barrels of apples to his com-

mission merchant, who sold them at $4 a barrel. The

merchant deducted $44 freight charges, $28 cartage, I2i a

barrel for cold storage, and 15 £ a barrel commission. Find

the amountthe commission merchant paid the grower.

7. A commission merchant received a shipment of 50 doz.

eggs and 250 lb. butter. He sold the eggs at 32^ a dozen and

the butter at 28^ a pound. How much did he pay the dairy-

man if his commission was 2^ a pound on the butter and 3^

a dozen on the eggs?



Chapter Test 37

Write the aumbers from I bo 25 on a papei\ After these numbers

write the words, or numbers, which belong in the corresponding

blanks below.

1. In 7,350, the 7 is in place, the 3 in.

—place, the

(1) (2)

5 in place, and the 6 in place.

(3) C)

2. In the example 5+8 = 13, 5 and 8 axe called the .

13 is called the— or(6) (7)

3. In the example 9-3 = 6, 9 is called the ,and 3 is

called the 6 is the or

(9) (10) (ID

4. In the example 7X9 = 63, 7 is called the ___ and 9 is

called the 63 is the 2(13) (14)

5. In the example at the right, 35 is called the 35)78

, 78 is called the , 2 is called the ,and 8 70

(15) (16) (17)—

is called the °

(18)

6. When we want to find how muchlarger one number

is than another, we(19)

7. When we want to find the combined value of several

unequal numbers, we(20)

8. When we want to find the combined value of several

equal numbers, we(21)

9. When we want to find how many times one number is

contained in another, we(22)

10. To find how many must be added to a smaller number

to make it equal to a larger number, we(23)

11. To divids a number into a number of equal parts, and

find the size of one of these parts, we(24)

12. When we want to find how many are left, when a smaller

number is taken from a larger, we (25)

Measuring Your Progress

Take Improvement Test Number One and record your score

on your Score Card. Did you improve? Find the class average

and bring your graph up to date.



L



CHA I'T E R

2

Improving your Work

with fractions

Fractions have been used and have caused people trouble

from the earliest times. The oldest book on arithmetic which

has come down to us is that written by Ahmes who lived

in Egypt about 1550 B.C. His work was written on papyrus,

which is a crude form of paper made from the papyrus reed

which grew along the banks of the Nile. Ahmes had so much

trouble with fractions that he worked only with simple ones

like J, \, \, i and ^.

Our method of writing a fraction with the numerator over

the denominator comes from the Arabs. In early American

arithmetics fractions were also called broken numbers.

Do you see why? Fractions like \ and § are often called

common fractions since they were in common use long

before the introduction of decimal fractions like .7 and .32.

In this chapter you will review the work with fractions

and receive help in overcoming your difficulties. Study care-

fully and you should no longer agree with the old rhyme:

Multiplication is vexation

Division is as bad;

The Rule of Three perplexes me,

And Fractions drive me mad.

The Rule of Three was an old name for proportion, which

you will study next year.

39



Two Meanings of a Fraction

Some pupils have trouble with fractions because they do

not clearly understand what fractions mean. A fraction means

a part of a whole thing. If you have f of a whole or unit, it

means that the unit has been divided into four equal parts

and that you have three of those parts.

The picture above shows a second meaning of fraction.

It shows that § = 2. But 6^-3 is also equal to 2, so we see

that f= 6^-3. A fraction indicates division.

1. Name the numerator and denominator of each fraction.

2. Which fractions in problem 1 are proper fractions?

Which are improper fractions?

3. Define a proper fraction. An improper fraction.

4. Draw a picture to illustrate J of a cake.

5. Draw a picture which illustrates f of 18.

6. Draw a picture to show that \ equals 7^-2, or 3\.

7. Copy the following and fill the blanks.

a. f of 9=_

b. £of 25 =

c.

£of 72=

a* 3

e.

f.

9 _

A —6



\ Third Fraction Meaning 41

RATIO

In many practical problems il is necessary to compare

one number or quantity with another. Suppose Mary has

$6 and John has $3. How does the amount of money Mary

has compare with the amount John has? Arithmetic gives

us two ways of comparing numbers.

You have learned how to compare by subtraction. Since

6 _ 3=3 we can say that Mary has $3 more than John, or

that John has $3 less than Mary. When we compare two

numbers by subtraction, the result is called the difference.

We can also compare numbers by division. Since

6^3 = 6 = 2 we say that Mary has two times as much money

as' John. Since 3-6 = | =i we say that John has one half

as much money as Mary. When we compare two numbers

by division, the result is called the ratio and is written in the

formof a common fraction.

1. Copy the following statements and fill the blanks.

a. 8 is __ of 16 c. 15 is times 5 e. 8 is _ of 12

b. 5 is_ of 9 d. 9 is_ of 5 f. 6 is __ of 9

2 Write the ratio of the first number to the second.

5,7 5,10 2,8 4,8 4,9 2,12

3, 4 3, 9 5, 8 6, 2 12, 3 3, 5

3. In example 2 write the ratio of the second number to

the first.

4. The seventh grade of Washington School sold 60 tickets

for a school entertainment. John and Robert each sold 1

ticket Julia 2, Louise 3, Joe 5, Mary 6, Agnes 10, Tom 12,

and the teacher sold the rest. What part of the whole number

of tickets sold did each onesell?

Numbers may be compared in two ways:

a. By subtraction. The result is called the difference.

b. By division. The result is called the ratio and is

written as a common fraction.



Six Principles of Fractions L3

Agnes discovered six important principles about fractions

by examining the fractional parts of an inch on her ruler. See

if you can find the answer to the following questions from your

ruler.

1. Write |. Without using a pencil, multiply both the

numerator and denominator by 2, and write the result. Find

|- of an inch on your ruler. Find | of an inch. What effect

does multiplying both terms by the same number have on the

value of the fraction?

2. Write -|. Divide both the numerator and denominator

by 2 and write the result. WT

hat effect does dividing both

terms by the same number have on the value of the fraction?

3. Write §-. Multiply the numerator by 3 but leave the

denominator unchanged. What effect does multiplying the

numerator by 3 have on the value of the fraction?

4. Write f. Divide the numerator by 4 but leave the

denominator unchanged. What effect does dividing the

numerator by 4 have on the value of the fraction?

5. Write ^. Multiply the denominator by 2 but leave the

numerator unchanged. What effect does multiplying the

denominator by 2 have on the value of the fraction?

6. Write f. Divide the denominator by 2 but leave thenumerator unchanged. What effect does dividing the denomi-

nator by 2 have on the value of the fraction?

7. State six general principles covering what you have

discovered in problems 1 to 6.

8. Which principle is used in each of the following?

a.Y8

= 3

• ^X^=

3

g. 3Xg-=-g- •520 e * 9 * ^ 9 n. OA

9—

3

u ^ A 8~ 4l

' 9 •^~ 18 l

' 7 •«-> — 7

9. Find each of the following in the easiest way.

b. ±+ 4 d. 3xf f. 4xf



44 Safety First. Checking Work with Fractions

The errors you make in working with common fractions are

probably due to two different causes. You may get the wrong

answer because you do not understand the method to be used,

or because you have not had sufficient practice.

Many of your errors, however, are probably not due to

lack of understanding or lack of practice, but to carelessness.

You can do two things to reduce the numer of errors made

through carelessness:

1. Becareful not to let your attention wander in the middle

of a step.

2. Check each step as soon as you have done it, by going

over it a second time.

In working with whole numbers, you have already learned

the value of checking each step before going on to the next

step. This habit is even more useful in working with frac-

tions. If you reduce ^ to lowest terms and get f as a result,

check immediately by again dividing both the numerator and

denominator by 3 to see if you get the same result as before.

If you change | to f-f , stop and again multiply the numerator

and denominator by 4.

Go over each step in the examples given below. If you

find any errors, copy. the example and work it correctly.

AH* ft-+ t+W

Measuring Your Progress

Take Improvement Test Number Two. Copy the Score

Card on page 341. Record your score in your notebook. Find

the class average and start a graph like the one on page 6.

In working with fractions, always check each step before

starting the next.



Improvement Test Number Two 15

COMMON FRACTIONS

Work these examples. Check each step by going over it a

second time. Time, 10 minutes.

1.



46 Avoiding Errors in Adding and

Subtracting Fractions

Watch for thefollowing trouble spots in adding and sub-

tracting fractions.

1 Finding the L.C.D. Fractions must be changed to JO*

fractions, or to the same denominators before they can be added

or subtracted. The new denominator is the smallest number

which can be evenly divided by each denominator. It is called the

Least Common Denominator, or L. C. D.

To find the L.C.D., use the plan Dick used in the problem

below.

Add -ft, A> and -T5-mat iS thC L 'C 'D '?

Dick started with 20, the largest of these denominators. Can

20 be divided evenly by 12 and by 15? Can 2X20 (40) be divided

evenly by 12 and by 15? Can 3X20 (60)? The L.C.D. is 60.

Turn to Practice Test Number 21, page 321.

2. Changing to L.C.D. Dick wrote the example given above

in two ways. Explain each step:

512—



Avoiding Errors in Adding and

Subtracting Mixed Numbers

17

Watch for the following trouble spots in adding and sub-

tracting mixed numbers.

%2 = ^g L Adding. Study the example at the

left carefully. Explain each step.

'6

a 3 ~ J6

45j = 45f = 44§

i3j=

m=m31|

2. Subtracting. Subtracting mixed num-

bers is no harder than adding them if you

know what to do when the fraction in the

minuend is smaller than the fraction in the

subtrahend. One method is shown at the left. Explain each step.

Saving Time by Using a Short Form

George shortened the work for the examples given above as

shown below. Explain to the class where he got each number

in the first example and what he thought. In the second example,

George thought, 4 from 9 = 5, 3 from 4=1, 1 from 4 = 3. Ex-

plain each step. You can save time by using this form, but it

is better to use the long form than to make mistakes.



48 Avoiding Errors in Multiplying

and Dividing Fractions

You can increase your accuracy and speed in multiplying

and dividingfractions by mastering the following points.

1. Cancellation. Use cancellation as much as possible

3 X *=T«-= J»- to shorten your work in multiplying and

dividing,

i Compare the two solutions at the left.

»x^ = _§_ Which is shorter? What principle do you

Is use when you cancel? Turn to Practice Test

Number 27, page 325.

2. Changing a Mixed Number to an Improper Fraction.

In multiplying and dividing mixed numbers it is often necessary

to change them to improper fractions. How can you change 8|

to an improper fraction? 7|? Turn to Practice Test Number 28,

page 325.

3. Multiplicationof Mixed Numbers. Always choose the

shortest method in multiplying mixed numbers. Study the three

examples below carefully and explain each step. You will find all

three methods useful. In what kind of examples would you use

each of these methods?

27| 85

8 1\ 5

7- T -1...1 I^w 13_ 65_ qoj.

216 595x

22l£ 637|

Turn to Practice Test Number 29, page 326.

4 Dividing Mixed Numbers. Study the example given

below. Explain each step. Turn to Practice Test Number 30,

page 326. m a _, ,

Locating Your Difficulties. Diagnostic Tests

The Diagnostic Tests on the following pages will help you find

your weak points in working with fractions.



Diagnostic Tests in Common Fractions r>

LOCATING yOUB DIFFICULTIES

Do not copy the examples unless directed to iId so. Write

the answers on your paper, ('heck each step. Work as t'.i-t as

you can without hurrying.

1. The numbers given in each group are the denominators

of fractions to be added or subtracted. Write the L.C.D. in each

case.

I 2, 4 3, 6 8, 6 5, 10 10, 4 4, 14

2, 8 6, 4 10, 8 9, 15 9, 6 12, 15

6, 12, 3 4, 2, 3 4, 12, 8 2, 4, 6 6, 5, 2 9, 15, 3

2. Write the numbers that belong in the blanks.

2 __3—

9

A —

_

5—

15

12

12

18

3~2

16

24

1 _2—

8

12—18

7__8 ~32

36

3. Reduce to lowest term§.

1 e32

161 215

15.252736

8321 218

9121 435

203642

1 2163542

1 8243540

3236

135243

4. Change to mixed or whole numbers.

73

1 2

5

1 8

6

9.

4

2 3

8

1 3

1 9

2

252437

10

1 7

3

1

3

214

209

5. Change to improper fractions.

3i

Z5

5*

8}

Z4

Z8

°3

13^-

4

y5

4f

8f


Study your errors carefully and note the trouble spots. Read

pages 46, 47, and 48 again, if necessary. Then turn to the Practice

Tests (pages 307 to 340) for further practice, as suggested below.

Test 1—21 Test Z— 23 Test 5— 2S

Test 2— 22 Test 4— 24



tyDiagnostic Tests in Common Fractions


Copy the examples and add, subtract, multiply, or divide, as

directed. If the result is an improper fraction, change it to a

mixed number. Leave all results in lowest terms. Check each

step. Work as fast as you can without hurrying.

1. Add.

1

87 7

8 123 5.

4 .6_



©

Using Fractions in Problems

1. On Saturday morning Betty made oatmeal cookies.

The recipe that she followed would make 48 cookies. Besides

other things it called for 1 cup sugar, -|- cup sweet milk, \teaspoon soda, and 2 cups flour. Betty made only half of the

recipe. Can you write the amount of each of these articles

she used? How much should she use for making 96 cookies?

2. John walked 2\ miles to town. On his way home he

rode f of a mile with a neighbor and walked the remainder

of the distance. How far did he walk altogether?

3. Tom lives on a farm. Last summer his father gave him

a field to raise potatoes and promised him half of all he raised.

What was Tom's share if he raised 65f bushels?

4. Mary had 1\ yards of cloth in one piece. From this she

cut 1^- yards for an apron and2-f

yards for a dress. How much

did she have left?

5. On a certain automobile map one inch represents ten

miles. How far apart are two towns if the distance between

them on the map is 6f inches?

6. A transcontinental passenger airplane averages 150 miles

an hour. How far will it go in 3^ hours?

51-V



Using Fractions in Problems

HOW MANY OF THESE PROBLEMS CAN YOU SOLVE?

1. Louis measured 100 feet along a level path and then

counted 'the number of steps he took in walking from one

end to the other. What was the average length of his step

if he took 40 steps? The next day Louis went on a hike with

another boy scout and counted his steps. How far did they

go if Louis took 1584 steps, assuming that his steps were the

same length as the day before? How many miles was this?

2. Julia made a gelatin dessert. The directions called for

2 cups of liquid for a tablespoon of gelatin. Julia measured

a tablespoon of gelatin and dissolved it in J cup of hot water.

She then added f cup of lemon juice. How much water did

she need to add to make the proper amount of liquid?

3. Bill and Harry got into an argument one day. Bill said

that multiplying a number by anything always made the

number larger. Harry did not think that this was true. Which

was right?

4. Bill also thought that dividing a number by anything

always gives a quotient smaller than the number divided.

Was he right?

5. In six months Mary's weight increased from 58o lb.

to 62J lb. What was the average increase per month?

6. Last Sunday Ruth Raymond's father drove his auto-

mobile 75 miles in 2\ hours. On the average, how far did he

drive in one hour? At the same rate how far could he drive

in 1 hour and 20 minutes?

7. During the second World War Dick studied a large map

of the Pacific. One inch on this map represented 276| miles.

When the Americans captured Saipan, in the Marianas Islands,

Dick measured the distance of this island from Tokyo. It

was 5J inches on the map. How many miles is Saipan from

Tokyo?

8. No scale is given on an automobile map, but two cities,

which are known to be 70 miles apart, are If inches apart on

the map. What is the distance between two cities that are

3^ inches apart on the map?



Saving Time by Cancellation

Harry Wright drove his father to a neighboring town. Mr.

Wright cautioned Harry to drive more slowly when he was

approaching a crossroad. When they returned home, Mr.

Wright asked Harry if he knew how far a car, going 50 miles

an hour, would go in one minute. Harry figured it out as

shown below. Mr. Wright then showed Harry a shorter wayof working this problem. Study both solutions.

Harry's Solution

5280 4400

50 60)264000

264000 240

240

240

Mr. Wright's Solution

5 880

#3X5280 = 4400

m

1. Why did Harry multiply 5280 by 50? Why did he

divide the result by 60?

2. An automobile is going 45 miles an hour. How far will

it go in one second? Work this problem both the long and the

short way.

3. How many feet will an automobile go in 10 seconds, if it

is going 60 miles an hour? Work the short way.

4. In 1932, at the Indianapolis race track Frederick Frame

drove 500 miles in 288 minutes. How many feet a second did

he average?

5. The Empire State Express makes the run from New

York City to Buffalo, a distance of 436 miles, in 510 minutes.This is at the rate of how many feet a second?

If the solution of a problem involves only multiplications

and divisions, you can often save time and work by indi-

cating the operations and cancelling.

53



54 Problem Test

1. If you knew the cost of a dozen oranges, how would

you find the cost of eight?

2. If you knew the number of boys and the number of girls

in a class, how would you find what part of the total number

in the class are boys? What part are girls?

3. If you knew how many hours an automobile took to run

a given number of miles, how would you find the average

speed per hour?

4. If youknew the weight of each member of a football

squad, how would you find their average weight?

5. A farmer sold a crate of chickens to a poultry buyer.

The chickens were weighed in the crate. What else would

you have to know in order to find the average weight of the

chickens? How would you find the average weight?

6. Eugene counted the number of steps he took in walking

from the house to the garage. What else did he have to know

in order to find this distance in feet? How could he then

find this distance?

7. What would you have to know to find out how many

doll dresses you could make from a given piece of goods?

How would you then find the number?

8. Lawrencehas been gaining in weight. What would you

have to know to find his average gain each month? How

would you then find the average gain?

9. What would you have to know in order to find the dis-

tance between two towns on an automobile map, if the dis-

tance was not given? How would you find the distance?

10. Thomas climbed the stairs to

the top of an observation tower. Hemeasured the height of one step.

What else did he have to find out in

order to find the height of the

tower? How did he then find

the height?



Diagnostic Tests in Whole Numbers 55

LOCATING YOUB DIFFICULTIES

Work each example. Check the addition examples by adding

again ill the reverse order. Check the subtraction examples by

addition. Go over each step in the multiplication and division

examples a second time. Work as fast as you can without hurrying.

I. a. Copy and b. Subtract: c. Multiply: d. Copy and

add: 84,



56 Diagnostic Tests in Common Fractions


Work these examples and check by going over each step a

second time. Work as fast as you can without hurrying.

1. a. I b. 3JX22| c. 2*+A <*• J e. 15-4^25.

+1

f- fXA g- n h. 8| i. f+f J- *Xf

+9f ^71

2. a. 4j+8| b. 8XH c-

28§ d'87X6*

e« *+

9§

+36|

r „ 19 • 8 Vi -*- i. J. 2^X21-f. g. 1-6—

-9n. 10

i. J 3' 5

7 4 Q_3_


Study your errors carefully and note the trouble spots. T/ien

turn to the Practice Tests {pages 307 to 840) for further practice as

suggested below.

la_ 25, 21, 22, 24 lh- 26, 21, 22 2d— *0, 24

lb- 29, 27, 28 li - 30, 24, 27 2e - 30, 24, 27

lc - 30, 24, 27, 28 lj - 29, 27 2f - &?, 21, 22

Id- *0, 21, 22 2a- SO, 24, 27, 28 2g - 30, 24, 27

le -_ 30, 24, 27, 28 2b- 29, 24, 27 2h- 25, 21, 22, 24

If — 29 27 2c — 25, 21, 22, 24 2i — 00, 21, 22

lg-^21,22,24 2j- 02, 24, 27, 28

MEASURING YOUR PROGRESS

Take Improvement Test Number Two and record your score

on your Score Card. Did you improve? Find the class average




( Ikapter Test 57

Write the numbers from 1 to 17 on a piece of paper. After

each number, write the words or numbers which belong in the

corresponding blanks below.

1. The numerator of the fraction i is

(i)

2. The denominator of the fraction 4 is .y

(2)

3. A proper fraction is one whose numerator is than

its denominator.(3)

4. To reduce an improper fraction to a mixed number

the numerator by the(5)

5.



k. ;



.V

; Improving \four Work

with 'Decimals

When people first began to count, they used counters such

as pebbles or the fingers. Our word calculate comes from

the Latin calculus meaning pebble. Because of their con-

venience, the fingers were the commonest counters. You

have probably heard the word digit used, meaning one of

the figures used in writing a number, as the digits of 37 are

3 and 7. Digit also means finger.

A shepherd, in counting his sheep, turned down a finger for

each sheep, until he had used all his fingers. Then he made a

mark in the ground and started over again. How many sheep

did he have, if he had two marks on the ground and 7 fingers

turned down when he was through?

Howmany, if he had 5

marks and 3 fingers down? How many sheep does each mark

represent? Each finger?

When the Hindus invented our method of writing numbers,

they used the same system and wrote numbers up to ten

(units) in the first place to the right, and tens in the next

place. Where did they write ten tens (hundreds)?

Since our number system is based on ten it is called a

decimal system, from the Latin word decern meaning ten.

Some early people, who did not wear shoes, counted to

twenty before starting over again. Do you see why? Westill sometimes count by twenties, or scores. What does

one score mean? Two score? Three score and ten?

59



•^r» *<r- f

Decimal Fractions

Although the decimal system of writing whole numbers was

introduced into Europe as early as the twelfth century, it was

a good many years before anyone discovered that the same

method can be used in writing fractions such as ^-,j

3^,183

tetc> The first complete discussion of this new method

of° writing fractions was given by Simon Stevin in a book

first published in Belgium in 1585.

As you know, each place in a whole number, as you go

to the left, has a value ten times as great as that of the pre-

ceding place. If you go to the right, each place has a value

just one tenth as great as that of the preceding place. For

example, the next place to the right of hundreds is tens and

^ of 100 = 10. The next place to the right of units is tenths,

since ^ of 1 =to-

1. What is the next place to the right of tenths? Why?

To the right of hundredths?

2. Learn the names of the first four places to the right of

units' place.

3. Why is a decimal point placed after units' place, or

before tenths' place? What does .2 mean? How would you

read it if the decimal point were omitted?

60



Reading Decimals 61

By using the places to the right of units' place, we can write

fractions like,

:i

,

1

'

U7

U , andI

,

„

s

u:,

„ in an easier way:

_a

= 3-1-7-

= .17' •'=.183

1 o -° 1I ooo

Fractions written in this way are called decimal fractions

or decimals. Usually it is easier to work with decimals than

with common fractions because decimals are written in the

same way as whole numbers. For this reason, they are being

used more and more in practical work.

1. Study the first fourexamples below carefully and then

read the other decimals. Notice that in reading a mixed

decimal (a whole number followed by a decimal fraction),

you use the word and between the whole number and frac-

tion. The and shows the location of the decimal point.

a. .07 is read seven hundredths.

b. .38 is read thirty-eight hundredths.

c. .247 is read two hundred forty-seven thousandths.

d. 32.41 is read thirty-two and forty-one hundredths.

e. .7 .04 .009 .0005 .97 2.7 3.84

2. Herbert read .217 as two hundred and seventeen

thousandths. He then read 200.017 in exactly the same way.

Did Herbert read .217 correctly?

3. Read these. Say and only at the decimal point in a

mixed decimal.

.537 2.8 .756 .07875 208.007

500.037 47.86 .0387 .378916 7159.2148

202 85.792 .5024 .005197 5100.72

200.002 379.71 .7381 .20603859.0038

4. Read the number at the top of page 60.

5. A decimal fraction such as 32.896 is sometimes read

three, two, point, eight, nine, six. Read the decimals in

problem 3 in this way. Read the one at the top of page 60.



IN AN AIRPLANE FACTORY

1. The parts of an airplane engine must be measuredvery

accurately. Some parts are measured to one ten-thousandth

of an inch. Can you write this number in figures?

2. Write the following in figures.

a. Three hundred b. Three hundredths

c. Twenty-seven and eight tenths

d. Two hundred forty-six and one hundred six thousandths

e. Two hundred and eight thousandths

f Two hundred eight thousandths

g . Seven thousand two hundred thirteen and five hundred

thirty-one thousandths

3. Your teacher will dictate other decimals for you to write.

Ask her to read some of them as in problem 1 and some as

in problem 5 on page 61.

4. Cents are often written as decimal fractional parts of

a dollar. Thus, 25* can be written $.25. Write the following

as decimals.

80* 12* 7* 5* 132* 289*

62



Problems about Automobile Trips 63

1. The table at the right is taken

from an automobile route book. If you

lived in Sandusky andwanted to drive

to Cleveland, could you find the dis-

tance? How far is it from Sandusky to

Lorain? From Clyde to Huron? Cas-

talia to Vermillion? Lorain to Clyde?

2. Agnes took an auto-

mobile trip with her father

last summer and made a

record of the speedometer

reading every morning. How

far did they drive each day?

On the entire trip? Can

you find the total mileage

in two different ways?

READING9,876.8

10,108.5

10,297.7

10,547.3

Mil.MS

0.0

40.5

51.457.5

67.9

77.8

5.5

116.3

TOWNToledo

Clyde

CastaliaSandusky

HuronVermillion

Lorain

Cleveland

TIMEMonday A.M.Tuesday A.M.Wednesday A.M.Thursday A.M.Friday A.M.Saturday A.M.Saturday P.M.

10,809.2

11,071.5

11,385.3

3. Mr. Arnold was driving his automobile from Albany,

N. Y., to Rochester, N. Y., a distance of 223 miles. When he

stopped for lunch, he found that he had driven 121.7 miles.

How many miles remained for him to drive after lunch?


Take Improvement Test Number Three. Record your

score on a Score Card in your notebook. Find the class aver-

age and start a graph like the one on page 6.

Check the location of the decimal point by making a rough

estimate of the result.



64 Improvement Test Number Three

DECIMAL FRACTIONS

Work each example. Check the addition examples by adding

again in the reverse order. Check the subtraction examples by

addition. Go over each step in the multiplication and division

examples a second time. Check the location of the decimal

point by estimating the answer. Time, 16 minutes.

1. Subtract 2. 9) .288 3. 4. Copy and

8.397 from add:

32.7. .38 7.8; 45;

X.07 38.9; 5.98.

5. Give result

to nearest

cent:

$28.72-^100

9. 28.7X2.79

6. Change to

decimals

and multi-

ply-

7t> X23^

10. To nearest

thousandth:

87.2)124

7. Write as a

common

fractional

part of a

dollar

$.12^.

11. 32XS.25

8. To nearest

hundredth

13. To nearest

tenth

.287)17

17. Change ff-

to decimal,

to nearest

hundredth.

14. 10X $2.87 15.

18. 19.

9.87

X3.05

Subtract

79.8 from

252.39.

Copy and

add:

389.7; 8.35;

39.87;.976;

158.06

.72)38.2

12. Copy and

add: 89.76;

387; 829.5;

7.588;

28.49.

16. To nearest

cent:

7.8X$19.35

20. Change^

to decimal,

to nearest

tenth.

21.

32.076

-27.296

22. To nearest 23.

thousandth

3.9).0688

You wall take this test again in a

practice as suggested, you should be

24. To nearest

8.39 cent:

X73.5 50X $.33J

If you study andew days

able to improve your score



Avoiding Errors in Adding and 65

Subtracting Decimals

In the addition and subtraction of decimals watch the follow-

ing points which sometimes cause trouble.

1. Lo< vn\<; the Decimal Point. Write the numbers to

be added or subtracted with decimal point directly

3-13 imcler decimal point. Why? Place the decimal point

.2 4in the result directly under the decimal points in the

3.3 7 original numbers. Why?

2. Zeros Understood. If there are more places to the right

of the decimal point in one decimal than in the others, zero is

understood in each vacant place. You may write these zeros if

you wish. Turn to Practice Test Number 31 and 32, page 327.

3.2(0) (0) 7.5(0) (0)

.3 8 4 .0 2 8

3.5 8 4 7.4 7 2

Avoiding Errors in Multiplying Decimals

We multiply decimals in the same way that we multiply whole

numbers. The only new difficulty is locating the decimal point.

1. Locating the Decimal Point. Dick forgot how to mul-

tiply decimals so he worked the

.5X .7 = -j%Xto =^^ = -35 first examPle at the lef* and formed

the rule given at the bottom of the

.31page.

Doyou see how he got the

.8 rule? Dick used his rule to work

248 the second example at the left. Is

his work correct?

2. Supplying Zeros. In this example there are four decimal

places in the multiplier and multiplicand combined,

•2 7 go a zero must be supplied between the decimal point

.0 8 aild the 2 in the product. Turn to Practice Test Number.0216 33, page 328.

Multiply decimals just as you multiply whole numbers and

point off as many decimal places in the product as there are

in the multiplier and multiplicand combined.



66 Avoiding Errors in Dividing Decimals

Study the suggestions below and learn to avoid errors in locat-

ing the decimal point when you divide with decimals.

1. Dividing a Decimal by a Whole Number. Dick's teacher

told him to watch two things in dividing a decimal by a whole

number.

a. Place the decimal point in the quotient di-

4 1.2 rectly above the point in the dividend.

b. Divide. Place each figure of the quotient

directly above the last figure of the dividerd used

in that division.

2. Dividing a Decimal by a Decimal. Dick's teacher sug-

gested the following method in dividing a decimal

4.2 by a decimal. Watch these four steps carefully

3.2 )l 3.4 4and you will avoid errors in locating the decimal

point.

a. Make the divisor a whole number, if necessary, by moving

decimal point to the right of the last digit. Use a caret to

show the new position of the point.

b. Move decimal point in the dividend to the right as many

places as you moved the point in the divisor. Mark the

new location of the point with a caret.

c. Place decimal point in the quotient directly above the

caret in the dividend.

d. Divide. Place each figure of the quotient directly above

the last figure of the dividend which you used in the division.

3. Supplying Zeros. Sometimes, in dividing decimals, it is

necessary to fill empty places with zeros. Study the examples

below. Notice the zeros in heavy type. Turn to Practice Test

Number 34, page 328.

.027 15 68. 32 00.

8) .216 .05A)78.40A .02A)64.00A



Safety First. Checking Work with Decimals 07

1 8.7

3.2

374

56 1

5 9 8.4

The work with decimals should be checked by the same methods

that you used with whole numbers. These do not give a check

on the location of the decimal point, however, as Dick discovered.

Dick multiplied 18.7 by 3.2, as shown at the left,

and placed the decimal point in the product directly

below the decimal points in the multiplier and multi-

plicand. He then multiplied 3.2 by 18.7 in the same

way, as a check, and obtained the same result. He

decided his result was correct. Was it?

The best wayto detect such errors as Dick made is to make a

rough estimate of the result. If Dick had stopped to think that

18.7 is approximately 19, and that 3.2 is approximately 3, he

would have known that the product is about 57, and not 598.

1. What is the correct answer to Dick's problem? State a

rule for multiplying decimals.

2. Study the examples below, then find the exact answers.

Be sure you place the decimal points correctly.

Example



68 Diagnostic Tests in Decimals


Add, subtract, multiply, or divide as directed. Check. Work

as fast as you can without hurrying.

1. Copy and add.

a. 3.27; .389; 84.2 b. 82.2; 75.86; 9.7

c. .8; .09; .057 d. 386.7; 18.39; 72.08

e. .375; .89; .75; .09 f. .3756; .75; .814

2, Subtract the first number from the second.

a. .45; .91



Saving Time. Short Methods in 69

Multiplying and Dividing

BY 10, 100, AND 1000

You can save time by using the short methods of multiplying

and dividing decimals by 10, 100, and 1000. Study the examples

given below:

10X.23 = 10X^=H=2^ = 2.3

5.7 4- 10 = 5yo4- 10 = yq X yo

=uTo

= -°

How could you get the result in the first example without using

common fractions? Does multiplying .23 by 10 change the

figures? Does it change the location of the decimal point? Howcould you get the result in the second example without using a

common fraction? State a rule for multiplying a decimal by 10.

For dividing by 10.

1. Find each answer by moving the decimal point

a. 10 X.384 d. 23.54-10 g. 845.2-f- 10

b. 10X.7 e. .74-10 h. .75 -4-10

c. 10X23.876 f. .0184-10 i. 10X3.265

2. Can you make a rule for multiplying a decimal by 100?

By 1000? Kemember, 100=10X10, and 1000=10X10X10.

3. State a rule for dividing a decimal by 100. By 1000.

4. Find each answer by moving the decimal point:

a. 100X.37 d. .74-100 g. 1000X5.2

b. 23.34-100 e. 1000X8.17 h. 7.54-1000

c. 1000X.27 f. 100X8.639 i. 576.74-1000

5. For further practice on these short methods turn to Practice

Test 85a, page 829.

6. The method of dividing decimals given on page 66 is based

on the principle that multiplying both the dividend and divisor

by the same number does not change the quotient.84-2

= ?

804-20 = ? Can you explain the method of dividing decimals?

To multiply a decimal by 10, move the decimal point one

place to the right. To divide a decimal by 10, move the

decimal point one place to the left.



b. .12j =^ =

70 Changing Decimals to Common Fractions

SAVING TIME

Mary works in a dry goods store on Saturday. Mrs. Flagg

bought 12 yards of ribbon at 25^ a yard. Mary rememberedthat 2H is the same as $.25, or \ of a dollar. She multiplied

12X J and charged Mrs. Flagg $3 for the ribbon. Was she right?

How else might she have found the total cost of the ribbon?

Which way is the easier?

Mary's experience shows that you can sometimes save time

by changing decimals to common fractions. Study the examples

below. Is the rule at the bottom of the page correct?

a 75- 7 5 _.15_3a. .io— 100 20 4

,i12^- = 2X12^ _

25 _ 5 1

100~~

2X100 200 40 8

1. Change these decimals to common fractions.

5. .66§ .16§ .75 .12j .2

.33j .50 .25 .20 .08$ .125

2. Express each of the following as parts of a dollar.

50^ 20^ 75j6 12^ 33^

3. Find the cost of each. Do not use pencil and paper.

a. Eight yards of muslin at 12^ a yard.

b. A dozen caps at 25^ each.

c. Nine pounds of butter at 33j^ a pound.

d. Eight golf balls at 75yf each.

4. Change the following decimals to common fractions. Sim-

plify and reduce to lowest terms.

.01 .4 .65 .021 .8$ .62j

.07 .5 .025 .075 .6j .87|

5. For further practice turn to Practice Test 35c, page 329.

To change a decimal to a common fraction, drop the decimal

point and write the denominator. Simplify the result and

reduce to lowest terms.



Changing Common Fractions to Decimals 71

SAVING TIME

Several seventh grade pupils multiplied l\ l)V $$• Most of

them did the work as shown in the first solution below. George

changed l\ and 3^ to decimal fractions and multiplied as shown

in the second solution. Which method do you prefer?

3.5

7.5

7iX3i =^X =HJL = 26j 175

245

2 6.2 5

It is often easier to work with decimal fractions, so you should

know how to change common fractions into decimals.

.75 Since § = 3-f-4, we can proceed as shown in the

4)3.00 first example, and find that % = .75.

The work of changing § to a decimal is shown in

1.8 the next example. #=1.8.

5)9.0 These examples illustrate the rule at the bottom

of the page.

1. Change the following common fractions to equivalent

decimals. Carry out the division to hundredths' place.

3 3 7 _i_ IS. 5. -UL8 5 T2 16 12 8 12

5 5 5 _9_ 2_3 1. 112 4 T 16 11 7 15

2. Use decimal fractions in working these problems:

a. Margaret made three aprons in 8 hours and 15 minutes.

How long did it take her to make one apron?

b. Mary had 13 ft. 6 in. of ribbon. How many rosettes

could she make if each rosette required 1 ft. 6 in.?

3. For further practice turn to Practice Test Number Sod, page

829.

To change a common fraction to a decimal, divide the

numerator by the denominator.



• a.50 r

.25= ±

.2 = ^

.20 c 4

/6irl

./1 4 = 4r

^^^•^^

Common and Decimal Fraction Equivalents

1. The decimal equivalents of the simple common fractions,

such as 5, J,etc., are used so often that they should be memorized.

Memorize the table at the top of this page.

2. Can you give the common fraction equivalents of the

decimals below, without referring to the table above? If you

miss any, look it up in the table and practice on it.

.33j.2 .16§ .66} .5 .50

.25 .75 .12£ -08j .06j .20

3. Give the decimal equivalents of the following common

fractions. Look up those you miss, for extra practice.

341

16

1

12

12

1,

5.

Find the cost of each, without using pencil:

a. Five books at 50<£ each.

b. Nine baseball gloves at 66§^ each.

c. Two dozen cans of tomatoes at 8j^ a can.

Use decimal fractions to solve these problems:

a. On their hike, Mr. Smith and four boys each carried a

pack weighing 3 lb. 8 oz. After walking several hours,

Mr. Smith offered to carry the boys' packs as well as

his own. How much did all the packs weigh?

b. Herbert wanted to paint a porch floor. In order to

find how much paint he would need, he had to find the

area of the floor. He measured and found the floor was

10 ft. 6 in. by 5 ft. 3 in. What was the area?

72



Finding Approximate Results 73

6.2 5 Mr. Etueger's coalbin was filled with coal costing

~V7 $6.25 a ton. It took 5100 pounds (2.7 tons) to fill

137 5 the bin. How much did the coal cost? The exact

1250* «** of the coal was $16,875, or $16.87^. As we

i()o - 5 do not have a half-cent coin, the dealer charged

Mr. Rueger $16.88.

We often get more decimal places in a result than we need.

In such cases we drop the figures that are not wanted. // the first

figure dropped is 5 or greater, the last figure saved should be increased

'by 1. Why is this done? Is $7,848 nearer $7.84 or $7.85? Is

$3,283 nearer $3.28 or $3.29? Is $6,735 nearer $6.73 or $6.74?

Mr. Marshall receives $725 a year, paid monthly, for part-time

work. How much does he receive each month?

6 °- * *In the last division you have one 12 in 20,

12)725.00 ^th a remainder of 8. The last figure in the

7 2 _ quotient is between a 1 and a 2, but is nearer

5 a 2, as the remainder 8 is more than half of

48 the divisor 12.

~~20

1 2 Mr. Marshall received $60.42 a month for

~^ his part-time work.

1. Write the value of each to the nearest cent.

$5,872 $3,598 $23,745 $8,794 $15,008

2. Write the value of each to the nearest tenth.

3.78 5.82 7.994 2.518 18.25

3. Write the value of each to the nearest thousandth.

.7861 1.5976 .00624 3.8975 7.0008

4. Find each of the following to the nearest hundredth.

3.4X6.29 $3.87-2 87-5-6 1h-7 .32x7.1

5. Find each of the following to the nearest tenth.

1.6X27.9 8X3.27 5.7-^.9 8-3 9.5X.38

When you get more decimal places in the result than you

need, drop the figures not wanted. If the first figure dropped

is 5 or greater, the last figure saved should be increased by 1.



74 Who Won the Pennant?

1. The Oakwood High School belongs to the Big Six League.

Last year they played seven

school won lost standing games, winning four and losing

Oakwood 4 3 three. What part of the games

Winston 4 4 played did they win? Express

Marion 3 4 the result as a decimal fraction

Woodside 5 3 to the nearest thousandth. Find

Carville 3 3 the standing of each of the other

Trenton 3 5 teams. Who won the pennant?

2. The number of games won and lost by each team in the

American Baseball League during the 1943 season is given below.

New York won 98 games and lost 56. They won -j^- or .636 of

the games they played. Find the standing of each team to the

nearest thousandth. Copy and complete. Arrange in order.

TEAM

New York

Washington

Cleveland

Chicago

Detroit

St. Louis

Boston

Philadelphia

3. The table below gives the number of games won and lost

by each team in the National League in 1943. Find the standing

of each team to the nearest thousandth. Copy and complete,

arranging the teams in order.

TEAM

St. Louis

Cincinnati

Brooklyn

Pittsburgh

Chicago

Boston

Philadelphia

New York

WON



70 Finding Parts and Ratios

TWO TYPES OF FRACTION PROBLEMS

Problem:Mrs. Anderson received two thirds of her hus-

band's estate when he died. What was the value of her share

if the total estate was valued at $8700?

Solution: 2900

§ of $8700 =j>x$870(r=$58OO

Problem: Mr. James owned 87.4 acres of land, of uniform

value. He sold 25.5 acres to Mr. Smith, who agreed to pay

the taxes for that year on his partof the land. What part of

the taxes on the whole tract should Mr. Smith pay?

Solution: The ratio of the part Mr. Smith bought to the

whole was ffff , or 25.5-87.4. This equals .29, to the nearest

hundredth. Since Mr. Smith bought .29 of the land, he should

pay .29 of the taxes. Find his share.

1. Find the following:

J of 32 fof32 fof9| of 96 | of 57 | of 12

2. Find the following to the nearest hundredth.

.2 of 3.7 .32 of 21.2 .372 of 146

3. Find the ratio of the first number to the second. Express

the result as a decimal to the nearest tenth.

3.2, 8.717, 49 17.4, 40.9

4. Alice Cummings teaches school. Her salary of $1100

for tne school year is paid in 9 equal installments. Find the

amount of each installment to the nearest cent.

5. The school board offered to pay Alice's salary (problem

4) in 12 equal installments, if she preferred. How much would

each installment be, to the nearest cent?

To find a fractional part of any number, multiply the

number by the fraction, common or decimal.

To find the ratio of one number to another, divide the

first by the second. The result may be expressed either

as a common fraction or a decimal.



Finding the Whole when a Par Is Known 77

A THIRD TYPE OF FRACTION PROBLEM

Mr. Adams, a farmer, was filling a bin with wheat. After

he had put in 24 bushels, he estimated that the bin was|

full. How many bushels would the whole bin hold if the

estimate was accurate?

Solution: As § of the whole, or f times the whole, equals

24 bushels, the problem is to find what number multiplied

by | equals 24. That is, we know the product of two factors

(24), and one of the factors (§), and we want to find the otherfactor. We must divide the product by the known factor in

order to find the unknown factor. Why? 24n-f=24xf = 36.

The bin holds 36 bushels.

1. Find the value of the whole number. Notice the form

of solution used for the first two examples.

a. fxa number=21; the number =21-j-§=21x|-=28

b. .32 xa number = $2.17; the number = $2.17 ^-.32 = $6.78

c. § of a number = 42 d. f of a number = 12^

2. Julia's older sister used N as an abbreviation for

a number. Study both solutions. Which is shorter?

Julia's Solution Her Sister's Solution

fxa number = $9 f xA/ = $9

The number = $9-h| =$9 X| = $12 A^ =$9-f =$9xf = $12

3. Find the value of the whole number to the nearest cent,

in each of the following.

a. .32xAr = $8.17 c. §xAr = $25.10 e. .07xA/ = $0.32

b..7

XAT =

$5.23d. .

189XN= $2.09 f. .79

XN= $17.91

To find a number when some fractional part of it is known,

divide the known part by the fraction it is of the whole.

fXJV=$9 iV=$9-f = $9X;f = $12



78 The Three Types of Fraction Problems

#:

W49

L'-<f• *'**&**

1. Mary picked strawberries

for her father. There were 12

equal rows in the patch. Bynoon she had picked 8 rows

and had 45 quarts. How many

quarts should she get from the

whole patch at that rate? How

many different ways can you

find to work this problem?

2. On a test in arithmetic, Alice worked 17 problems cor-

rectly out of a total of 20. What part of the problems did

she work correctly? What part did she miss? Express the

results as decimal fractions.

3. The A. B. C. Furniture Company advertised the follow-

ing articles during a sale. Find the sale price of each to the

nearest cent.

a. Desk. Regularly $87.25. J off.

b. Table. Regularly $39.50. § off.

c. Rug. Regularly $115.00. \ off.

4. In 1930, the population of Bettsville was 8756. In 1940,

it was 9182. Theincrease

waswhat fractional part of the

population in 1930? Express your result as a decimal to the

nearest thousandth.

5. In 1930, the population of Mason was 12,091. In 1940,

it was 8756. The decrease was what part of the population

in 1930? Express your result to the nearest thousandth.

6. Mr. Thompson was driving his car from Cincinnati to

Columbus. He looked at his speedometer when he reached

South Charleston and found that he had driven 73.4 miles.

He estimated from the map that South Charleston was about

| of the distance from Cincinnati to Columbus. From this

he found the approximate total distance from Cincinnati to

Columbus. What was the approximate distance?

1A



IUsing Decimals in Problems

1. Twelve boys had a steak roast and shared the expenses

equally. John bought the meat, which cost $2.75; Harry

bought the buns, at 75^; Morris bought the baked beans, at

70^; and Robert bought the ice cream, at $1.25. John acted

as treasurer. What was each boy's share of the expenses?

How much did John collect from each of the boys who did not

buy any of the food? How much did he pay each of the other

boys who bought food? How much did he pay himself?

2. Roger Owen earned $2.75 last week. He spent $1.50

for a sweater, 35^ for a moving picture show, 15^ for candy,

and saved the rest. What part of his earnings did he save?

Express the result as a decimal to the nearest hundredth.

3. The Jones family try to save at least .15 of their income.

Their income is $250 a month. How much should they save

each month?

4. Last year the Jones family saved .23 of their income.

This is how many times (nearest hundredth) the least amount

they try to save? (See problem 3.) Can you work this prob-

lem without finding how much they saved last year?

79



80 Avoiding Errors with Compound Numbers

In many problems like those on the next page it is necessary

to work with denominate numbers expressed in more than one

unit, as 2 ft. 3 in., or 3 weeks 4 days. Such numbers are called

compound numbers. Study the points given below and you will

avoid errors.

1. Tables. Tables of denominate numbers are given on pages

342 to 344. You should memorize those used most commonly

and know where to find the others.

2. Changing to Smaller Units. Change 5 hoursto minutes.

5 hrs. = 5X60 min. = 300 min. For practice turn to Practice Test

45a, page 836.

3. Changing to Larger Units. Change 17 inches to feet.

17 in. =^ ft. = l-& ft. = 1.4 ft. (nearest tenth). For practice turn

to Practice Test 45 b, page 836.

4. Changing from One Unitto Two. Change 31 days to

weeks and days. Divide 31 by 7. Why? What is the quotient?

The remainder? 31 days = 4 weeks 3 days. Why? For practice

turn to Practice Test 46, page 336.

5. Changing from Twro Units to One.

a. Change 2 lb. 4 oz. to ounces. 2 lb. 4 oz. = 2Xl6 oz.+

4 oz. = 36 oz.

b. Change 2 lb. 4 oz. to pounds. 2 lb. 4 oz.^2^- lb.=

2-L lb. = 2.25 lb.

For practice turn to Practice Test 47, page 837.

6. Adding. See page 10. For practice turn to Practice Test 48a,

page 838.

7. Subtracting. See page 16. For practice turn to Practice

Test 48 b, page 338.

8. Multiplying.

a. Multiply 3 qt. 1 pt. by 3. Explain the three methods shown

at the top of page 81. Are the results equal?



Avoiding Errors with Compound Numbers 81

3qt, 1 |>t. 3 qt. I

3

= 7 pi-



s-J Problems Using Compound Numbers

1. Last summer Dick worked in a grocery store. One week

he worked 2 hr. 20 min. on Monday, 1 hr. 25 min. on

Tuesday, 3 hr. on Wednesday, 2 hr. 10 min. on Thursday,

1 hr. 50 min. on Friday, and 9 hr. 30 min. on Saturday. He

was paid 20^ an hour. How long did he work altogether

that week? How much did he earn?

2. Josephine had 5 yd. 9 in. of braid. She used 23 inches

to trim a dress. How much did she have left?

3. Lucille made doll dresses to sell for Christmas. She had

2 yards and 1 foot of material and used 8 inches for each

dress. She paid 30^ a yard for the material and sold each

dress at 25^. How many dresses could she make? How much

was her gain on all of them?

4. Jane worked 8 addition examples in 3 minutes and 20

seconds. What was the average time it took her to work one

example? At the same rate, how long would it take her to

work 12 examples?

5. How many square feet are in the area of each of the

following rectangles? Work each problem in two ways.

a. 12 ft. 9 in. by 8 ft. 6 in.

b. 2 ft. 8 in. by 9 in.

c. 15 ft. 3 in. by 12 ft. 4 in.

6. How many boxes, each holding 20 oz., can be filled from

8 lb. 12 oz. of candy? Work this problem in two ways?

7. The pages of June's scrapbook at 11 in. by 1 ft. 3 in.

How many square inches are there on each page?

8. John's Scout Troop were making up 24 bags of candy

for a Christmas tree. They wanted 12 ounces of candy in

each bag. How manypounds of candy will they need to buy?

9. Mr. Adams ordered the A. B. Jones Coal Co. to fill his

coalbin with coal costing $7.50 a ton. It took three loads.

The first load weighed 2350 pounds, the second 2615 pounds,

and the third 2187 pounds. How many tons was this alto-

gether? Find the cost to the nearest cent.



Problem Test 83

1 . The school system in which Jane teaches pays its

teachers every two weeks. Find the amount of Jane's check

if she receives 16 equal installments and her yearly salary

is $950.

2. Last spring Harold Simmons wanted to reseed the

back yard. His garden book recommended the use of .002 lb.

of seed for each square foot of lawn. He measured the back

yard and found it was 50 ft. 9 in. by 42 ft. 3 in. How much

grass seed did he need?

3. After the grass had been up about three weeks, Harolddecided to apply some fertilizer. His book recommended the

use of .03 lb. for each square foot. (See problem 2.) Howmuch fertilizer did Harold need?

4. John's father wants to build a barbed wire fence around

the barn lot. How much wire will it take to run one strand

around the lot if it is square and measures 42 yd. 2 ft. on

each side? How much will it take for 3 strands? How muchwill 3 strands cost at 3j£ a yard?

5. Find the cost of each of the following:

a. 7582 pounds of coal at $8.25 a ton.

b. 873 pounds of hogs at $8.40 a hundred pounds.

c. 2385 feet of lumber at $65.50 a thousand feet.

d. 758 pounds of potatoes at $1.19 a hundred pounds.

e. 6 yards 9 inches of cloth at $2.50 a yard.

6. Dick made a model airplane which flew | mile in 3

minutes 15 seconds. This was at the rate of how many feet

in a minute to the nearest foot?

7. A submarine is 100 feet under water. How many cubic

feet of water are there above each square foot of its top sur-

face? Find the total weight of the water in each square foot

of top surface. One cubic foot of sea water weighs approxi-

mately 64.3 lb.

8. The heights of the five regular members of the Harrison

High basketball team are 5 feet 8 inches, 5 feet 10 inches,

6 feet, 5 feet 6 inches, and 5 feet 9 inches. Find the average

height.



Chapter Tesl 85

Write the numbers from 1 to 2(1 on a piece of paper. After

each number write the words or numbers which belong in the


1. In the number 532.78, the 5 is in place, the 3 in0)

place, the 2 in place, the 7 in place, and the 8 in

(2) (3) (»)

place.(5)

2. The number .469 is read Four hundred sixty-nine

(6)

3. Two hundred and seven thousandths is written(7)

4. Two hundred seven thousandths is written(8)

5. The number of decimal places in the product is equal to

the number of decimal places in the multiplier the number(9)

in the multiplicand.

6. Of the numbers .29, 2.9, and 29, the answer to the example

95.124-32.8 is nearest to(10)

7. 7.2^100 = on

8. The common fraction equivalent of .663 is

(12)

9. The decimal equivalent of £ is*

(13)

10. To the nearest thousandth the decimal 2.78682 is equal

to(14)

11. To the nearest cent, $17,725 is usually called

(15)

12. To the nearest tenth, 5.987 is

(16)

13. To find .39 of $7.25 you $7.25 by .39.

(17)

14. If .71 of a number equals 9.2, in order to find the whole

number you divide by(18) (19)

15. To find the ratio of 9.35 to 12.82, you divide by(20) (21)

16. 4 ft. 8in. = in.

(22)

17. 4 ft. 8in. = ft.

(23)

18. Subtract 1 yd. 2 ft. from 3 yd. The remainder isJ J

(24)

19. Multiplv 3 yards 2 feet by 2. The answer is feet, or

yards.

*

(25)



CHAT T I R

h\ teaming to Work with

|U Per Cents

After the decimal system of writing numbers was

introduced into Europe, it became the common prac-

tice for merchants and others to use one hundred as

a base. For example, a man's tax might be $7 for every $100

of his property, or 7 hundredths of the value of his property.

In Latin, 7 hundredths was written 7 per centum, or 7 per cent.

The % sign was introduced in the fifteenth century as an

abbreviation for per cent.

Dick saw the advertisement on the opposite page in a news-

paper. You have probably seen many statements like the

one in the paper. Can you explain its meaning? Remember,

per cent is simply another name for hundredths. Thus, 25%means 25 hundredths; 17% means 17 hundredths; \% means

\ of a hundredth.

b.

f.

1. Copy the following and fill the blanks

71 hundredths = %3 hundredths = %

2\ hundredths = %900 hundredths = %

| hundredths = %7.8 hundredths = %

g.



88 Three Ways of Writing Hundredths

You should be able to write any number of hundredths in

three different ways—as a common fraction, as a decimal fraction,

and as a per cent. You will find thatall three of these methods

are useful.

1 Study the first three examples carefully. If you do not

understand them, ask your teacher for help. Then copy the

remaining examples and fill the blanks.

_ 367 —7%=T*T= 07 —I™'—39%=t^=-39 189%=—:;r214%=tU = 2 -14

-

9%=_=— 79%r~zr~

= =.45—To~o

—

_ 81 - = =1-15—100

2. Write each of the following in three ways.

Fifty-nine hundredths

Five hundredths

Forty-six hundredths

One hundred eight hundredths

Six hundred seventy-five hundredths

One hundred hundredths

One hundredth

Nineteen hundredths

3. Write the following per cents as decimal fractions.

17% 85% 23% 5% 65% 2%

3% 49% li% 90% 31% f%

4. Write the following per cents as common fractions or mixed

numbers.

3% 9% 13% 27% 1% i% 93% 125% 210%

5. Write each of the following in two other ways.

8% M -17 ^ 125% -°7

6 Do you know the difference between 2 hundreds and

2 hwidredths? Write each as a decimal. As a common fraction.



Finding Per Cents ofNumbers Using Decimals 89

Henry boughl a suit of clothes a1 a Bale. The suit was marked

$18 but during the sale this price was reduced by 1.5%. How

much did Henry have to pay for the suit?

18 Original price=$18

.1 5 Reduction=15% of $18 = .15X$18 = $2.70

9~o Sale price= $18-$2.70= $15.30

18

2.7

Henry's solution is shown above. It shows you how to work

a very important type of problem in percentage, that is, finding

a per cent of a number. Henry wrote the rule at the bottom

of the page. Is it correct?

1. Find the following.

a. 35% of 95 c. 65% of 18 e. 2% of 320

b. 8% of 25 d. 74% of 56 f. 95% of 132

2. Find the following to the nearest cent.

a. 12% of $18 c. 25% of $12.65 e. 1% of $7.25

b. 80% of $7.32 d. 6% of $25.50 f. 97% of $25

3. Mrs. Baird's gas bill for December was $5.19. The com-

pany gives a discount of 3% if the bill is paid by January 10.

How much did Mrs. Baird save by paying before that date?

How much did she have to pay the company?

4. Find the sale price of each of the following.

Article Former Price Per Cent of Discount

Overcoat $18.50 20%

Suit $22.50 15%Hat $ 4.50 25%

To find any per cent of a number, write the per cent as a

decimal fraction and multiply.



Sh% ~ 7x 25 % =-2f

3o7o = To

3 3 5% =3

6o% *§j

Memorizing Per Cents as Common Fractions

\lbert Carter works in his father's furniture store during

$12 vacations. Last summer they held a sale and

.2 5 $12 gave 25% off on all marked prices. Albert

6 __3 Sokl a table marked $12 for $9. Was this correct?

2 4 •$ 9 His solution is shown at the left.

$3.0

Albert's father showed him another way, making use of the

fact that 25% =tW=i-

His solution is given below.

25% of $12=\ of $12 = $3 $12- $3 = $9

Mr Cartersuggested that Albert would be able to do the

work in his head if he memorized the common fraction equivalents

of the per cents commonly used in business.

1 Study the first column in the table at the top of the page.

When you think you know it, cover the common fractions with

a paper and say them from memory. Then cover the per cents

and give them from memory. Study each of the other columns

in the same way. Putextra time on any that give you trouble.

2. Give the common fractions that are equal to each

12j% 25% 50% 10% 30% 62j%

33l% 90% 80% 87j% 20% 37j%

3. Give the per cents that are equal to each:

75%

16f%

66|%

40%0

5

6

1

10

1

121

4

31018

4. Give the common fraction or per cent equal to each:

n _ . ^w 9

25%5

30%

12j%

16§%12

1

10 66f% 80%

90%5.

8

40%

87i%1

6

50% 33^%

75%

2.

5

20%1

3

10%r¥

62j% 37£%

90



Finding Per ( <-nis of Numbers 91

Using Common Fractions

You have Learned to find per cents of numbers by firsl writing

the per cent as a decimal fraction and then multiplying. You

have also found that ill sonic cases it is easier to write I lie per cent

as a common fraction. Always use the easier method. A^nes

worked the examples below. Did she use the easier way?

66§% of $18= f X$18 = $12

65% of $23 = .65X 23 = $14.95

1. Find the following without using a pencil. Change the

per cents to common fractions.

a. 33j% of 24 d. 25% of 120 g. 66§% of 21

b. 16f% of 72 e. 10% of 95 h. 37^% of 32

c. 12j% of 64 f. 75% of 12 i. 87j% of 16

2. Find the following. Change the per cents to decimals.

a. 17% of 84 b. 32% of 125

3. Find the following in the easiest way. If the answer is in

dollars and cents, express it to the nearest cent.

a. 82% of $8.25 d. 50% of $17.30 g. 97% of $3.20

b. 25% of $37 e. 75% of 84 h. 62j% of 16

c. 8j%of$144 f. 81% of 7 i. 67% of $61.17

4. Mr. Allen receives a salary of $2400 a year. He plans to

spend 30%, of this for food, 25% for rent, 10% for clothing, 25%for miscellaneous expenses, and to save 10%. How much does

he plan to spend for each item? How much does he plan to save?

Check your results.

5. During a sale, one music store advertised a saxophone for

$12, with a discount of 16§%. Another store advertised a similar

saxophone for $15, with a discount of 33^%. Which is cheaper?

6. Which is cheaper, $24 with a discount of 25%, or $27 with

a discount of 33j%?

To find a per cent of a number, write the per cent as a

common fraction or decimal, and multiply.



-J

The Meaning of Per Cent

You have probably seen many statements such as the following.

Read them carefully and explain what each means.

1. We pay 2% on all savings deposits. (From a bank adver-

tisement.)

2. All suits marked down 25% to 33j%. (From the adver-

tisement of a clothing sale.)

3 #

99_4_4_<v pure. (From a soap advertisement.)

4. In 1930 approximately 13.1 per cent of the wheat grown

in the United States was exported to foreign countries. (From

a magazine article.)

5. Sale. Discounts up to 50%. (From a jewelry store adver-

tisement.)

6. Canned Milk Price Is Up 20 Per Cent, (Headline in

newspaper.)

7. Contains 80% wool. (From the advertisement of a wool

and cotton blanket.)

8. It's realcoffee—with 97% of the caffein removed. (From

a coffee advertisement.)

9. 100% Pure Cane Sugar. (From label on box of sugar.)

10. 98% Camel's Hair Sport Coats. (From the advertisement

of a coat sale.)

11. With 3 per cent starch added to prevent caking. (From

label on box of sugar.)

12. LOANS up to $300, Charges 2^% on unpaid monthly

balance. (Advertisement in newspaper.)

13. Alcohol 5%. (Label on bottle of vanilla.)

14. See how many similar statements you can find and report

them to your class.

92



Selling on Commission <J3

Jim Ryerson sold paper

products during his vaca-

tion last summer. He re-

ceived a commission of 12%

on the amount sold. One

week Jim's sales amounted

to $85. He found his com-

mission as shown below.

Commission = 12% of $85 =

.12 X$85 = $10.20

When one person sells

goods for another and is

paid according to the amount

sold, the amount received

for selling the goods is called the commission

figured as a per cent of the selling price.

1. Find the commission on each of the following

(

It is usually

Rate of

Commissionales Commission Sales

$3000 16|% $4800

$150 25% $70

$375 15% $925

2. Mr. Phillips travels for a large grocery firm,

Rate of

Commission

8|%

12J%3%

He receives

a commission of 12J% on the amount he sells. Last year his

sales totaled $17,250. What was the amount of his com-

missions for the year?

3. Mr. Reeves sells farm machinery on commission and

receives 8% of his sales. How much would his commission be

on a tractor which he sold for $850?

4. Hazel Strong works in a ladies' clothing store. Shereceives $12 a week and a commission of 3% on all she sells.

What would be the total amount of her earnings for a week

during which she sold $350 worth of clothing?

5. The sales for Jim Ryerson's best week amounted to

$107.38. How much was his commission for that week?



IVr Cents on Tests and in Games «.»:,

1. On an addition test Marion finished 12 examples and

had 10 right. What per cent of the total number worked did

she have right?

2. On the same test James finished 10 examples and had

9 right. What per cent of the total number worked did he

have right?

3. If the time spent was the same, did Marion or James

add more rapidly? Which was more accurate?

4. Marion took the same test a second time. She worked

14 right out of 16 attempted. Did she improve in speed?

In accuracy?

5. On his next attempt James had 12 right out of 12 at-

tempted. Did he improve in accuracy? In speed?

6. Alice Henderson plays on the

girls' basketball team. Every day

she practices free throws. One day

she threw 15 out of 25 attempts.

What per cent did she throw? The

next day she threw 20 out of 32 at-

tempts. What per cent did she make?

Did she improve her record?

7. Last year Alice's team played

12 games and won 8. What per cent

of the games did they win? The year

before they played 10 games and

won 7. What per cent did they win

that year? Which record was better? 4^ ^8. Work these examples without pencil. First think of the

ratio; then simplify and change to a per cent. The correct

answer to the first one is 6 is -|, or f , or 75% of 8.

a. 6 is what 9c of 8?

b. 9 is what % of 18?

c. 7 is what % of 42?

d. 8 is what % of 12?

e. 3 is what % of 30?

f 5 is what % of 40?

g«



•Mi Per Cent of Increase and Decrease

The population of Owensburg, where George Simmons lives,

was 6000 in 1930. In 1940, it was 8000. In order to find how

rapidly the town had grown in ten years, George figured the

per cent of increase as shown here.

Increase = 8000 - 6000 = 2000

2000 is what per cent of 6000?

Per cent of increase = f{HHf=

3= 333%-

Owensburg increased 33j% in population.

George's cousin Tomlives in Unionville. In 1930, the popu-

lation of Unionville was 10,000. In 1940, it was only 9000.

George found the per cent of decrease.

Decrease = 10,000-9,000 = 1,000

1,000 is what per cent of 10,000?

Per cent of decrease =11

<&$r= To = 10%

Unionville decreased 10% in population.

1. State a rule for finding per cent of increase; per cent

of decrease. Compare your rules with those given below.

2. A year ago Robert weighed 90 pounds and James

weighed 64 pounds. Robert now weighs 99 pounds and James

72 pounds. Find the per cent of increase in weight for each.

3. Did Robert or James (problem 2) increase his weight

by the larger amount? Which increased more rapidly, or at

the greater rate?

4. What does the per cent of increase show you that is

not shown by the amount of increase?

5. Tom grew from 5 feet to 5 feet 5 inches, while his sister

Mary grew from 4 feet 2 inches to 4 feet 7 inches. Find the

per cent of increase of each. Which grew more rapidly?

To find the per cent of increase or decrease, first find

the amount of the increase or decrease by subtracting.

Then find what per cent this increase or decrease is of

the original value.



Problems. PerCenl of Increase and Decrease 91

1. Find the per cent of increase or decrease' <> each of the

owing.



Working with Large Per Cents

Month

January

February

March

Number of

Subscriptions

16

12

Per Cent of January

Subscriptions

100%

200%

150%

In comparing numbers by division, the ratio is often a

whole or a mixed number.

Mrs. Sheridan started taking magazine subscriptions last

January. The first month she secured eight subscriptions. In

order to know how rapidly herbusiness was growing, she

compared the number of subscriptions she secured each month

with the number she obtained in January.

In filling out the last column in her summary at the top

of the page, Mrs. Sheridan thought as follows:

8 is*, orl times 8. 8 is 100% of 8. (1 =100. = 100%)

16 is -^, or 2 times 8. 16 is 200% of 8. (2 = fj$=200%)

12 is 3f,or lj times 8. 12 is 150% of 8. (lj =^_o = 150%)

1. Explain how Mrs. Sheridan got the 100%,. The 200%.

The 150%.

2. Copy Mrs. Sheridan's summary and make your state-

ment cover the remainder of the year, as given below.

April, 10 subscriptions; May, 18; June, 22;

July, 14; August, 20; September, 19; October, 23;

November, 21; December, 24.

3. Copy and complete.

300% =3 d. %=3Je. %=4f. 1000%=

98

a.

b. 125%

c. 750%=li

g. 175%=

h. 100%=

1. 70—^8



Working with Large Per Cents 99e

L. Read the following comparisons in terms of per cents.

The correct answer to the first one is 12 is 300% of 4.

a. 12 is 3 times 4 d. 30 is 3 J limes 9

1). 28 is 4 times 7 e. 9 is 2\ times 4

c. 20 is 2\ times 8 f. 14 is 7 times 2

2. Read these comparisons in terms of whole or mixed

numbers. The correct answer to a. is 7 is 3| times 2.

a. 7 is 350% of 2 d. 14 is 175% of 8

b. 9 is 100%, of 9 e. 40 is 1000%, of 4

c. 15 is 500%, of 3 f. 20 is 166|% of 12

3. Find the following without using pencil and paper.

a. 21 is what % of 7? d. What % is 4 of 3?

b. 15 is what % of 20? e. What % is 44 of 8?

c. 5 is what % of 5? f. What % is 15 of 2?

4. Find the following without using pencil and paper.

a. 200%, of 8 d. 100%o ofl3

b. 150% of 16 e. 125%, of 12

c. 500%, of 7 f. 350% ofl0

5. In 1940, there were about 25,000,000 children attending

school in the United States. In 1890, there were about

12,500,000. The number attending in 1940 was what per

cent of the number in 1890? The increase between 1890 and

1940 was what per cent of the attendance in 1890?

6. In June, 1940, there were 900 airplane engines produced

in the United States. In June, 1941, there were 1800 pro-

duced. The number produced in June, 1941, was what per

cent ofthe

numberin June, 1940? This was what per cent

increase?

7. In June, 1940, the United States built 20 light army

tanks, while 260 were built in June, 1941. The number pro-

duced in June, 1941, was what per cent of the number in

June, 1940? Find the per cent of increase.



MA6A7/NE-

qUsing Per Cents in Figuring Expenses and Profits

Eugene Sanders owns a newsstand. Last month his total

sales were $250. The newspapers and magazines he sold

cost him $175. He paid $15 for rent, $20 to a boy who helped

him, and $15for express charges. Find his profit.

The money a merchant takes in is used for three purposes:

to pay the cost of the articles sold, to pay the operating ex-

penses, and to pay the merchant a profit.

Selling Price = Cost+ Expenses+Profit

To find whether expenses are too great and whether profit

is reasonable, most merchants express cost, expenses, andprofits as per cents of the sales. Eugene did this.

Expenses = -££$ = \ =20% of the sales.

Cost =m = -^ = 70% of the sales.

Profit=^&^ = 10% of the sales.

1. How could Eugene check his results? 20%+70%+10% = 100% . Why should this sum equal 100%?

2. Express cost, expenses, and profit as %'s of sales. Check.

Sales Cost Expenses Sales Cost Expenses

$ 1,000 $ 600 $ 800 $ 800 $ 600 $ 100

12,000 8,000 3,000 7,200 3,600 2,400

5,000 3,500 1,000 9,200 6,900 1,480

100



Using Per Cents in Problems 101e

1. The total enrollment in the seventh grade of the

Washington School is 30 pupils. Last Monday there were

27 pupils present. What per cent of the pupils were present?What per cent were absent? Check.

2. In order to test his seed corn, Mr. Smith planted 40

grains in a box. Thirty-five of these grains sprouted. What

per cent of the corn sprouted?

3. On a test containing 24 questions, Mane Owens

answered 18 correctly. What per cent did she have right?

What per cent did she answer incorrectly or omit? Check.

4. A farmer weighs his corn in the fall and finds that he

has 50,215 lb. How many pounds will he have in the spring

if the corn loses 20% of its weight through drying?

5. During one month Mr. Elling sent to the creamery 7500

pounds of milk, testing 4% butterfat. How much butterfat

was there in the milk? What did Mr. Elling receive if he was

paid at the rate of 22^ a pound for the butterfat?

6. Last year Mr. South picked 900 bushels of peaches

from his orchard. Of these, 600 bushels were first grade.

What per cent of the total crop was first grade?

7. During the second World War, Hopeville sold $60,000

worth of bonds during the first week of the Fifth War Loan

Drive. This was what per cent of its quota of $75,000?

8. Mr. Morris owns a shoe store. Last year he sold $15,000

worth of shoes, which cost him $9,000. He paid $900 for

rent, $3,100 for salaries to himself and clerk, $200 for adver-

tising, and $300 for heat, light, and other expenses. Express

the expenses, profit, and cost each as a per cent of the sales.

Can you check these results?

9. Mr. McComb had a deposit of $857.18 in a bank that

failed during the depression. Six months after the bank

closed, it paid the depositors a dividend of 25% of their

deposits. How much did Mr. McComb receive? The bank

later paid dividends of 10% and 3%. How much did Mr.

McComb receive altogether? How much did he lose?



102 Finding the Whole when a Per Cent

of It Is Known

THE EQUATION

Albert Carter gave his father this problem: Two thirds

of the pupils in the eighth grade are girls. How many pupils

are in the eighth grade, if there are 18 girls in the class?

Mr. Carter's solution is given below.

fxAf = 18

9

N = 18 -r- J =iSfxJ>= 27 pupils

A statement such as §XN = 18 is called an equation.

The two members of an equation are equal. We show this fact

by using the sign =. If we add or subtract, multiply or

divide in one member of an equation, we must do the same

in the other member to keep them equal. What did Mr.

Carter do to both members of the equation above?

Mr. Carter divided § XN by §. The quotient is N; just

as when you divide 5x7 by 5, the quotient is 7. Then he

divided 18 by J.The quotient is 27. So, N = 27.

Mr. Carter explained to Albert that this kind of problem

often occurs in working with per cents and gave the follow-

ing illustration from his furniture store. He sells his furniture

so that 60% of the selling price covers the cost. The rest

of the selling price covers expenses and profit. He showed

Albert how to find the selling price of a chair that costs $12.

Mr. Carter used S.P. for selling price.

60% of S.P. = $12

|XS.P. = $12

4

S.P. =$12-hf =$HTxJ>= $20

1. State a rule for finding the whole when a per cent of it

is known. Compare your rule with the one on page 103.

2. Find the whole number, N, in each of the following.

a. 66f%ofAr = 15 b. 37i%ofN = 21 c. 25%oftf = $7



Problems. Finding tin- Whole 103

USING COMMON FRACTIONS

1. Margueritetakes

subscriptions for a maga-

zine and receives a com-

mission of 8|%. How

much must her subscrip-

tions amount to for a

week in order to earn

$1.50?

2. Howard read that

5,180,538 people in Cali-

fornia live in towns and

cities and that this is ap-

proximately 75% of the

population of that state. From this, Howard found the

population of California. Canyou?

3. Harriet made the following problem: I spelled 35 words

correctly on a spelling test. This was 87J% of the total

number of words. Can you tell me how many words were

in the test? Find the answer to Harriet's problem.

4. Tests show that a certain breakfast food loses 10% of

its weight after packing. How much will the manufacturers

have to put in a box so that the contents will weigh 18 ounces

after standing?

5. Find the whole number, N, in each of the following.

33i%ofiV= 12 12i%o(N = 7 66f%ofiV = $8

75% of N = 30 87j% of N =U 37j% of Ar = 9

16f%ofAT = 5 60%ofAr = 15 200% of iV = $17

300% of AT = 18 250% of A

r

= 25 125%, ofAT

=25

To find the whole when some per cent of it is known,

write an equation expressing the per cent as a common

fraction or whole number. Then divide both members of

the equation by this fraction or whole number.



Finding the Selling Price

Mr. Hopper owns a jewelry store. He knows that 45% of

the selling price of every article goes to pay expenses (rent,

insurance, wages to clerks, advertising, light, heat, etc.); and

that in order to make a living he must get 15% of the selling

price as his profit. At what selling price must he mark a

watch for which he paid $30?

As the expenses and profit together equal 60% of the selling

price, only 40% of the selling price is left to cover cost. Mr.

Hopper must sell the watch so that 40% of the selling price

(S.P.) equals the cost price (C.P.), or $30.

40%, of S.P. = $30

5

S.P.= = $30X4 = $75

Check

40% of S.P. =

fX$75 = $30

1. Find the selling price. Check.

Profit

10% of S.P.

Cost Expenses

$ 5.40 15% of S.P.

17.15 33^% of S.P.

8.05 25% of S.P.

21.36 30% of S.P.

9.50 35% of S.P.

2. In Mr. Whitaker's furniture store last year the total

sales were $72,000. The expenses were $18,000. The furniture

which was sold cost $48,000. What was the profit? Express

the expenses, profit, and cost of furniture each as a per cent

of the sales. How can you check your results?

16f% of S.P.

5% of S.P.

of S.P.

of S.P.

10%

15%

104



Writing IVr Cents as Decimals L05

You know that in finding a per cent of a number it is often

easier to write the per cent as a decimal fraction instead of a

common fraction. As per cent is just another name for

hundredths, simple per cents such as 17',

and 32 r;

are

easily written as decimals. In many practical problems,

however, you will meet more complicated per cents such as

317%, i%, 2|%, .7%, and 32.7%. In working such prob-

lems you must be able to write these per cents as decimals.

1. Study the following examples carefully. Be sure that

you understand all of them.

a. 23% = 23 hundredths = .23

b. 7% = 7 hundredths = .07

c. 1% = 1 hundredth = .01

d. 317% = 317 hundredths = 3 wholes and 17 hundredths =

3.17

e . 1% = i of a hundredth = .00j = .005

f . 2J% = 2J hundredths = .02\ = .025

g. 3J% = 3J hundredths = .03

h. .7% = .7 of a hundredth = 7 thousandths = .007

i. 32.7% = 32.7 hundredths = .327

2. State a rule for writing a per cent as a decimal. Com-

pare your rule with the one at the bottom of the page.

3. Write the following per cents as decimals.

35% 486% f% 81% 560%

393% .5% 8i% i% 19-9%

45.2% 6.6% 19.5% 160% |%9.4% 56.1% 2i% 37.7% 82.2%

\%170% 67% 5i% 9%

To write a per cent as a decimal, think of it as hundredths

and write it as a decimal. Remember that a tenth of a

hundredth is a thousandth, so tenths of a per cent are

written in thousandths' place.



106 Finding Per Cents of Numbers

USING DECIMALS

1. Find the following.

.2% of 875 i% of 810 .7% of 1250

37.2% of 8.7 88.7% of 49 17.1% of 45

117% of 93 2.5% of 112 lj% of 105

i% of 756 349% of 18 85.4% of 70

2. Find the following to the nearest cent.

32.2% of $8.75 19.2% of $17.80 307% of $19.20

3.8% of $25.18 \% of $72 91.5% of $43.15

231% of $5.25 2j% of $36 .7% of $125

3. There were 48,833,000 persons in the United States in

1930 engaged in gainful occupations. Of these, 21.5% were

engaged in agriculture, 2.5% in mining, 29.3% in manufac-

turing andmechanical industries, 9.0% in transportation,

15.4% in trade, 2.3% in public service, 7.0% in professional

service, 9.8% in domestic service, and 3.2% in all other occu-

pations. How many were engaged in each type of work?

Can you check your results? Do they check exactly?

4. The population in 1940 and the per cent of increase, or

decrease, from 1930 to 1940 are given in the table below for

the eight largest cities in the United States. If they continue

to change at the same rate, what will be the population of

each of these cities in 1950?

City



Finding the Whole when a IVr Cent

of It Is Known

In finding the whole when some per cent of it is known, it

is often easier to use a decimal instead of a common fraction.

In one year the people of the United States bought 2495

tons of shelled peanuts from China. This was 91.9% of the

total amount bought from all foreign countries. How many

tons were bought from all countries?

91.9% of N= 2495

.919XAr = 2495

Divide both members of the equation by .919. Explain.

N = 2495 -f- .919 = 2715 tons (approximately)

1. State a complete rule for finding the whole when a per

cent of it is known, using either common or decimal fractions.

2. Horace Jennings read

that Texas produced 4,038

bales of cotton in 1930, and

that this was 28.9% of the

total produced in the United

States. How many bales

were produced in the United

states in 1930? How many

pounds was this if the bales

averaged 500 pounds? What

was its value if it was worth

9.5 cents a pound, on the

average?

3. Find the whole to the nearest cent.

17% of AT = $2.04 330% of iV = $9.24 27.3% of N25% of TV = $16 3.4% of N = $2.72 85.9% of N

8% of AT = $2.50 149% of A^ = $25.10 .7% of A^ = $

$8.25

$31.21

.83

To find the whole when some per cent of it is known,

write an equation expressing the per cent as a common or

decimal fraction. Then divide both members of the equa-

tion by this fraction.



Finding the PerCent of Games Won

The football team of the Connellsville High School played

7 games and won 4. Eugene said they won j- or .571, of their

games. He thought, .571 equals 57 hundredths and 1 thousandth,

or 57.1 hundredths, or 57.1%.

In changing a decimal to a per cent it is necessary to read

the decimal as so many hundredths. This is easy in the case of

.39, or .72, but harder in such cases as .571, .3, or 2.32.

1. Eugene used the rule at the bottom of the page. Study

and explain each step.

2. Can you explain each step of the rule below?

3. Read the following as hundredths.

.17 .4 .185 1.324 .0032

1*38 3.5 .497 7.019 .1709

To Read a Decimal as Hundredths:

a. Read the figures up to and including the second decimal

place as a whole number. Fill in hundredths' place with

a zero if necessary. This is the number of whole hun-

dredths.

b. If there are three decimal places, read the figure in the

third place as tenths of a hundredth.

c. If there are four decimal places, read the figures in the

last two places as hundredths of a hundredth.

108



Writing Decimals as Per Cents L09

When Eugene found the per cenl of games won by the Connels-

ville High School, he had to know how to write ;i decimal fraction

as a per cent. He used the rule a1 the bottom of the page. Ex-

plain. Remember that per cents and hundredths arc differenl

names for the same thing.

1. Road the following decimals as hundredths and write

them as per cents.

2.176

2.019

5.307

11.122

1.007

7.301

2. The standings of the teams in the two major baseball

leagues on August 12, 1943 are shown below. The standings are

given as decimal fractions. Write these decimals as per cents.

National League American League

Stand- Stand-

.29



1 10 Changing Common Fractions to Per Cents

You have already memorized the per cent equivalents of certain

common fractions. You know that | = 66§%, J = 75%, and so

on. Sometimes you meet other commonfractions and must

know how to change them to per cents.

In the Woodward School there are 786 pupils in all, 363 boys

and the rest girls. What per cent of the total enrollment is boys?

.4 6 1

The boys are f§i of the total enrollment, 7 8 6)3 6 ,3.0

|H = 363+ 786= .462 (nearest thousandth) —

_

= 46.2% (nearest tenth of a per cent)4 7 16

14407 86

654

In changing a common fraction to a per cent, unless it comes

out even like 17%, or with an easy fraction like 13j%, it is

customery to find the result to the nearest tenth of a per cent.

This means that the division must be carried to the nearest

thousandth.

1. State a rule for changing a common fraction to a per cent.

Compare your rule to the one at the bottom of the page.

2. Change the following common fractions to per cents.

Express the results to the nearest tenth of a per cent.

f « « ft T*T f

¥ m u u a m

To change a common fraction to a per cent, first change it

to a decimal by dividing the numerator by the denominator.

Be sure to carry this division to the nearest thousandth.

Then write the decimal as a per cent.



Saving Time* Using Short Methods ill

K.illnyn Phillips' father showed her several short methods

of changing common fractions to per cents. He explained thai

it was often easy to change the denominator of a fraction to 100without changing the value of the fraction. Study hisjllustrations

below.

20 100 200 1002 °

1. Can you explain Mr. Phillips' first illustration? How did

he get the -j3^? How did he get the 35%?

2. Can you explain his second solution? How did he get the

48——2? The 48*%?100

2

3. Change the following to per cents. How many can you

work without using pencil and paper?

7

25



112 Finding What Per Cent One NumberIs of Another

The school nurse found that 8 of the children in John's

room were overweight, 17 were normal weight, and 7 were

underweight. John found what per cent of the pupils were

overweight, normal weight, and underweight. His solutions

are shown below. Study and explain.

The number overweight=^ = i = 25% of the total.

The number normal weight = ^| = .531 =53.1% of the total.

The number underweight=^ = .219 = 21.9% of the total.

1. State a rule for finding what per cent one number is of

another. Compare your rule with the one given at the bottom

of the page.

2. Find the following without using a pencil.

24 is what % of 48? What % of 2 is 7?


15 is what

%of 20? What

%of 3 is 15?

3. Solve to the nearest tenth of a per cent.



4. An overcoat marked $22.50 was sold for $18. The

reduction was what per cent of the original price?

5. In one school year there wereapproximately 919,400

students enrolled in American colleges and universities. Of

these 563,200 were men and 356,200 were women. What

per cent of the total enrollment was men? What per cent

was women? The number of men enrolled was what per cent

of the number of women? The number of women was what

per cent of the number of men? Figure all per cents to the

nearest tenthof a per cent.

To find what per cent one number is of another, write

the ratio of the one to the other as a common fraction. Re-

duce this fraction to lowest terms and change to a per cent.



Problems^ Using Per Cents L13

1. In 1911, Hay Ilarroun won the 500 mile automobile

race at Indianapolis. His average speed was 74.59 miles per

hour. Floyd Roberts established the record for this race in

1938, with an average speed of 117.2 miles per hour. How

many more miles an hour did Roberts average than Harroun?

Find to the nearest tenth of a per cent the per cent of in-

crease in speed in this race from 1911 to 1938.

2. In the first half of a recent year the air lines of the

United States scheduled 42,005 trips. Of these, 39,441 trips

were started. What per cent of the trips scheduled werestarted? Figure to the nearest tenth of one per cent.

3. Of the trips started, 36,823 were completed. What per

cent of the trips scheduled were completed? What per cent

of the trips started were completed? Give results to nearest

tenth of a per cent.

4. A farmer wishes to have 25 bushels of seed corn in May.

How many bushels should he save in October, allowing 14.3%

for shrinkage during the winter?

5. Mr. Andrews, a real estate agent, sold a house and lot

for Mr. Johnson for $8000. He charged a commission of

5% of the selling price. How much did he get for making

the sale? How much did he turn over to Mr. Andrews?

6. Tell what you understand by this statement: Mr.

Johnson's expenses last year were equal to 125% of his in-

come. What were his expenses if his income was $1200?

7. Last year the Armstrong family saved $250. What per

cent of their total income of $2000 did they save?

8. Find the sale price of each of the following:

Name of Article Former Price % of Discount

Davenport $150 33j%

Chair 40 25%

Table 25 20%

Bed 30 25%

Lamp 12 16f%



114 Safety First. Estimating Results

In working with per cents, a common. error is getting the

decimal point in the wrong place. The best way to detect

such errors is to form the habit of making an estimate of the

result before working the problem out more accurately.

1. Read and explain each of these examples.

a. 17% of 79 = ?

Estimate: J of 79 = 13 (approximately)

Solution: .17x79 = 13.43

b. 29 is what per cent of 76?

Estimate : ^ is about f£ or §. f = 37j%Solution: ff

= 29 +76 = .382 =38.2%

c. 68.2% of a number is $17.13. What is the number:

Estimate: N=l7+£> =^- =24f-

Solution: N = $17.13 -.682 = $25.12

2. Estimate the result in each of the following; then work

it out more accurately. See who can find the easiest method of

estimating.

a. 32.7% of $8.70 = ? e. 39 is what % of 18?

b. 19 is what % of 28? f. 86.7% of $2.50 = ?

c. 7 is what % of 98? g. 532 is what % of 786?

d. 215% of N = $17.21 h. 35% of N = $7.25

N = ? N = ?

3. A traveling man received a salary of $2500 a year plus a

commission of 3% on all sales above $50,000 a year. Last

year he sold $78,000 worth of goods. What was his total in-

come? Estimate the result before working this problem.

4. There are 2500 pupils enrolled in the Woodward High

School. One day .8% of the pupils were absent. How manypupils were absent?

Check the location of the decimal point by first estimating

the result.



Problems. Estimating Results LIS

In each of the following problems, first estimate the result;

then solve the problem.

1 A silver dollar contains 14.25 grains of copper and 398.25

grains of silver. What per cent is pure silver? What per cent

is copper? Check your result.

2. It is estimated that the weight of a steer shrinks about

46% of its live weight, when dressed. What will be the dressed

weight of a steer weighing 1350 pounds when alive?

3. The population of St. Paul in 1940 was 288,000. In

1930, it was 272,000. Find the per cent of increase to the

nearest tenth of a per cent.

4. The population of Minneapolis in 1940 was 490,000.

In 1930, it was 464,000. Find the per cent of increase to the

nearest tenth of a per cent. Did St. Paul or Minneapolis

grow at the faster rate? (See problem 3.)

5. The weights of various substances are often given by

comparing them with the weight of water. The average

weights of several kinds of well-seasoned woods are given below

as per cents of the weight of water. Find the weight of a cubic

foot of each kind of wood to the nearest tenth of a pound.

A cubic foot of water weighs 62.5 pounds.

Redwood 41.6% American Elm 52.8% Walnut 59.2%

White Pine 46.4% Sugar Maple 67.2% White Oak 76.8%

6. In a city containing 7853 qualified voters, 5821 voted in

the presidential election of 1940. What per cent of the quali-

fied voters actually voted? Find the result to the nearest

tenth of a per cent.

7. Meadville voted on building a hospital. Only 4027 of

the 6152 qualified voters voted. Of these, 3297 voted for the

hospital. The number voting for the hospital was what per

cent of the number voting? Of the total number of qualified

voters? Find results to the nearest tenth of a per cent.



110 Saving Time. Round Numbers

Harold read in his geography that the circumference of the

earth at the equator is 24,902 miles. The book stated that

for most purposesit is sufficiently accurate to call the cir-

cumference 25,000 miles. What is gained by this? Is the

circumference nearer 25,000 or 24,000 miles? Why?

In working with large numbers it is often possible to save

time and work by using only the first few figures and filling

in the other places with zeros. The resulting numbers are often

called round numbers. Harold found other illustrations of

round numbers in his geography.

The diameter of the earth at the equator is 7,927 miles, or

in round numbers, 8,000 miles.

The distance to the sun from the earth is 92,897,416 miles,

or in round numbers, 93,000,000 miles.

1. Round each number below to two figures.

34 281 141,295 465,149 75,893,471

2,756 58,548 1,572,093 8,430,286

2. Round each number in problem 1 to three figures.

3. Harold tried to round 39,782 to two figures. He called

the result 39,000. His teacher said it should be 40,000. Why?

Is 39,782 nearer 39,000 or 40,000?

4. Round each of the numbers below to three figures.

189,628 75,976 839,581

5. Harold wanted to round the decimal 3.1416 to three

figures. He wrote 3.1400 and then changed it to 3.14. Is

this correct? Round 3.1416 to two figures. To four.

To round a number, save as many figures at the left as

you want to use and put zeros in place of the other figures.

If the first figure dropped is 5 or more, increase the last

figure saved by 1.



7 8,3 9 24.8



118 Saving Time. Round Numbers in Division

Louise worked out a rule to save work in division. She divided

24 by 3.1416. Then she rounded 3.1416 to three figures and divided

again. Her work is shown below.Are the two results the same

to two figures?

First Solution Second Solution

7.6 7.6

3.1 4 1 6A)2 4.0 A 3.1 4A)2 4.0 A

219912 2 198

200880 2020188496 1884

12384 136

Louise divided 192,395 by 23,421. She rounded each number

to three figures and divided again. Her work is shown below.

Are the two results the same to two figures?


8.2 8.2

2342 1)19 2 39 5.0 2 3 4)19 2 0.0

187368 1872

50270 480

46842 468

3428 12

In the second solution, Louise first wrote V^Vo =Wr-She divided both the numerator and denominator of the first

fraction by what number? Does dividing the numerator and

denominator of a fraction by the same number change the result?

1. Round each number below to three figures. To four.

187,391-52 154,739^78,569

2. Find the results in problem 1 approximate to two figures.

To three figures.

In dividing large numbers, round them to one more figure

than you wish in the result.



Population Problems ii«.<

In each of the following problems first estimate the result;

then work it out to the desired number of figures. Save time

and work by using round numbers.

1. The population of the United States from 1890 to 1940

is given below. Find the per cent of increase (nearest tenth of

a per cent) for each ten-year period. How many figures do you

want in your results? How many figures must you use?

During which period did the population increase most rapidly?

Year Population Year Population

1890 62,947,714 1920 105,710,620

1900 75,994,575 1930 122,775,046

1910 91,972,266 1940 131,409,581

2. If the population of the United States continues to in-

crease at the same rate as it did from 1930 to 1940 (prob. 1),

find the probable population in 1950 to the nearest million.

3. The area of the British Islesis

94,284 square miles.This is .71% of the total area of the British Empire. Find

the area of the British Empire correct to two figures. Be

sure you get the decimal point in the right place.

4. The population of the British Isles is 46,196,945. This

is 9.5% of the population of the British Empire. Find the

population of the British Empire to two figures. Be sure to

get the decimal point in the right place.

5. The number of illiterates (persons unable to read and

write) in the United States and the total population over ten

years of age are given below for 1910, 1920, and 1930.

Year



Diagnostic Tests in Fundamental Processes i_i


Work these examples and check. Work as fasl as you can

without hurrying.

1. a.



122 Diagnostic Tests. Changing Fractions


In working with per cents you have found that it is often

necessary to change per cents to decimals or to common frac-

tions, and to change common fractions and decimals to per cents.

Further practice in making these changes will help you in working

the problems in the rest of this book, and in making a better

score on Improvement Test Number Four.

1. Write the common fraction equal to each of these per cents.

60% 37j% 40%66f%

30% 62j% 70%

75% 16f% 20% 87|% 10% 25% 80%

2. Write the per cent equal to each of these fractions.

5 7 1 3 4 3. JL

8 TO 5 8 5 4 21 3 1 1 2 i 7

8 TO 4 12 3 6 8

3. Write as whole or mixed numbers (common fractions).

250% 800% 375% 366§%175% 300%

233j%150% 600% 100% 125% 1000% 525% 116§%

4. Write the following per cents as decimals.

23% 318% 7.2% 70% 400% J% .7%

21.4% 282% 250% 76.9% 2.8% 4±% 279%

5. Write the following decimals as per cents.

.17 .8 .376 .007 .5 .6 3.18

.002 .8562 5.025 2.1 .009 .07 .0593

6. Change these common fractions to per cents.

8 5 2J7. _7_ 3. .18. 138 25 7 31 20 7 11 257

19 75 25 2 3 5 1 1 155~0 29 300 200 1221 13 9


Study your errors carefully and note the trouble spots. Then

turn to the Practice Tests (pages 807 to 80) for further practice as

suggested below.

1.—37 2.-87 3.-38 4.-35 5. — 6. — U



DiagnosticTests.ThreeProblems ofPercentage 123


1. State a rule for finding a per cent of a number.

2. Find the following without using pencil or paper.

16§% of 32 200% of 19 150% of 7 87^% of 56

3. Estimate, then find the results to the nearest cent.

2.8% of $85. 10 17% of $9.85 .9% of $9.25

31.2% of $25.50 232% of $5.15 81.7% of $12.00

4. State a rule for finding what per cent one number is of

another.

5. Find the following without using pencil or paper.



6. Estimate, then find to the nearest tenth of a per cent.

7 is what % of 299 What % of 17 is 13?

41 is what % of 35? What %of 31 is 150?

7. State a rule for finding the whole when a per cent of it is

known.

8. Find the whole number. Do not use a pencil.

16§% of the number = 9 250%o of the number = 25

66§% of the number = 48 500% of the number = 85

9. Estimate the whole number, then find to nearest cent.

7.6% of the number =$1.17 149.2% of the number = $81.25

85% of the number = $19.83 84.7% of the number = $49.50

113% of the number = $31.29 5.2% of the number = $8.20




suggested below.

2.-42 1.—42 5.-43 6.-43 8.-43 9.-44


Take Improvement Test Number Four. Record your score

on a Score Card in your notebook. Find the class average and

start a graph like the one on page 6.



124 Improvement Test Number Four

THE FUNDAMENTALS OF PERCENTAGE

Check each example by going over it a second time before

starting the next. Whenever possible, estimate the result and use

this as an additional check. Time, 12 minutes.

1.

Find 66|% of

36.

>3/0

5.

Write .876 as a

per cent.

9.

195 is

whatper cent of 73?

(Nearest tenth

of a per cent)

13.

Give the

per cent equal

tof.

17.

Write 2.6

as a per cent.

21.

Find 217.3%

of $8.50.

(Nearest cent)

2.

Write 325% as

a common

fraction or

mixed number.

6.

Find the whole

if 75% of it is

27.

10.

Find 76.4%

of $89.25.

(Nearest cent)

14.

12 is what

per cent of 8?

18.

Find the

whole if 37^%of it is 12.

22.

Write 7.389

as a per cent.

3.

Change-f-f-

to

a per cent.

(Nearest tenth

of a per cent)

7.

Write 37.9%

as a decimal.

11.

Write 387%

as a decimal.

15.

Write 900%

as a whole or

mixed number.

19.

Write |as a per cent.

23.

Write the com-

mon fraction

equal to 62j%.

4.

37 is what per

cent of 89?

(Nearest tenth

of a per cent)

8.

Write the com-

mon fraction

equal to 12j%.

12.

Find the

whole if 7.9%

of it is $5.13.

(Nearest cent)

16.

Write .005

as a per cent.

20.

Write .3%

as a decimal.

24.

24 is what

per cent of 16?

You will take this test again in a few days. If you study and

practice as suggested, you should be able to improve your score.



Chapter Test 125

Write the numbers from 1 to L8 <>n a paper. After these numbers

write the words or numbers which belong in the corresponding

blanks below.

1. 39% means 390)

2. 72 hundredths can be written in three ways, , and(2)

, and(3) (4)

3. To find any per cent of any number, write the per cent

as a common or decimal fraction, and then(5)

4. The common fraction equivalent of 37^% is

(6)

5. The per cent equivalent of -^ is

6. 75% of 24 =

6(7)

(8)

7. 6 is %of9.(9)

8. 12 is %of6.(10)

9. A's population is 450% of B's population. This means

that A's population is times B's population.(id

10. 25% of a number = 8. The number is(12)

11. 300% of a number = 36. The number is

(13)

12. The decimal equivalent of 87.2% is

(14)

13. The per cent equivalent of 1.793 is

(15)

14. Of the three numbers 37 £, $3.69, and $36.86; 39.7% of

$92.85 is nearest(16)

15. In finding (to the nearest tenth of a per cent) what per

cent 7,895,971 is of 9,321,456, it is sufficiently accurate to call

the first number and the second number(17) (18)



(.11 A IT I. R

Using Arithmetic

in theMome



1 JS Personal Accounts

Dick Cunningham received an allowance of three dollars a

month from his father. His allowance usually was gone

before the end of the month, and he never knew where it

went. One Christmas Dick's father gave him an account

book and suggested that he keep an account of all of the

money he received and spent. Below is his account for the

first month.

I

:

£ l<?45



Family Accounts 129

Many families keep a cash account. Do you think all

should? Why? Below is Mr. Cunningham's account for May.

JHrt

lJJWI

1. What was Cash on hand called in Dick's account?

How is it obtained? Check.

2. Total the Received column.

Does the account balance?

The Paid column.

3. Compare the form used in this account with the one

Dick used. Which do you like best? Make out Dick's ac-

count on a form like the one above.



Making a Personal ltu<l-< I L3I

After Dick had been keeping an account for several months

he decided that it was not enough to know where his money

went after it was gone. He decided he would probably get more for his money if he planned ahead for a year. His

father showed him how to make an estimate of his earnings

and expenses for a year. Such an estimate is called a budget.

£g&r>^Ct&C *3/riXLcrmJi-. 4jS

3/r*-o. &&&

36

24

Oo,

oo

6<J Oo

i&ittsH^ds&f*** ^

/O. /m-o. <<? 4 1.So

7

5

5/o

/o

go%

So

oo

oo

oo

oo

oo

OO

b9 oo

%

U

^j^m g Hb-^^.^w^«>-w^'s^iT ~-.„^^sj

1. How did Dick find the balance? Check to see if it is

correct. Check the addition of each column. Did Dick add

correctly? Does Dick's budget balance?

2. Why did Dick include savings with estimated expenses?

3. Give as many reasons as you can why you think it is a

good plan to make a budget.

4. The United States government, state governments, and

large business organizations all make yearly budgets. Why?

5. Make a budget of your personal income, expenses, and

savings for a year. Keep a personal account so you won't

spend more than you planned.



132 Making a Family Budget

During one year the Cunningham's income was $2500.

They spent $700 for food; $600 for shelter; $450 for clothing;

$100 for education and recreation; $300 for operating ex-

penses such as fuel, electricity, and telephone; $350 for other

expenses; and saved nothing.

At the end of the year Mr. and Mrs. Cunningham talked

over their financial problems and decided they would have

to make and follow a budget if they wanted to save any money.

Mrs. Cunningham found the table given below in a book

on household management, suggesting how a family of four

might plan their expenditures. She showed it to her husband,

and they studied it carefully.



Making a Family Hu<l^<'t 133

1. What per cent of their income did the Cunninghams

spend for each item? According to the table given on page

132, which items ought they reduce in order to save some-

thing?

2. The Cunninghams decided to try the budget suggested

in the table on page 132, for a yearly income of $2500. Their

expenditures for the first year are given below. Classify

them under the seven headings in the table. Where would

you put money spent for cleaning and pressing clothes? For

Christmas and other gifts? For Dick's and Agnes' allow-

ances? For War Bonds? The Cunninghams put cleaning

and pressing under Clothing, gifts and allowances under

Other Expenses, and bonds under Savings. Why?

Rent $360 Church $24 Fuel $80Clothes 275 Cleaning and Gas 36

Pressing 65

Meat 100

Savings bank

deposit 25

Magazines 8

Groceries 400

Telephone 24

Education 30

3. What per cent of their total income did the Cunning-

hams spend for each of the seven groups? Find each to the

nearest tenth of a per cent. Do you think they were reason-

ably successful in following their budget?

4. Mr. Cunningham said their rent was rather low because

they lived in a small town. He suggested that the saving in

shelter should be added to their savings. Did they succeed in

doing this?

5. A family budget should be based on the size of tne in-

come as well as the number in the family. As the income

increases, for which items are smaller per cents allowed in

the table on page 132? Which items increase with the income?

Can you explain each of these?

Amusements. . . 47

Life Insurance. 67

Gifts 40

Shoes 45

Furniture 50

Milk 75

Doctor, Dentist 80

War Bonds. ...375

Charity 25

Automobile

Expenses. ... 100

Allowances .... 72

Books 12

Electricity 50

Cash on hand

at end of year 35



134 Making a Family Budget

1. Agnes found

the table at the

right in a maga-

zine article about

a typical family

consisting ofmoth-

er, father, and two

sons. What per

cent of their in-

come do the Jones-

es spend for each

item in the table

on page 132? On

what items are

they spending less

than suggested?

More? Does it pay

them to own their

home?

How the Joneses Spend Their

MoneyFood $ 600

Clothing 200

Boys' allowances 50

Taxes on home 150

Gas, water, electricity, telephone 180

Health 50

Life Insurance150

Recreation 100

Contributions 10

Automobile 150

Newspapers, magazines, lectures 50

Insurance on home 30

Interest on mortgage on home .. 60

Maintenanceof home 120

Miscellaneous 100

$2000

How Banks Serve Us

One day Dick went to the bank with his father. On the

way home he asked, What do banks do for us? Wouldn't

we be just as well off if there were no banks?

Mr. Cunningham explained that banks serve the people

of a community in many important ways. They receive

money on deposit for safekeeping. They pay out the money

you have on deposit by means of checks, or written orders for

payment, and collect cash for the checks you receive from

others. This makes it possible for you to make and receive

payments without handling the actual money and gives you

a written record of each transaction.

Banks also lend money for use in business and for other

purposes, and advise their depositors on how to invest their

money and on other business questions. Mr. Cunningham

said we could not get along without banks and Dick agreed.



Doiii Business with a Bank L36

When Mr. Cunningham decided to open an account with

the bank, he was receiving a salary of $200 a month. As soon

as he received his pay for the month, he deposited it in the

bank for safekeeping and to give him an easy way of paying

his bills. The bank required Mr. Cunningham to give refer-

ences and to leave a copy

of his signature. The bank

then gave him a bank-

book and a checkbook.

DEPOSITED WITH

THE BANK OF WOOD COUNTYBOWLING GREEN. OHIO



136 Paying a Bill by Check

One of the reasons why Mr. Cunningham deposited his

money in the bank was that it gave him a convenient way

of paying his bills by check. The firstbill

hepaid was for rent.

™>/JAXfi or Wooit Cor.xn' (>o.

Ch**-4~CC . V*l*_*_ «-<-w£, ^^/feo n»u.\ns

This check is a written order for the bank to pay to Mr.

J. H. Roberts the sum of $25.00 out of the amount deposited

by Mr. Cunningham. In order that he might have a record

of the way in which he spent his money and of the balance

remaining in the bank, Mr. Cunningham filled out the stub

as shown. He tore out the check and gave it to Mr. Roberts

but kept the stub in his checkbook.

*H /L*^o^*y

Mr. Roberts then took the

check to the bank to have it

cashed. In order to receive his

money hehad to endorse the

check by writing his name across

the back of the left end, as

shown here. X^^VJXTvTvJV^S

1. Why did the bank require

Mr. Cunningham to leave a copy of his signature?

2. Why is it better to pay bills by check than by cash?

3. Why did Mr. Roberts endorse the check?

4. When will banks cash checks for strangers? Why?

5. Try to get a copy of a blank check with an attached

stub. The balance in the bank is $312.14. Write a check

for $4.65 and fill in the stub correctly.



Writing and Gashing ChecJ&s 137

1. Mr. Cunningham also

paid these bills by check: W.

O. Cox, groceries, $17.23; Hol-

man's Meat Market, $8.15; P.

A. Allen, milk, $2.40. Write a

check for each. Fill out the

stubs.

2. To pay his grocery bill,

Mr. Allen used the check he

received from Mr. Cunning-

ham. Before giving it to Mr.

Cox, he endorsed it as shown.

Why did he do this instead of

simply writing his name? Can

you find out?

3. On February 10, Mr. Cunningham drew $25 for his

own use. He wrote a check payable to Cash. Write this

check. Such checks are sometimes written to Self.

4. Who cashed the check Mr. Cunningham wrote to

Cash ? Is it necessary to endorse such checks? Why?

5. Imagine you have a balance of $50 in a bank. Write

a check in payment for a new suit. Fill out the stub.

6. It is against the law to write a check on a bank in

which you have no money, or for more than you have on

deposit. How can you avoid overdrawing your account?

7. Mr. Smith expected to be away from home for a week.

Before leaving he signed his name to a blank check and told

his wife to fill it out for any amount she might need. Is it

safe to sign one's name to a blank check? Why? Would it

have been safer if he had made the check payable to Mrs.Smith and left the amount blank?

8. Is it safe to write a check in pencil? Why?

9. Will a bank cash a check drawn on another bank? Howdoes it get the money? Appoint a member of the class to

find out and make a report.



138 Merchants' Bills and Statements

Most of the merchants with whom Mr. Cunningham dealt

mailed him a bill, or a statement of the amount he owed, on

the first of each month. On April 1, he received the statementshown. MONTHLY STATEMENT



Receipts 139

One month Mr. Cunningham paid his rent in cash. He

asked Mr. J. H. Roberts, the landlord, to give him a receipt.

The receipt is shown below.

[Ai/r//Y//f/'<M ob. Jj. Gu^n^iJi-ria'h'tzmT/

Instead of writing a separate receipt, merchants often

write or stamp PAID at the bottom of their bill, with the

date and initials of somerepresentative of the firm.

If Mr. Cunningham had paid his rent by check, he would

not have asked Mr. Roberts for a receipt. The cancelled

check, when returned to Mr. Cunningham by his bank, would

have been evidence that he had paid his rent. Do you see

why this is true? This is one reason why the bank required

Mr. Roberts to endorse the check.

1. Many people pay all bills only by check. Why?

2. Is it necessary to ask for a receipt at a drugstore if

you pay cash at the time the purchases are made? Why?

3. On October 1, 19—, Mr. M. E. Moss paid Mrs. C. A.

Owens $4.50 for garage rent for the month of October. Make

out a receipt.

4. On June 3, 19—, Mr. T. O. Mclntyre paid Marion

Wells $7.25 for dressmaking. Write a receipt.

5. On September 5, 19—, Mrs. Cox asked George Stevens

to pay a bill for her, amounting to $4.65. George wished to

give Mrs. Cox a receipt. What should he do?



140 Postal Money Orders

Mr. Cunningham paid most of his out-of-town bills by-

mailing a check. Often, when he was not well known to the

person or firm he owed, he would mail a postal money order.

The post office charges a small fee for money orders,

depending upon the amount. The fees are given on the back

of the application blank.

Fees for Postal Money Orders

From $ 0.01 to $ 2.50 6 cents

From 2.51 to 5.00

From 5.01 to 10.00

From

From

From

From

From

10.01 to

20.01 to

40.01 to

60.01 to

80.01 to

20.00

40.00

60.00

80.00

100.00

8 cents

11 cents

13 cents

15 cents

18 cents

20 cents

22 cents

1. On April 26, 1944, Mr. Cunningham sent a postal moneyorder for $4.75 to Morrison Book Company, Chicago, Illinois,

to pay for some books. He bought the money order at his

post office. How much did it cost, including fees?

2. What did Mr. Cunningham do with the order after

buying it at the post office?

3. Inwhat way

could the Morrison Book Company get

its money after the money order was received?

4. You must fill out an application blank when buying

a postal money order. Have some member of the class get a

sample blank from your post office. Show how you would fill

out the application for Mr. Cunningham's money order.

5. Find cost, including fees, of money orders for the fol-

lowing amounts: $17.25, $81.36, $25, $.72.

6. Dick Cunningham bought a model airplane outfit by

mail for $4.75. In payment he sent a postal money order.

What did the order cost? What was the total cost of the

airplane outfit, including the money order?



BanU Drafts, Express, and Telegraph Money 141

< Orders

m> i — i l— r .-..r»«r r „-*»».»-. n— -r—rw

O.COl NTY-Co«-4*-< The B.\Nk.ov.Ww()j).• . • • • •

• • • ... . • ,•

cpwuno elaJL. o^kk. fyb>*-Afy\ii4 n<> 2801

J y4 £• •

^TJl^tZf - ~ft>~<i*->w <2^£-*_- ^2 _ Doil.AltS£

Q \ i > ^^ / ~

u-Third Union TRt;sr Co ' /y An \,\ ^ >

On April 12, 1942, Mr. Cunningham wished to pay a bill

of $54 which he owed The Winslow Company of Cincinnati.

He sent them a bank draft, which he secured from his local

bank. A draft is a check written by one bank on another.

1. Study the bank draft above, and answer these questions.

a. Who signed the draft?

b. On what bank was the draft made, or to whom was

it addressed?

c. To whom was the draft made payable?

d. The bank charged yq% of the amount of the draft as

a fee. Mr. Cunningham paid the banker by check. Write

a check exactly like Mr. Cunningham's.

e. Mr. Cunningham mailed the draft to The Winslow

Company. How should the company obtain payment in

money?

f. To whom should the Fifth-Third Union Trust Co.

look for payment after it has cashed the draft?

2. Find the cost of bank drafts, including fee at yo%, for

the following amounts: $139.50, $89.17, $937, $2500.

3. Express money orders are similar to postal money

orders. Where may they be bought?

4. Money may be sent quickly by telegraph. Have some

member of vour class find out how this is done and report.



142 Buying in Larger Quantities

Mrs. Cunningham decided that her groceries were costing

too much. She had a talk with her grocer and he suggested

that she savemoney

by buying in larger quantities. He took

her around the store and gave her comparative prices on

small and large amounts. She made a list of some of the

things on which she might save.

Mrs. Cunningham's Grocery List

Small LargerArticle Amount Price Amount Price

Breakfast food. . . . 8 oz. package . . . 10^ 13 oz. package .... 13 ji

Cocoa \ lb. package . . . lb{. 1 lb. package 25j£

Soap flakes 5 oz. package . . . 10^ 1 lb. package 25^

Soap 2\ oz. bar hi 8 oz. bar 10^

Canned soup 1 can 10^ 12 cans $1.05

Canned pears 1 can 25j* 12 cans $2.65

When she returned home, Mrs. Cunningham figured the

per cent she could save by buying the larger quantity of each

of the articles she had priced. She did this first for the break-

fast food. Her solution is shown below.

a. ^ = 1^ an ounce, d. x-li- = i--f = i =20%

b.

\%£= l<t an ounce e. 20% of higher price saved.

c. Saving = \£ an ounce.

1. Why can manufacturers and merchants afford to sell

larger quantities at a cheaper rate than smaller quantities?

2. What per cent of the total price, figured at the higher

rate, could Mrs. Cunningham save by buying the larger

amount of each of the articles she priced?

3. On which article on the grocery list was there the larger

per cent of saving?

4. Mrs. Cunningham bought a case (24 cans) of canned

peaches for $8.00. What per cent of the higher price did she

save if the peaches sold for 35^ a can?



Buying in Larger Quantities

After leaving the grocery store, Mrs. Cunningham asked

Mr. Durham, the druggist, if she could save money on drugs

and toilet articles by buying larger packages. Mr. Durham

suggested several articles on which she could save money,

and Mrs. Cunningham made another list.

j

1



Buying at Sales

One January afternoon Mrs. Anderson, a neighbor, told

Mrs. Cunningham about a coat which she had just bought

at a sale for $25, and for which the store asked $40 before

Christmas. What per cent did Mrs. Anderson save on the

coat?

Mrs. Cunningham said that she did not believe in sales,

that stores first marked up the prices and then marked them

down again for the sale. Mrs. Anderson was sure that this

was not true of the coat she had bought. She had priced it

herself several weeks before, when it was marked $40.

That evening Mrs. Cunningham asked her husband about

sales. He gave her several reasons why stores find it a good

policy to hold sales.

Many kinds of goods are sold only during certain seasons

of the year. Mrs. Anderson's coat belonged to this class. It

is usually poor business to keep seasonal goods from one year

to the next, so most merchants hold sales at the end of the

season. They are glad to sell such goods at a lower price in

order that they may use the money to buy other goods that

can be sold at a profit within a shorter time.

Sales are also held to get rid of goods that are shopworn,

or that may go out of style, or that are not selling well. Some

merchants hold sales for advertising purposes. The purpose

of these sales is to gain new customers.

144



Buying al Sal<\s I r>

1 Write a list of all the reasons Mr. Cunningham gave for

holding sales. Can you give an illustration of each?

2. What other reasons can you think of for holding sales?

Ask your parents and some of your local merchants. Make

a list of as many reasons as you can find.

3. Bring to class as many sales advertisements as you

can find. Does each advertisement give the reason for hold-

ing the sale? Does the reason seem to be a good one?

4. Mrs. Cunningham read an advertisement offering

POSITIVE SAVINGS OF 25% TO 50% on all women'swearing apparel. The following items were included in the

advertisement.

$ 39.50 coats reduced to $29.75

49.50



How Gas Is Measured

The Cunninghams used gas for cooking. One month Mrs.

Cunningham thought her bill was too high. She knew that

the gas meter measured the amount of gas that they had

burned, but she did not know how to read this meter. Dick

offered to show her.

Dick explained to his mother that a gas meterrecords the

number of cubic feet of gas that passes through it.

There are three dials. The right-hand dial reads in hundreds.

When the needle on this dial has made one complete revolu-

tion, 10 hundred or 1000 cubic feet of gas has passed through

the meter. For this reason this dial is usually marked 1

thousand. The center dial reads in thousands and the left-

hand dial in ten thousands. Dickexplained that in reading

the left-hand dial and the center dial you must always take

the last figure that the needle has passed, but in reading the

right-hand dial you must take the figure that is nearest to the

needle.

1. In what unit does the right-hand dial read? Why is it

marked 1 thousand ? Answer the same questions for each

of the other dials.

2. On some meters the three dials are marked 10,000's;

1000's; and 100's. Can you explain why this is done?

3. After Dick finished his explanation, Mrs. Cunningham

read the meter. She read it 49,500 cubic feet. Did she read

it correctly?

146



Problems. Reading the (ias Meter 147

1. Read the following meters:

a b.

2. Draw meter dials showing the following readings:

13,600 cubic feet 6,900 cubic feet

47,000 cubic feet 20,800 cubic feet

3. A gas meter is always read to the nearest hundred cubic

feet. Do you see why, in reading the right-hand dial, you

take the number nearest the needle?

4. Why do you read the smaller number instead of the

nearer number on the other dials? If you had read meters

b and d in problem 1 to the nearer figure on each dial, what

would the readings have been? Why?

5. If there is a gas meter in your home, make a drawing

showing the position of the hands on the dials. Read it.

6. Most gas meters have a small dial marked

Two Feet, or Five Feet. This is used to test

the meter and to detect leaks. What does it mean

if this hand moves when all the gas outlets in

the house are turned off?



THE OHIO FUEL GAS COMPANYSEE REVERSE SIDE FOR OFFICE ADDRESS

OFFICt HOURS: (40 A. M. TO 1)00 P. M.

tAIURDAY TO I* NOON

METW RCADINO OATK

Jan. 18, 19U

THE OHIO FUEL GAS COMPANY

LAST DAY TO PAY NET AMOUNT

GAS USEDHUNOPLUSCU. FT.

396 37 2.98 2.84

Feb. 8



Problems. Figuring Gas Hills i ft

1. Read the following meter and find how many cubic feet

of gas were consumed during May.

Reading May 1 Reading June 1

2. Find the amount of the bill in problem 1, using the rates

given on page 148.

3. One of the Cunningham's neighbors had a gas furnace.

One winter month they burned 26,400 cu. ft. Find the net

and gross amount of their bill, using the rates on page 148.

4. Mrs. Cunningham burned natural gas which is obtained

from wells. In many parts of the country there is no natural

gas, so that all the gas used must be manufactured. The rates

for manufactured gas are usually considerably higher than

for natural gas. Find the amount of the bill in problem 1,

using the rates given below.

Service charge per month $ .50

First 1,000 cu. ft. per month 1.20 per thousand

Next 4,000 cu. ft. per month 1.15 per thousand


Over 20,000 cu. ft. per month 85 per thousand

Discount for prompt payment. 5^ per 1000 cubic feet

5. If there is a gas company in your community, secure a

copy of their rates and figure the amount of the bill in problem

1 according to these rates.

6. If possible, bring to class copies of gas bills and verify

the amounts.



150 How Electricity Is Measured

KILOWATT HOURS

The Cunninghams used electricity in their home. Electric

current is measuredin kilowatt hours. A kilowatt hour (or 1000

watts for one hour) is the amount of electrical energy that is

needed to run ten 100-watt lamps for one hour, or one 100-

watt lamp for ten hours. Kilowatt hour means 1000 watts for

one hour.

Dick also showed his mother how to read an electric meter.

It is read just like a gas meter, except it has four dials instead

of three. The right-hand dial reads in kilowatt hours, thenext

in 10 kilowatt hours, the next in 100 kilowatt hours, and the

left-hand dial in 1000 kilowatt hours. On the right-hand dial

the figure nearest the needle is read. On the others the last

figure the needle passed is read. Why?

1. The dials of the Cunningham's meter are shown above.

Mrs. Cunningham read them 857 kilowatt hours. Did she

read the meter correctly?

2. Read the following meter:

KILOWATT HOURS

3. If there is an electric meter in your home, make a draw-

ing showing the position of the hands. Read the meter.



ill ron rut mo H M K •'•

t or nmvi.iNo green. 01110 pn.

;tric department

omn noma

»30A. MTO500PM.

L. D. CUNNINGHAM

310 N. MAIN BT

BOWLING GREKN, OHIO

o, ... »i.ocK~n

887 £0 3.19

fu'lV' * h. i' 01° r'otneVMAKE CHECKS PATABLE TO

bffi «SIlfUulMb11ptJ; OTi 0F BOWUNC GREEN. OHIO

brim; THIS BILL A.VD ATTACHED STVB WHEN MAKLNO SETTLEMENT AT OCK OfTlCE

III. .1 nil

July 1

July 10

.A<T DAT TO TAT HET AMOrWT

3-04

BALMKMVVm

rieu* EnctoM Slab Win BonllUim

Figuring the Cost of Electricity

When she received the electric light bill for June, Mrs.

Cunningham asked Dick to verify it and make sure it was

correct. The bill showed that the meter read 887 kilowatt

hours on June 1 and 931 kilowatt hours on July 1.

The amount of the bill was $3.04. Dick obtained the follow-

ing schedule of rates from the company.

7^ a kilowatt hour for the first 40 kilowatt hours per month

6 (if a kilowatt hour for the next 40 kilowatt hours per month

3jt a kilowatt hour for all additional per month

Minimum charge $1.00. Penalty of 5% of net amount if

not paid by date due.

1. Dick figured the bill as follows:

First 40 kilowatt hours $2.80

Next 4 kilowatt hours .24

Net Bill $3.04

Penalty .15

Gross Bill $3.19

Check Dick's figures. Explain each step. Was the bill

correct?

2. Figure the bill in problem 1 on the basis of rates charged

by your local electric company.

3. Secure a copy of an old electric light bill and take it to

class. Compare the bill with the one at the top of this page.

Use the solution in problem 1 and the rates in your city to

check the correctness of the bill.

151



152 The Cost of Electric Lights

Dick Cunningham left two 40-watt lamps burning all

night in the cellar. When his mother discovered this the next

morning, she told him that if he had to pay for the current

out of his allowance, he would not be so forgetful. Dick

promised to be more careful. He wondered how much the

light cost and figured it out as follows:

80 watts for 12 hours =960 watts for one hour

960 watts for one hour = .96 kilowatt hours (1 kilowatt hour =

1000 watts for one hour)

.96X7^=7^

(to the nearest cent)

Dick showed his mother his work and offered to pay her

7i, but she would not accept the money.

1. Mr. Cunningham liked to light the whole house every

night. Dick made a list of all the lamps and the number of

watts each used. He found the number of watts printed on

the end of each lamp. He then figured the cost of lighting the

whole house for three hours each night. Can you find out howmuch it cost for one night, at 7j£ a kilowatt? What did it cost

for a month of 30 days?

Room



Problems. Using Electricity in the Home

1. On the back of their radio, Dick found the statement

that it used 125 watts an hour. He estimated that the radio

was in use about 70 hours a month. Find the cost of operating

the radio for one month at li a kilowatt hour.

2. Dick found that their refrigerator used 300 watts an hour.

The motor ran about 6 hours a day. What was the cost of

operating it for one day at 7^ a kilowatt hour?

3. Before buying an electric refrigerator, the Cunninghams

bought 50 pounds of ice a day at 60^ a hundred pounds.

Which cost more to operate, the ice or the electric refrigera-

tor? How much more?

4. Find the cost of operating each of the following for

one hour, at 1$, a kilowatt hour.

Watts Used Watts Used

Appliance per hour Appliance per hour

Iron 600 Small fan 40

Vacuum cleaner .... 150 Waffle iron 500

Washing machine.. .

200 Dishwasher 175Electric toaster 550 Heating pad 50

Sewing machine. ... 75 Electric pump 160

5. Make a list of the electrical appliances used in your

home and the amount of current each uses. Find the cost of

operating each at the rate charged by your local company.

153



DICK CHECKS THE WATER BILL

The Cunninghams received their waterbill

every month.Dick tried to check the bill. He found that the water meter

had six dials as shown above and that it was read just like a

gas meter.

Dick read the meter 2,749 cubic feet. He did not read the

dial marked One Foot as this is used only to detect leaks.

One complete revolution of the bottom dial means that 10

cubic feet of water has passed throughthe meter, so each

space on this dial equals 1 cubic foot. In the same way, each

space on the dial marked 100 equals 10 cubic feet.

1. Did Dick read the meter correctly?

2. The bill showed that the previous reading of the meter

was 1,818 cubic feet. How many cubic feet had they used

since the previous reading?

3. The rate was 25^ for 1000 gallons. One cubic footis

equal to approximately 1\ gallons. Find the amount of the

bill.

4. In some communities water is charged for by the cubic

foot, instead of by the gallon. What would be the cost ot

2700 cubic feet at $1.50 per 1000 cubic feet?

154



Problems. The Cost of Water 1 55

1. The diagrams below show the reading of a water meter

on May 1 and on June 1. Find the cost of water for May at

the rate of 25^ per 1000 gallons.

^ou^,

2. Water rates differ in different communities. What

would be the cost of 3,400 cubic feet if the rate is 30^ for the

first

1000 gallons, 20^for

the second 1000gallons, and 15£

for each 1000 gallons after that?

3. Figure the amount of the bill in problem 1, using the

rates given in problem 2.

4. In some communities water is not measured by meters,

but the user pays a flat rate, that is, a fixed amount for each

outlet for water. Find out how water is sold in your com-munity. Figure the bill in problem 1, using your local rates.

5. Which is cheaper, 25^ per 1000 gallons, or $1.50 per

1000 cubic feet?

6. Secure several old water bills and check them.



156 Interest and Amount for One Year

Mr. Cunningham loaned $2000 to Mr. E. R. Sanders, a

farmer, who agreed to pay it back at the end of three years.

Mr. Sanders agreed to pay interest at the rate of 6% a year.

This means that, for each year's use of the money, he promised

to pay 6% of $2000, or $120. Mr. Sanders had to pay for the

use of Mr. Cunningham's money just as he would pay rent

for the use of a house.

The $2000 which Mr. Sanders borrowed is called the prin-

cipal; the $120 which he pays for the use of it is the interest;

and 6% is the rate of interest. At the end of the first year

Mr. Sanders pays Mr. Cunningham $120 interest. At the

end of the second year he again pays $120. At the end of the

third year he must pay $120 interest and $2000 principal, or

a total of $2120. The interest plus the principal is called the

amount.

1. You can sometimes shorten your work by using can-

cellation. Study the twosolutions below for finding the in-

terest for 1 year on $900 at 7%. Which do you prefer?


$900 9

.0 7 Interest =^rx^= $63

$ 6 3.0

2. Find the interest for 1 year, to the nearestcent:

$500 at 4% $930 at 7% $3000 at 7%$275 at 7% $890.75 at 3j% $3187 at 6j%

$387 at 4|% $132.25 at 6%, $5864 at 4%

3. Mrs. Cunningham inherited $3500 which she loaned to

J. C. Smith for five years at 5% interest. How much interest

was due each year? How much would the interest have been

if it had not been paid until the end of the five years?

Interest for 1 year= Principal multiplied by Rate

Amount= Principal+ Interest



Promissory Notes lf)7

£>A6*je>0_ ,

_ &<**£~_ *-p//__qjz_ GEEARS

fynW^, v4^i<^-/Ota*

ppfttrm tint/ fourttfA'protrss and twt/kss aj

__ Jiereiyaultiortyam/jVtornrt/ atIan' /u appearmany fourt qf/iccord in the United Stat™ a/ter the ahotv

l^*it odligahon tieromes dueami naive tne Issuingandservice efpmxrssand tan/ess aJudgmmiapainstJJO^J^r injhiorof

rfsuit/fihitherrih&n lo releast aiterrors and wa/ve aitthe holder hereof.[for tne amount then appearingdue, together

rightufappeai.

7h

When Mr. Sanders borrowed the $2000, he gave Mr.

Cunningham the promissory note shown above. A promissory

note is simply a written promise to pay a given amount, at a

given time and place, with or without interest.

The individual who promises to pay the money is called the

maker of the note. The one to whom the money is to be paid

is the payee. The amount borrowed is the face of the note.

1. Name the maker of the above note; the payee; the face.

2. The maker of a note may not agree to pay interest.

Such a note is called a noninterest-bearing note. Is the note

above interest bearing, or noninterest bearing?

3. Write the following notes. Use fictitious names and

write Sample across each note so that your copy cannot

by used by anyone for collection on the note.

Face Time Rate of Interest

$2500



158 The Interest Formula. Time in

Years and Months

When money is borrowed for several years, the interest is

usually paid at the end of each year. Sometimes it is not paid

until the principal is due. If Mr. Cunningham had agreed

to wait for his interest until the end of 3 years (page 156),

how much interest would Mr. Sanders have had to pay? The

interest for 3 years would be how many times as much as the

interest for one year?

Dick wrote the rule below:

Interest = principal X rate X time. Mr. Cunningham sug-

gested that Dick's rule could be stated in shorter form by

using i for interest, p for principal, r for rate, and t for time.

Dick wrote i=pXrXt.

An abbreviated rule, written as an equation, using letters

and mathematical signs instead of words, is called a formula.

In writing formulas, the multiplication sign is usually omitted

between letters. Thus the interest formula is usually written

i=prt. Can you write a formula for the amount? Can the

sign for addition be omitted? Why not?

1. Dick found the interest Mr. Sanders would have to pay

20 at the end of three years as shown

* =,2000X^X3 = $360 at the left. Check and explain.

2. If Mr. Sanders wishes to repay Mr. Cunningham at

10 the end of 9 months, how much in-

JX3 terest will he owe? Dick's solution

i =^06CXj^X4^=$90 is shown at the left. Check and ex-

* plain. Why did Dick change 9

months to -^ year?

3. In using the interest formula given below, the time

must be expressed as years or part of a year. Why?

Interest formulas:

i=prt a=p+i



The Interest Formula. Time in L59

Years and Months

1. Find the interest and amount, to the nearest cent:

Principal Rate Time Principal RateTime

a. $800 5% 2 years d. $2750 4% 9 months

b. $875 6% 3 years e. $480 5% 1 year 3 months

c. $72.50 7% 4 months f. $257 7% 2 years 6 months

Time in Days

One day Dick asked his father, How do banks earn money?

What do they do with the money that is deposited with them

for safekeeping? Can you answer Dick's questions?

Banks lend money and charge interest. On small loans the

time is usually 30, 60, or 90 days. The banks count 360 days

to the year and express the number of days as a fraction of

a year. 30 days = ^°o year^ year.

Express 60 days as a fractional part of a year; also 90 days.

Count 360 days to the year. Banks count 360 days to the

year, instead of 365, because it is more convenient in figuring

interest for 30, 60, or 90 days. Can you show why this is true?

2. Find the interest on $300 for 90 days at 5%. Check

3 and explain each step in

i=3<&X^xMr= 1*L = §3 -75 the solution at the left '

4

3. Find the interest, to the nearest cent.

Principal Rate Time$125 7% 60 days

$850 5% 90 days

$1200 4% 30 days

$925 7% 120 days

4. Banks usually collect interest in advance. Mr. James

gave his bank his note for $500 payable in 90 days. The bank

deducted 6% from the amount they gave him. How muchwas the interest? How much did Mr. James receive? How

much did he have to pay the bank at the end of 90 days?

5. How much would the borrower receive in each part of

problem 3 if the interest is deducted in advance? How much

would he pay back?



Opening a Savings Account

At Christmas Agnes received $10 from her grandmother.

Agnes also had money in a toy bank. Her father suggested

that she deposit all of it in a savings bank so that it would

earn more money for her by drawing interest. He took her

to the bank and helped her open a savings account.

The banker told Agnes they would pay interest at 2% a

year, and that this interest would be added to her account

every six months, on January 1 and July 1, and would draw

interest from that time.

1. Agnes asked her father if he had a savings account at the

bank. He replied that he had two accounts, a checking ac-

count and a savings account. Explain the difference.

2. Mr. Cunningham received no interest on his checking

account. Can you explain why banks pay interest on savings

accounts but not on checking accounts?

3. Some banks charge small fees for checking accounts.

Why do they do this? Appoint a classmate to find out what

fees your local banks charge on checking accounts.

4. Mr. Cunningham told Agnes that the rate paid on

savings varies from 3% to \\%. He also said that some

banks compute the interest annually, some semiannually, and

others quarterly. Explain.

5. How often did Agnes's bank compute the interest?

160



Finding Interest on Savings Accounts L61

1. Agnes opened her savings account on December 31.

She deposited $75. If Agnes made no more deposits and the

rate of interest was 2% per annum, how much interest should

the bank add to her account on July 1?

2. Savings banks pay no interest on parts of a dollar. On

what principal did Agnes draw interest for the second six

months if she made no more deposits? What was the interest

for this period?

3. How much did Agnes have in the bank at the end of

three years if she made no more deposits?

4. A few years ago Mr. Dunlop deposited $500 in a sav-

ings bank paying 3% interest, computed annually. How much

did this amount to at the end of three years?

5. Mr. Brown deposited $500 in a savings bank paying

3% interest, computed semiannually. How much did this

amount to at the end of three years?

6. Mr. Kennedy deposited $500 in a savings bank paying

3% interest, computed quarterly. How much did this amount

to at the end of three years?

7. Did Mr. Dunlop or Mr. Brown have more in the bank

at the end of three years? Why? Did Mr. Brown or Mr.

Kennedy have more? Why?

8. What rate of interest do your local banks pay? Do

they compute the interest annually, semiannually, or quar-

terly?

9. How much would Agnes have had at the end of three

years if she had deposited her $75 (problem 1) in one of

your local banks?

10. June Norton's grandfather gives her $50 every Christ-

mas. June deposits this in a savings bank paying 2% interest,

computed semiannually. How much will June have in the

bank at the end of 3 years? At the end of 5 years?

11. Do you have a savings account? How many reasons

can you give in favor of having such an account?



162 Postal Savings Accounts

Dick Cunningham had also saved some money. He decided

to put it to work by depositing it in the Postal Savings Sys-

tem. Dick went to the post office for information. The post-

master stated that the rules are as follows:

a. Any person, ten years old or over, may open an account.

b. He may make deposits at any time by buying postal

savings certificates. These are issued for amounts of

$1, $2, $5, $10, $20, $50, $100, $200, and $500.

c. Interest at the rate of 2% a year is computed starting

on the first day of the month following the month in

which the deposit is made. Unless called for earlier, this

interest is paid when the money is withdrawn.

d. Interest is not paid on any fraction of a year.

e. Money may be withdrawn at any time.

f. The deposits of any one person may not exceed $2500.

1. The purchase of postal savings certificates is one of

the safest ways of putting money to work. Why?

2. Why are the deposits of any one person limited to $2500?

See if you can find out if you do not know.

3. Dick bought a $50 certificate. He drew out his money

at the end of three years. How much did he receive?

4. How much would Dick have had at the end of three

years if he had deposited the $50 in the savings bank, with

interest at 2%, computed semiannually? How much more

is this than the amount he received for his postal savings cer-

tificate?

5. How much would Dick have received for his postal sav-

ings certificate if he had not withdrawn his money until the

end of five years? How much would he have received if he

had deposited his money in the savings bank for the same

time?

6. Dick and Agnes talked over the merits of savings ac-

counts and postal savings certificates. Give the chief advan-

tages and disadvantages of each.



MRS. CUNNINGHAM BUYS A WAR BOND

United States Savings Bonds were first placed on sale by

the government on May 1, 1935. By April 30, 1941, more

than two and a half million Americans had invested their

savings in almost four million dollars worth of these bonds.

This is the largest amount ever invested in a single security.

During World War II, the name of these bonds was changed

to United States War Savings Bonds. Mrs. Cunningham

decided to buy some of these war bonds. She gave two reasons.

My money will be safe because the credit of the United

States is pledged for payment of both principal and interest,

and my money will be working in the national defense pro-

gram to protect thefreedom and safety of the United States.

Do you think her reasons were good?

War Bonds, Series E, are dated the first of the month in

which purchased, and are repayable by the government at the

end of 10 years. They may be purchased in five denomina-

tions, as shown below.

Issue Price Maturity Value

$ 18.75 will increase in 10 years to $ 25.00

37.50 will increase in 10 years to 50.00




1. What is meant by maturity value?

2.

Mrs.Cunningham first bought a $100 bond (maturity

value). How much did she pay for it? How much will the

government pay her for the bond at the end of ten years?

3. Dick bought a $25.00 bond. How much did he pay?

How much will he receive at the end of 10 years?

4. Agnes did not have enough money to buy a bond, so

she bought War Savings Stamps. Find out about these.



; -',4i

THE CUNNINGHAMS BUY A HOME

The Cunninghams bought a house for $7500. Mr. Burch,

the owner, agreed to take $3000 in cash and the balance in

monthly installments of $60 a month. In this way the Cun-

ninghams could pay for the house like paying rent.

Mr. Cunningham gave Mr. Burch a mortgage for the balanceof $4500 and agreed to pay 6% interest on the unpaid balance.

This mortgage gave Mr. Burch the right to sell the house,

if Mr. Cunningham failed to pay, and to pay himself out of

the money received from the sale.

1. Secure a blank mortgage form and bring it to school.

Study it carefully and be ready to explain.

2. Part of the $60 Mr. Cunningham pays each month goes

to pay the interest; the rest is used to reduce the amount

owed. Find the interest on $4500 at 6% for one month.

3. How much of the $60 paid the first month was needed

for interest? How much was deducted from the amount

owed? How much did Mr. Cunningham still owe?

164



Problems. Buying a Home L66

4. On what principal did Mr. Cunningham have to pay

interest for the second month? (Use your results for problem

3 on page 164.) How much of the $60 payment went to pay

interest? How much to reduce the principal? How much did

he still owe?

5. Make a table like the one below for four months. Com-

pute the interest for each month to the nearest cent. Fill

each blank as you proceed. What is the amount of the debt

after the payment at the end of the third month?

Total Reduction on

Month Principal Payment Interest Principal

1 $4500 $60

2 $60

3 $60

4 $60

6. Before deciding on the plan for making payments, Mr.

Cunningham had gone to his bank. They agreed to loan

him $4500, at 6% interest, if he would pay $40 on the principal

each month and pay interest monthly on the remaining debt.

The table below shows how the bankers' plan works. Copy

the table and fill each blank.

Month



\0~~

£>

ft

DICK LENDS MONEY TO HIS CLUB

Dick's club needed $10 to help buy baseball uniforms.

They told Dick that if he would lend them $10 they would

pay him $11 at the end of a year. Dick wanted to know what

rate of interest he would be getting. Do you know?

Dick's father helped him. He explained that, since Dick

would get $1 interest on $10 for one year, his interest would

be <£$, or 10% of the principal. Dick wrote the rule at the

bottom of the page. Is it correct?

1. Find the rate of interest on each of the following.

Principal Interest for 1 Year

a. $ 750 $ 45

b. 800 56

c. 1000 90

d.420

23.10

e. 1200 144

f. 3500 297.50

2. Dick's club repaid his loan at the end of six months.

They offered to pay $11 but Dick said that wouldn't be fair.

One dollar for the use of $10 for six months would be at the

rate of $2 for the year. 2-r-10=T2o=20%. Dick accepted

$10.50. What rate of interest did he receive?

To find the rate of interest, divide the interest for one

year by the principal.

i

r= —

P

166



Problems. Finding the Kale of Interest L67

I. Find the rate of interest on each of the following:

Amount Borrowed



Buying on the Installment Plan

Mr. Cunningham needed an automobile. The car he

wanted cost $1200 and Mr. Riggs,the dealer, offered him

$400 for his old car. Mr. Cash price $1200Cunningham had only $200 Down payment andcash to pay on the car so

allowance 600Mr Riggs explained how he

d balance—^

could pay he balance in 12c { ch 36

equal monthly installments. qT«Q«

The carrying charge would be ^ otal * b3b

only 6% of the unpaid bal- T^. = $53 monthly payment

ance. Mr. Riggs figured the 12

monthly payments, as shown above. Explain each step.

Mr. Cunningham figured the rate of interest he would be

paying. He would owe the dealer $600 during the first month,

$547 during the second, and so on, the amount decreasing by

$53 each month. He totaledall

these amountsand found

he would owe the dealer the equivalent of $3702 for one month.

$36 for the use of $3702 for one month is at the rate of

$432 for the year. 432-T-3702 = .117 = 11.7%. Mr. Cunning-

ham decided he would be paying 11.7% interest.

1. Check each step in Mr. Cunningham's figuring. Do

you understand?

2. Is a carrying charge of 6% on the unpaid balance the

same as 6% interest? Why? Did Mr. Cunningham owe the

full amount of the unpaid balance for a year? Why not?

3. Mr. Cunningham decided to buy the car as he needed

it immediately in his business. Do you think he was justified?

168



IS BUYING ON INSTALLMENT EVER A GOOD PLAN?

1. The Cunninghams had a long discussion about install-

ment buying. Agnes and Dick drew up the following rules.

Do you think they are good ones? Wouldyou add others?

a. Do not buy any article on the installment plan if you

can wait until you have saved enough to pay cash.

b. Be sure you can meet the payments when due.

c. Don't pay too high a rate of interest.

d. Don't buy luxuries you could do without.

2. Agnes Cunningham became interested in buying on

payments and obtained cash and installment prices on various

articles from local merchants. She found that she could buy

a radio for $50 cash, or for $10 down and $7 a month for six

months. Find the rate of interest.

3. An electric refrigerator sold for $250 cash, or $55 down

and $35 a month for six months. Find the rate of interest.

4.

Apiano sold for $350 cash, or $75 down and $25 a

month for twelve months. Find the rate of interest.

5. A watch sold for $25 cash, or $1 down and a dollar a

week for 30 weeks. Find the rate of interest.

6. If possible, obtain cash and installment prices from

local dealers and figure the rate of interest.

169



170 The Cost of Running an Automobile

1. Mr. Cunningham paid $1250 for his automobile. Thefirst year he had it he kept an accurate account of all the

operating expenses, as shown below. Find his total operating

expenses for one year.

Gasoline $162.72 Insurance $52.25

Oil 27.30 Repairs 35.10

License 10.00 Miscellaneous. . . 17.30

2. At the end of the year, Mr. Cunningham estimated

that his car was worth only $900. He called the difference

between the cost of the car and the value at the end of the

year, depreciation. Find the amount of depreciation for the

year.

3. He figured that if he had not bought the car he could

have loaned the money and received interest on it. Find

how much interest he lost if he could have loaned the moneyat 6%. He called this the interest on his investment.

4. To find the total cost of the automobile for one year,

Mr. Cunningham added the operating expenses, depreciation,

and interest on his investment. Find the total cost.

5. The Cunninghams drove their car 11,900 miles during

the year. Find, to the nearest tenth of a cent, the total cost

per mile.

6. Operating expenses were what per cent of the total

expenses? Find to nearest tenth of a per cent.

7. Sometimes Mr. Cunningham drove his own car while

engaged on business for the company for which he worked.

The company paid him 6£ for every mile he drove on com-

pany business. Did he make or lose money? How much did

this amount to if he drove 2750 miles on company business?

8. Mr. Simmons paid $1400 for his car, drove it 40,000

miles and sold it at the end of two years for $600. His operat-

ing expenses were $900. Find the cost per mile, charging 6%interest on his investment.

9. When Mr. Simmons drove his car for business, his em-

ployer paid him 5^ a mile. How much did he gain or lose

per mile?



Finding the Cost of a Vacation

One summer the Cunninghams took an automobile trip

to Boston. Before starting they estimated the cost.

1. Dick found from the road map that it was about 800

miles to Boston. With some side trips to points of interest,

they would probably drive about 2500 miles on the trip. Find

the cost of running the automobile at 5j£ a mile.

2. Agnes figured that meals would cost them about $1.50

apiece, each day. Find the total cost of food for the four

members of the family if they were away 15 days.

3. They expected to be away 14 nights. Mr. Cunningham

said they could get rooms at a dollar a night each for himself

and Mrs. Cunningham, and fifty cents each for Agnes and

Dick. Find the total cost of sleeping accommodations.

4. Mrs. Cunningham suggested that they would probably

spend some money for newspapers, magazines, candy, and

souvenirs. The family thought $1.50 a day would be enough

for this purpose. How much would this be for the trip?

5. Find the total cost of the vacation as estimated above.

This would be how much a day for each person?

171



172 Travelers' Checks

U.S.DOLLAR TRAVELERS CHEQUE *uo,ooo,ooo*

Mr. Cunningham decided to take $350 with him on his

vacation trip. He took $50 in cash and the rest in travelers'

checks, which are sold by banks and by the American Express

Company. These are printed checks, in denominations of

$10, $20, $100, and $200.

1. The purchaser signs each check when he buys it and

signs it a second time in the presence of the person cashing it.

This serves as an identification. Explain.

2. Mr. Cunningham did not use his personal checks to

pay expenses on the trip because people will seldom accept

such a check from a stranger. Why?

3. There is seldom any difficulty in cashing travelers'

checks. Any bank will cash them and most business menwill accept them in payment for articles bought. Why?

4. Mr. Cunningham paid a fee of f% of the face value of

the checks he bought. This is how much per $100?

5. He bought 4 checks of $10 each, 8 of $20 each, 2 of

$50 each. How much did he have to pay for these checks,

including the fee?

6. Mr. Cunningham checked out of the hotel in Chicago.

The clerk at the cashier's window gave him his bill which

amounted to $8.30. Mr. Cunningham gave the clerk a $20

travelers' check. Where did Mr. Cunningham sign the

travelers' check?



Problem Teal 17:;

1. In keeping a cash account, what must you know to

find your balance? How do you find it?

2. The Osburn family did not want their expenditures

for food to exceed 35% of their income. What would you

have to know to find out whether they kept within this limit

last year? How would you find out? Give more than one

way.

3. What must you know to find the balance in your

checking account at the bank at the end of the month? Howwould you proceed to find the balance?

4. Mr. Anderson bought a suit at a sale. What would you

have to know to find the per cent of discount he received?

How would you then find it?

5. On March 31, the Smith's gas meter read 23,200 cubic

feet. What else would you need to know to find the amount

of their bill for March? How would you find it?

6. The Joneses pay 50^ for each 1,000 gallons of water

they use. At the end of August their meter read 37,823 cubic

feet. What would you need to know, and how would you find

the amount of their bill for August?

7. Mr. Owens got a 90-day loan. What would you need

to know, and how would you find the interest?

8. Mr. Collins bought a new car. At the end of the yearhe said that the depreciation on his car was $350. Whatwould you need to know to find the per cent of depreciation?

9. Mrs. Madison deposited $300 in a savings bank. Whatwould you need to know, and how would you find to howmuch this amounts?

10. Mr. Murray borrowed some money and paid $32

interest at the end of a year. What would you need to know,and how would you find the rate of interest?

11. Write a formula for finding the interest when the prin-

cipal, rate, and time are known.

12. Write a formula for finding the rate of interest when

the principal and interest for one year are known.



174 Problem Test

1. The school enrollment of a city is about 20% of the total

population. What is the probable population of a city whose

school enrollment is 13,752? Estimate the result before working

this problem.

2. Mr. Higgins owns a prize winning cow which produced

18,523 pounds of milk last year, averaging 4.1% butterfat.

How many pounds of butterfat did this cow produce? Estimate,

then find result to nearest pound.

3. Mrs. Johnson bought an electric stove. At li a kilowatt

hour, how much did it cost to cook a meal on this stove if it took

2 hours and the stove used 600 watts per hour?

4. Electric motors use 746 watts per hour for each horse-

power. At 1£ per kilowatt hour, how much would it cost to run

a 2-horsepower motor for 8 hours?

5. A furniture dealer marks all articles with the installment

price and allows from 10% to 20% discount from this price if

cash is paid. Find the cash price and the amount saved by paying

cash for each of the following:

Article Installment Price Discount

Table $ 15 10%

Chair 25 10%

Davenport 125 15%Bedroom set 200 20%

6. Mrs. Owens attended a sale of canned goods at a grocery

store and made the following purchases:

Regular Price Mrs. Owens Bought

Peaches 35^ a can 4 cans for $1

Pineapples 25^ a can 3 cans for 50^

Plums 20^ a can 2 cans for 35^

Pears 30^ a can 3 cans for 55^

What per cent of the regular price did she save on each? What

per cent of the regular price on her entire purchase?



Diagnostic Tests in Fundamental Processes


Work and check. Work as fasl as you can without hurrying.

Multiply



176 Diagnostic Tests in Percentage


Work each example and check. Work as fast as you can

without hurrying.

1.

Write the

common

fraction

equal to

87j%-

e.

Write .219

as a per

cent.

b.

16 is what

% of 87?

(Nearest tenth

of a per cent)

f.

Write 750% as

a whole or

mixed number.

c.

Write

329% as a

decimal.

g-

Find 75%

of 96.

2. a.

Write .3

as a

per cent.

e.

Write

166§% as

a mixed

number.

c.

Write the

% equalto

d.

Find the whole

if 66§% of it

equals 86.

h.Change §-J to

a %. (Nearest

tenth of a per

cent)

d.

What % of

32 is 64?710-

Write

71.3%

as a

decimal.

h.

Write 3^as a per cent.

b.

Find 86.2% of

S8.27?

(Nearest cent)

f.

Find the whole,

if 17.4% of it

is $2.14.

(Nearest cent)



turn to the Practice Tests (pages 307 to SIfi) for further practice as

suqqested below.

le _ 40 2a— 40 2e — 38la— 37

lb— 43

lc — 39

Id— 44

If — 38

lg-42

Ih— 41

40

2b— 42

2c — 37

2d— 43

2f — 44

2g-39

2h— 41


Take Improvement Test Number Four. Record your score on

your Score Card. Did you improve? Find the class average and

bring your graph up to date.



( ihapter Test 177

Write the numbers from I to L3 on a paper. Alter these numberswrite the words, or numbers, which belong in the correspondingblanks below.

L. In keeping an account, money is entered on the

left-hand side, and money - on the right-hand side.

(2)

2. The Jones family have a yearly income of $3000. Last

year they spent $600 for clothing. This was % of their(3)

income.

3. In everyday language, currency means money.(4)

4. When you cash a check, you must write your name across

the back. This is called the check.(5)

5. , , and are convenient and safe methods of(6) (7) (8)

paying bills at a distance.

6. The gas meter shownbelow reads cubic feet.(9)

7. The electric meter below reads kilowatt hours.(10)

8. The general rule for figuring interest can be abbreviated

into a formula as follows:(id

9. provide a convenient and safe method of carrying(12)

money when traveling.

10. When you borrow money to buy a home, you are usually

required to give a(13)



<: ii a p r 1; R

6^^V ?*

Helps in Problem Solving

One summer the Cunninghams visited Chicago. In plan-

ning their trip, Mr. Cunningham laughingly remarked that

they needed to know three things: (1) their destination, (2)

their starting point, and (3) the connections between their

starting point and their destination.

Agnes, who liked arithmetic, suggested that planning the

solution of a problem in arithmetic is just like planning a trip.

You must know the destination, or what you want to find.

You must know the starting point and other necessary facts.

Finally, you must plan the solution. To make the plan, you

must investigate the relations between the things to be found

and the things known. These relations correspond to the

roads in planning a trip. Just as in the case of the trip, more

than one solution may be possible.

179



180 The Steps in Solving a Problem

It is 245 miles from Bowling Green to Chicago. On a long

trip Mr. Cunningham usually averages about 35 miles an

hour. How many hours should the journey take? At what

time must he start so as to reach Chicago by noon?

Solution

To Find: (1) Number of hours it will take for the trip.

(2) Time he must start.

Given : Distance = 245 miles

Speed = 35 miles an hour

Time of reaching Chicago = 12 o'clock noon

Plan: (1) To find the number of hours to reach Chicago,

divide 245 miles (total distance) by 35 miles

(distance traveled in one hour).

(2) To find the starting time, subtract the number

of hours spent on the journey from 12 (the

time he wants to reach Chicago).

Computations: _7 12 Check: 5 35

35)245 ^7 _7 __]_

245 5 12 245

Mr. Cunningham must start by 5 o'clock in the morning.

1. Explain the first step in The Plan. If you know the

total distance and the average distance in one hour, how do

you find the number of hours the trip will take?

2. Can you explain the check? What is the purpose of

adding 7 to 5? Of multiplying 35 by 7?

Follow these steps in solving a problem:

1. To Find. What does the problem call for?

2. Given. What facts are known?

3. Plan. How are the known facts related to the un-

known? Make a plan. This plan should be kept in your

head, but sometimes it helps to write it out.

4. Computation. Carry out the computations as planned.

5. Check. Are your results correct?



FINDING THE COST OF THE CHICAGO TRIP

Read each problem carefully and write down the fact or

facts to be found and the things given, or known. Refer to

page 180 for information, if you need it. Plan the solution

before you work the problem. It may help you to write out

the steps. Try to find some way of checking your results in

each problem.

1. Mr. Cunningham's car averages 14 miles on a gallon

of gasoline. How much would the gasoline cost for the round

trip to Chicago, at 19^ a gallon?

2. Mr. Cunningham said he would need to change the oil

in the car before starting. If he buys 5 quarts at 35^ a quart,

how much will it cost?

3. The Cunninghams planned to leave home on Thursday

morning and to return Sunday afternoon. They can rent a

small apartment in Chicago for $1 a night for each person and

50£ a night for the garage. They allowed $15 for meals. Find

their expense for the apartment, garage, and meals.

4. Agnes asked how much they could have to spend while

on the trip. They decided to allow Agnes and Dick each 50^

a day and Mr. and Mrs. Cunningham $2 apiece each day.

Find the total amount allowed for spending money.

5. Dick found the total estimated cost of the trip and the

average cost for each person. Can you find how much these

were?

181



182 Help in Planning the Solution

Agnes Cunningham often had trouble in planning the solu-

tions of arithmetic problems. One day her teacher suggested

thatshe draw a picture, or diagram, to illustrate the rela-

tions between the facts she knew and the facts she wanted

to find. Miss White showed Agnes how to draw the picture

for the problem given below.

1. Mr. Mason owns a furniture store. To determine the

selling price of an article, he increases, or marks up, the

cost price by one third of itself. What did he pay for a desk

which he sold at $36?

SELLING PRICE =*'36

>- v- ^COST PRICE =P MARKUP--?

2. Study the figure. What does the whole rectangle repre-

sent? Since you know the selling price and want to find the

cost price, you must discover the relation between these two.

The cost price is what part of the selling price? How can you

find the cost price? Complete the solution.

3. Dick showed Agnes another way of solving the problem.

He asked her, The markup is what part of the selling price?

How can you find the markup? Knowing the markup and

the selling price, how can you find the cost price? Can you

answer these questions? Complete the solution.

4. A year ago MissOwen's salary was cut 20%. By what

per cent would her present salary have to be increased to

restore her to her original salary? Draw a figure.

5. During the Christmas rush a store increased its sales

force by one third. What fractional part of its force would

it have to drop to reduce it to its original size?



Problems^ I sin^ Diagrams 183

i. Mrs. Jackson boughl a coat advertised at One Fifth

Off Regular Price. She paid $56 for the coat. What was

the regular price? Agnes chew the diagram given below.

What part of the regular price equals the sales price? Howcan you find the regular price? Complete the solution.

_ a,

2. Mr. Madison's expenses average 20% of his sales. At

what amount must he sell a table that cost him $14, in order

to make a profit of 10% of the selling price? What per cent

of the selling price goes to gross profit, or expenses and profit

combined? What per cent of the selling price must equal

the cost? How can you find the selling price? Complete the

solution.

r



184 Problems. Using Diagrams

1. Three boys, Harry, Joe, and Ned, ran a refreshment

stand at the county fair and agreed to divide the profits in

proportion to the time they worked. Harryworked 12 hours,

Joe worked 8 hours, and Ned worked 10 hours. The tota

profits amounted to $30.45. Find each boy's share.

2. Three men, A, B, and C, bought a business together

and agreed to share the profits in proportion to the amount

each invested. A invested $2000, B invested $1500, and C

invested $3000. At the end of the first year they divided

profits of $8500. Draw a diagram. Find the share of each.

3. A and B invested equal amounts in a store and agreed

that A was to manage the store and to receive twice as large

a share of the profits as B. At the end of the first year the

profits were $2550. Find the share of each.

4. In his will Mr. Doane directed that his money should

be divided among his wife, his daughter, and his son. The

wife was to receive three times as much as the daughter, and

the son twice as much as the daughter. Find the share of each

if the total amount to be divided was $12,000.

5. Agnes and Dick Cunningham ran a lemonade stand in

their front yard one summer during vacation. Agnes worked

20 hours at the stand and Dick worked 15 hours. Their profits

amounted to $7.00. How much should each receive? Can you

work this problem without drawing a diagram?



Problems. Using Diagrams

In each of these problems the value of part of the whole is

known and you are to find the value of the whole. Draw a

diagram to illustrate each problem. Write a complete solu-

tion, as shown on page 180, if it helps you to do so. Include

the diagram in the third step, or plan.

1. Mr. Jones sold a horse at a profit of 20% of the selling

price. The profit was what per cent of the cost price?

2. Louise obtained a sample of the goods she planned to

use in making a dress. She washed the sample and found

that it shrank one tenth of its original length. Her pattern

called for 4^ yards of material. How much goods should

she buy if she washed it before making the dress?

3. How many pounds of wheat are needed to make a barrel

of flour weighing 196 pounds if wheat loses 25% by weight

when made into flour?

4. Last winter Ralph Becker helped his father cut ice and

store it to sell in the summer. Mr. Becker said that the ice

could lose about 25% from the time it was stored until it was

sold. He usually sold about 1200 tons each summer. Howmany tons of ice should he store?

5. One Saturday morning Morris started hoeing potatoes

at 8 o'clock. By noon he had hoed 6 rows. At the same rate,

how long would it take him to finish if there were 8 rows in

the field? Can you find more than one solution?

6. When the potatoes were ready, Morris dug them and

got 1\ bushels from the first 3 rows. At the same rate, how

many bushels should he get from the whole field?

185



Problems. Using Diagrams

Write a complete solution for each problem below. Draw a

diagram for each problem as part of the third step, or plan.

1. Mr. Baylor shipped some hogs to Chicago. How much

pork was obtained from a hog weighing 360 lb. when loaded

on the train, if it lost ^ of this weight on the trip and the

dressed weight was 75% of the weight just before killing?

2. Mr. Baylor received $3.45 a hundred pounds, live

weight at Chicago. How much was this per pound of pork?

3. Mr. Baylor wanted to build a new fence 135 feet long.

How many posts would he need if he wanted to set them

15 feet apart? There must be a post at each end.

4. Mr. Baylor planned a second fence, 168 feet long, be-

tween two fences that were already built. He did not need a

new post at either end of this fence as he could use the posts

in the existing fences. How many new posts would he need

if he set them about 15 feet apart?

5 Can you make a rule for finding the number of posts in

a fence, including posts at both ends? Including a post at

one end only? At neither end?

6. Mrs. Cunningham made a patchwork quilt. She made

20 blocks, as it was to be 4 blocks wide and 5 blocks long.

She then decided to add an extra row of blocks to both the

length and width. How many more blocks did she need?

7. A gross profit of 33j% of the cost price is what pei cent

of the selling price?

8. A gross profit of 20% of the selling price is what per cent

of the cost price?

186



Checking Problems. Working in Two Ways

One of the best ways of checking a solution of a problem is

to work it in two different ways and see whether you get the

same result both times. Solve each of the following in twoways. Draw a diagram whenever you find it helpful.

1. Two passenger trains leave New York for Chicago at

the same time on parallel tracks. One goes 40 miles an hourand the other 42±- miles an hour. How far apart will they beat the end of 5 hours?

2. A pupil pays $45 tuition in a term of nine months of

twenty days each. He is absent from school sixteen days.

Find the amount of tuition lost to him by his absence.

3. During one quarter of last year a household used 15,000

gallons of city water. The bill was $8.25. The next quarter

they used 7000 gallons. What should be their bill for the

second quarter at the same rate for each 1000 gallons?

4. A certain automobile uses 16 gallons of gas to run 180

miles. At this rate how much does it cost for gas for a season's

run of 7650 miles if gas costs 20|^ a gallon?

5. A farmer used 48 tons of hay to feed 32 head of cattle

through the winter. At this rate, how much hay will he need

to feed 75 head of cattle the next winter?

6. In October a farmer was offered 76 £ a. bushel for 2640

bushels of corn. However, he held it until January, whenhe sold it for 81^ a bushel. How much did he gain by waiting?

7. The distance from New York to Chicago by one route

is 960 miles, and from Chicago to El Paso by one route 1460

miles. Find the cost of a railroad ticket at 2i a mile by this

route from New York to El Paso. Find the length of time

for the trip at an average speed of 50 miles an hour.

187



188 Checking Problems. A New Method

One evening Dick tried to check his solution for a problem

by working the problem in another way. Mr. Cunningham sug-

gested that Dick could check bystarting with his answer and

working for one of the facts given in the original problem. Study

the work below.

Problem: The population of the town of Southwick was 7500

in 1930 and 9000 in 1940. Find the per cent of increase.

Dick's Solution

Per cent of increase

Population in 1930 = 7500


(1) Increase = 1940 population- 1930 population

(2) Per cent of increase = increase -r- 1930 population

9000

7500

To Find:

Given :

Plan:

Solution : 150 -JL-20%7 5 5

AU /o

To Find:

Given :

Plan:

Solution :

1500

Mr. Cunningham's Check

Population in 1940


Increase = 20% of 1930 population

(1) Increase = 20% of 1930 population

(2) 1940 population =1930 population+increase

1500 7500

^X 7500 = 1500 1500

9000

1. Why did Mr. Cunningham find iX7500? What does the

1500 represent?

2 Why did Mr. Cunningham add 1500 to 7500? How does

the'result, 9000, give a check on Dick's solution? Was Dick's

solution correct?

3. Describe two ways of checking the solution of a problem.



Problems. Checking by Starting with Answer

Solve each problem and check by starting with your result

and working for one of the facts in the original problem.

Write a complete solution for both the original solution and

the check.

1. A mechanic hadhis wages increased

10%.Find his

wages before the increase if he now receives $4.95 a day.

Draw a diagram.

2. The Bronson School gave a play. Tickets to pupils

were 15^ each and to outsiders, 35^ each. There were 175

pupils who promised to buy tickets. How many tickets must

be sold to outsiders to take in $100 altogether?

3. There are 45 pupils in the seventh grade of the OwenSchool, and there are f as many boys as girls. How many

are there of each?

4. A dealer in used automobiles sold a car belonging to

Mr. Calhoun. He mailed Mr. Calhoun a check for $409.50,

after deducting 10% of the selling price as commission. How

much did the dealer receive for the car?

5. Mr. Dwight drove his automobile 36 miles in 45 minutes.

How many miles an hour did the car average?

6. A family is renting a house for $55 a month. How much

money could they borrow and put into a new house without

increasing their yearly expense, if interest is 6% and if they

figure taxes at $70 a year, and repairs, etc. at $125 a year?

189

^9

m



190 Checking Problems

AGNES MAKES A MISTAKE

Agnes spent twenty minutes in working and checking the

problem given below, only to find that her solution was incorrect.

Study her solution and check carefully. Can you find the errors?

Problem :

Mr. Adams and Mr. Murray are planning an automobile trip

together. Mr. Adams' car averages 12 miles on a gallon of gasoline

and Mr. Murray's averages 16 miles. How much money can they

save by taking Mr. Murray's car, if the total distance is 2500

miles and gasoline averages 20^ a gallon?

Agnes' Solution

208^-



Checking Problems 191

AGNES LEARNS TO SAVE TIME

Agnes' mother said that she could save considerable time

if she would form the habit of checking each step in the solu-

tion of a problem before starting the next. Mrs. Cunningham

wrote out the following suggestions for Agnes.

a. Write down what you are to find. Check.

b. Make a list of all the facts given in the problem. Check

carefully to make sure you have omitted none and that

you have copied each number correctly.

c. Plan the solution. Write the plan out if it helps you.

Check carefully to be sure your method is correct. Adiagram will often help you to make and check your

plan.

d. Carry out the first step in your plan. Check by going

over the work a second time or by estimating the

result. Do this with each step before going to the next.

e. If possible, check your final results in some way.

1. There are four common errors in solving problems.

Which do you make?

a. Mistakes in what is to be found and given.

b. Mistakes in copying numbers.

c. Mistakes in plan.

d. Mistakes in computation.

2. Show how Mrs. Cunningham's suggestions would help

you prevent each of these errors.

3. What methods have you had for checking the result?

4. Mr. Bronson borrowed $500 from his bank, with 6%interest, payable in advance. How much was the interest?

How much cash did Mr. Bronson receive? Check.

5. On the next page you will find a list of problems. Follow

Mrs. Cunningham's suggestions and see how many of the

problems you can work correctly the first time you try.



Problems. Checking Each Step

Solve each of the following and check each step before

starting the next. If possible, also check your final answer.

1. Henry made a bookcase which he sold for $5. The

material cost him $3.72, and it took 5 hours and 20 minutes

to make the case. How much did he make per hour for his

labor?

2. John wanted to estimate the number of words in a

composition he had written for his English teacher. He found

that he averaged about 9 words to the line and 18 lines to

the page. About how many words were there in his composi-

tion if it contained 6f pages?

3. Mrs. Crane wants to average not more than $5 a week

for groceries. Last month she spent $5.32 thefirst

week,$4.29 the second, $6.03 the third, and $4.18 the fourth. Did

she average more or less than she planned? How much?

4. It is about 14,000 miles by sea from New York to San

Francisco by way of Cape Horn, and only 5250 miles by way

of the Panama Canal. How many days can a steamer save

by way of the Canal if it averages 350 miles a day?

5. Mrs. Owens was canning fruit. She burned one burner

of her gas stove for 6 hours. Find the cost of the gas used

if the burner consumed 6^ cubic feet of gas an hour, costing

90fi per thousand cubic feet.

6. A truck full of coal weighed 4150 lbs. Find the cost

of the coal, at $9.50 a ton, if the empty truck weighed 1975 lbs.

192



Problems. Checking Each Step

7. Mrs. Adams bought a remnant of silk goods for $8.50.

How much was this a yard if the remnant contained 5 yards

25 inches? What per cent of the regular price did Mrs. Adams

save if the goods sold regularly for $2.50 a yard?

8. The basketball team of the Jones High School won

8 games and lost 4. What per cent of its games did it win?

What per cent did it lose?

9. Mr. Anderson bought 20 dozen lemons for $5.25. He

sold them at 3 for a dime. How much did he make if 9 of

the lemonsspoiled?

10. Harold Summers bought a used car for $25. He spent

$17 for new parts and 16 hours in repairing the car. He then

sold it for $65. His profit was what per cent of the selling

price, counting his labor at 75^ an hour?

11. Mr. Bengston bought a farm for $7500 and sold it for

$8250. How much did he make on the transaction if he paid

an agent 5^% commission on the sale?

12. Mrs. Munson bought an electric refrigerator for

$189.60. She paid $75 down and agreed to pay the rest in

12 equal monthly payments. In addition she agreed to pay

$1 a month as carrying charge on the deferred payment.

How much did she have to pay each month?

193



Problem Helps. Telling How to Solve Problems

On this and the three following pages you will find four

Problem Helps. Practice on these will help you in solving

problems. More practice on these Problem Helps is given

at intervals throughout this book.

Probably the hardest step in solving a problem is making

the plan. Do you find this true? Practice in Telling How to

Solve Problems will help you on this step.

1. How do you find the difference between two numbers?

2. How do you findthe ratio of one number to another?

3. Morris is saving money to buy a bicycle. If you know

how much the bicycle costs and how much he has saved,

how can you find how much more he needs to save?

4. Ernest wants to find what per cent 8,279,758 is of

9,876,498. He wants the result to the nearest whole per

cent only. How can he find it with the least work?

5. How do you find the value of the whole when youknow the value of some per cent of it?

6. If you know how much a month a man earns, how

much a month he spends, and that he saves the remainder,

how can you find how much he should save in a year?

7. If you know how far you have driven and for how

many hours, how can you find your average speed?

8. If you know how fast you are driving and the distance

you must go, how can you find how long it will take?

9. If you know your average speed per hour, how do

you find how far you can drive in a given number of hours?

10. If you know the cost, and what per cent this cost is

of the selling price, how can you find the selling price?

194



Problem Helps. Supplying ih<- Missing Facts L95

Often in working a problem in life you do not know enough

facts, and you have to decide what additional information

you must get before you can work the problem.

In each of the following problems one or more facts that

are needed in solving the problem are missing. Look up the

facts needed, or have your class agree on probable figures,

and solve the problems. Check. Write complete solutions

and draw diagrams whenever you

find it helpful.

1. Last summer Ernest sold alu-

minum cooking utensils on commis-

sion. His total sales on the cooking

utensils amounted to $325.50. Howmuch did he earn?

2. Miss Norris borrowed $350

and agreed to pay 6% interest.

How much did she have to pay?

3. Mrs. Daly left the large burner

on her gas range burning for 8

hours one night by mistake. Howmuch did it cost her?

4. What does it cost to run an

electric fan for 3 hours at 7^ per

kilowatt hour?

5. Mr. Atkins had $175.22 in the

bank on May 1. During the month he withdrew $42.50,

$13.20, $112.10, $59.57, and $74.25. Find his balance on June 1.

6. The Atkins family planned to spend only 20% of their

income for rent, 25% for food, 15% for clothing, 20% for

miscellaneous expenses, and to save 20%. Last year they

spent $600 for rent and $800 for food. Were these items kept

within the budget?

7. The energy-producing value of foods is measured in

calories. Sirloin steak contains about 957 calories per pound.

Halibut steak contains about 458 calories per pound. Find the

cost of 1000 calories of each.



Problem Helps. Choosing the Facts You Need

In a life problem you usually know many things about the

situation which you do not need to use in solving your prob-

lem. It is often difficult to select the facts you need.

In each of the problems on this page one or more facts are

given that are not needed in the solution of the problem.

Decide which facts you need and solve the problems. Check.

Write complete solutions, if helpful.

1 Harry worked on a farm 42 days last year. He worked

275* hours and earned $72.50. This was how much an hour?

2. Mr. Adams bought a house for $4500. He paid $2000

cash and borrowed theremaining $2500 from a bank. At

the end of the year he paid the bank 6% interest and $1000

on the principal. How much was the interest?

3 A real estate dealer sold a farm for $7200. He charged

a commission of 3% and sent the former owner a check for

$6984. How much was his commission?

4. The pupils of the Smithville Junior High School gave

a play They sold 312 tickets to school pupils at 15* each

and 276 adult tickets at 25* each. They gave away 125 com-

plimentary tickets. What were the total receipts?

5 Robert raised 150 bushels of potatoes. He sold half

of them at 25* a bushel, 30 bushels at 30*, and 45 bushels at

35*. How much did he receive for the potatoes?

196



Problem Helps. Making Problems i'»7

Making problems of your own will help you in solving

other problems. Enough information is given in each para-

graph below to enable you to find one or more facts thai arenot given. Make one good problem out of each paragraph.

Solve each of your problems and check.

1. Mary Doane wishes to buy a radio that will cost $57.50

cash or $27.50 down and $5 a month for 7 months.

2. Mr. Thomas bought a house for $4500. During the

first year he spent $210 on repairs, $157 for taxes, and $62

for insurance. He received $35 a month rent.

3. In 1930, the population of Jonesboro was 11,287 and

the population of Brandon was 9,352. In 1940, their popula-

tions were 13,198 and 12,756 respectively.

4. On a recent date there were 2,333,000 telephones in

New York City and 717,468 in London. The population of

New York was 7,380,259, and of London, 8,202,818.

5. Alice Winters went to the drugstore to buy a bottle

of lotion. The clerk showed her two sizes. The small bottle

contained A^ ounces and cost 50^. The large bottle contained

11 ounces and cost a dollar.

6. Seven high schools belong to an athletic league known

as the Big Seven. Last year the record of games won and

lost in football was as given below:

School



198 Probem Test

Solve each problem on this page. Check.

1. Marjory took a standard test in arithmetic and made

a score of 5 examplescorrect out of 8 attempted. A month

later she took the same test and had 7 correct out of 9 at-

tempted. Find her per cent of accuracy on each test.

2. Morris plays on the basket-

ball team. Every day he practices W(

A

throwing fouls. One day he made

17 out of 30 attempts. The next

day he made 13 out of 19. What

per cent did he make each day?

Did he improve?

3. The International Limited, on the Canadian National

Railroad, makes the run from Montreal to Toronto, a dis-

tance of 334 miles, in 5 hours 40 minutes. This is an average

speed of how many miles an hour? Find the result to the

nearest hundredth of a mile.

4. Mrs. Clark bought a new automobile for $1285. It

was worth $815 at the end of the first year, $600 at the end of

the second year, and $515 at the end of the third year. Find

the per cent of depreciation each year.

5. When Mr. Fitzgerald bought his home, he borrowed

$5000 at 5% interest and gave a mortgage as security. He

agreed to pay $75 a month. How much of the first monthly

payment did it take to pay the interest? How much was

applied to the principal? Answer the same questions for the

second, third, and fourth months.

6. Mr. McCoy bought a used car. The dealer asked $120

cash, or $50 downand $10 a month for 10 months. Mr.

McCoy chose the second plan. How much did he owe the

dealer during the first month? The second? During each

succeeding month? This was equivalent to how much for

one month? For one year? Can you find the rate of interest

Mr. McCoy paid?

A



Diagnostic Tests in Fundamental Processes 199


Work ami check. Work as fast ae you can without hurrying.

1. a. b. c. <1.

Add: 5386; 759; Give result to §X^ Subtracl

878; 2647; 7056; nearest hund redth. $179.92 from

649; 538; 1077. 5.76)408.816 $398.72.

e. f. g. b.

h Give result to nearest cent. s-^

9| i S89.50

-3^q +]_ X.805

2. a. b. c. d.

(To nearest Subtract 5^ Multiply

cent) 5.8738 from 2 .7985

$132.57-^100 27.396. +4-1- by 100.

e. f. g. h.

6f-h2§ Subtract 2§X2j 2497.2

66§ from 396.7

100. 1786.9

+ 790.7




suggested below.

la — 5, 1, 2 If — 25, 21, 22, 24 2d— 35

lb— 34, 36, 17, 18 lg— 33, 36, 11, 13 2e — 30, 24, 27, 28

lc— 29 lb.— 30 2f— 26, 8

Id— 32, 6 2a— 35, 36 2g— 29, 27, 28

le— 26, 22, 21 2b— 32, 6, 8 2h— 81, 1,2

2c — 35, 21, 22, 23


Take Improvement Tests Numbers One, Two, and Three.

Record your scores on your Score Cards. Did you improve?

Find the class averages and bring your graphs up to date.





Work these examples and check. Work as fast as you can

without hurrying.

1. a. b. c. d.

875 is what per Write 1200%

cent of 1027? as a whole

(Nearest tenth or mixed

of a per cent) number.

2.

a.

Find the

whole, if

85.1% of it

equals $74.39.

(Nearest cent)

e.

Write 7.5%

as a decimal.

a.

Change -ff 2 to

nearest

hundredth of

a per cent.

e.

Write the

common frac-

tion equivalent

of 90%.

b.

Write

the %equivalent

off.

f.

Change £f£to a per cent.

b.

$2.50 is

what % of

g-

Find 37J%of 92.

c.

Write .9% as

a decimal.

h.

Write .1982

as a per cent.

d.

Write 13.97

as a per cent.

f. g. h.

Find the Write 166§ Find 18.6%

whole if as a whole of $2.87.

33^% of it or mixed (Nearest

equals 89.7. number. cent)


Study your errors carefully and note the trouble spots. Turn

to the Practice Tests (pages 307 to SIfi) for further practice as sug-

gested below.

la — 44 le — 39 2a— 41 2e — 37

lb_ 37 If— 41 2b— 43 2f— 44

lc _ 43 lg— 42 2c — 39 2g— 38

Id— 38 lh— 40 2d— 40 2h— 42



your Score Card. Did you improve? Find the class average




( ihapter Test 201

Write the numbers from 1 to 20 on a paper. After these numbers

write the words, or numbers, which belong in the corresponding

blanks below.

1. There are steps in the solution of a problem. They(i)

are and

2. The diagram at t la-

left shows thai a gross profil

of % of the selling price

(7)

is equal to a gross profit of'

, of the cost price.

TOTAL-\

H

(8)

3. The diagram at the left

shows that Harry's share of

the profits is times John's(9)

share. Harry receives per(10)

cent of the total profits and

John receives per cent.

on

4. Three ways of checking the solution of a problem are

, and(13) (14)

5. Practice in Telling How to Solve Problems helps you in

(12)

(15)

the solution of problems.

6. Alice spelled 8 words correctly on a spelling test. To find

wThat per cent she spelled correctly you would have to know(16)

7. The Barberton basketball team played 10 games last year.

They won 7, lost 2, and tied 1. To find what per cent of the

number of games played they won, you do not need to use the

number they or(17) (18)

8. Mr. White owns a clothing store. Last year his sales totaled

$22,700. The goods sold cost him $15,890. His expenses totaled

$4,540. He made $ profit. His profit was % of his sales.

(19) (20)



=&

Z II A l» T E R

\

Practical Measurements

Measuringis part of our daily lives. Almost everything we

use must be measured. Think of trying to buy supplies in a

grocery store if we had no means of measuring. We might

buy eggs and bananas by the dozen, although these are fre-

quently weighed today; but flour, sugar, butter, and many

other articles must be weighed. Even if you buy these in

packages, someone had to weigh them. Things we buy in

other stores are also measured. Nails are sold by the pound,

drugs by the ounce, dress goods and ribbon by the yard. In

fact, almost everything we buy is measured by someone.

Without accurate measurements we could not make a pie,

or a cake, or bake bread. We could not construct highways,

dams, bridges, railroads, skyscrapers, ships, automobiles, air-

planes, or many of the other things in our modern civilization.

We could not even build houses or make furniture for them.

Our age is often spoken of as a scientific age. Such

sciences as astronomy, physics, and chemistry and their

many practical applications would be impossible without

accurate methods of measuring. This is so true that we might

well call this the age of measurement.

203



Measuring in Everyday Life

1. When North America was settled by the white men,

did the Indians have much need to measure things? Why?

2. Can you describe at least one practical situation in

which we need to measure time? Length? Area? Volume?

Capacity? Weight? Temperature?

3. What reasons can you give for the fact that we need

to measure more than the Indians did?

4. What things have you measured lately? Why did you

measure them?

5. Ask your father and mother what things they have

measured lately, and why they measured them.

6. Describe at least one way in which each of the following

needs to measure. If you do not know, try to find out.

Cook Carpenter Bricklayer Clerk in dry goods store

Nurse Plasterer Electrician Clerk in hardware store

Tailor Painter Paper hanger Clerk in grocery store

Doctor Plumber Dressmaker Prescription clerk

204



x \

Natural Units of Measure

BOY SCOUTS MEASURE BY PACES

In order to measure anything, we must have a unit of

measure, that is, a quantity of the same kind with which the

given quantity may be compared. In order to measure the

length of this page, you must use some length as a unit. In

this country we usually measure length in inches, feet, yards,

and miles. To measure the length of this page, you would

probably compare its length with the length of an inch on

your ruler.

Originally people used the lengths of various parts of the

human body as units. A few of the many units employed

were the distance from the elbow to the tip of the middle

finger, the distance from the end of the thumb to the end of

the little finger of the outstretched hand, the breadth of the

thumb, the breadth of the four fingers, the length of the

stride or pace, the distance from the tip of the nose to the

end of the thumb, and the length of the foot.

1. Dick Cunningham, a boy scout, found that he took 20

steps in walking 50 feet. What was the average length of

his pace?

2. Last Saturday Dick's troop took a hike. Dick took

8,248 steps. How many miles did he walk?

205

i



206 The Span, Thumb Width, and Pace

1. The distance from the end of the thumb to the end of

the little finger of the outstretched hand is called the span.

Measure two of the following in spans:

The length and width of your desk

The length and width of the teacher's desk

The width of the door

The length of the blackboard

The height of the blackboard above the floor

The width of a window

2. Use the width of your thumb to measure the following:

The length and width of this book

The length and width of a piece of paper

The length of this line of printing

The length of your pencil

3. Measure at least two of the following in steps or paces:

The length and width of the schoolroom

The length and width of the schoolyard

The distance from your seat to the teacher's desk

The distance from your seat to the door

The length and width of the school building

The distance from school to your home

4. Measure the span of your hand to the nearest half inch.

Change your results in example 1 into inches.

5. Measure the width of your thumb to the nearest six-

teenth of an inch. Change the results you obtained in ex-

ample 2 into inches.

6. Take twelve steps along some straight line such as the

edge of a sidewalk, or a crack in the floor. Measure this

distance to the nearest foot. Find the average length of your

step or pace. Change your results in example 3 into feet.

7. If you will remember the width of your thumb and

the lengths of your span and pace, you will have a convenient

method of measuring lengths approximately when you do not

have a ruler. Use these to measure various lengths.



Standard Units

Measurements made by using the length of some part of

the body as the unit could not be very accurate, as individuals

differ in size. As more accuratemeasurements were needed,

the units of length were gradually standardized, or defined so

that they meant the same to everyone. In this way our

present units of length came into use. Some of the old names

have been retained. The foot comes from the length of the

human foot, the inch from the width of the thumb, and the

yard from the distance between the tip of the nose and the

end of the thumb. The mile originated with the Romans and

was a thousand double steps in length.

The early attempts to standardize the units of length were

very crude. Henry I of England defined the yard as the dis-

tance from the tip of his nose to the end of his thumb. At

another time the English Parliament denned the inch as the

length of three grains of barley laid end to end. In Germany

the foot was once defined as the average length of the left

feet of sixteen men taken at random.

Today the yard is defined as the distance between the

center marks in two gold plugs set in a large bar which is kept

in the Standards Department of the British Board of Trade.

The other units, such as the inch, foot, and mile are defined

in terms of the yard.

207



208 Diagnostic Tests in Denominate Numbers


1. What standard units do we commonly use in measuring

time? Length? Area? Surface? Volume? Weight? Capacity?

2. Do you know the relationship between the different units

of length? Copy and complete the following table.

inches = 1 foot

feet = inches= 1 yard

yards = feet = 1 rod

rods = feet = 1 mile

3. Can you make up similar tables for time, surface, volume,

weight, and capacity?


a. 5 hr. = min. g. 5 hr. 17 min.= min.

b. 4 bu. = pk. h. 2 gal. 1 qt, = qt.

c. lOoz. = lb. i. 2 T. 372 1b. = lb.

d. 13 pt, = qt, j. 3 wk. 4 days= wk.

e. 27 mo.= yr mo. k. 4 lb. 6 oz. = lb.

f. 60 in. = yd ft, 1. 2 yr. 9 mo. = mo.

5. Carry out the following operations. In how many ways

can you work eaeh?

a. Add 3 ft, 7 in. and 4 ft. 8 in.

b. Subtract 2 lb. 7 oz. from 5 lb. 2 oz.

c. Multiply 4 hr. 27 min. by 3.

d. Divide 7 ft, 2 in. by 4.

e. Divide 3 lb. 6 oz. by 9 oz.


If you had trouble with problems 1 to 3, study the tables given

pages 342 to 343. If you had difficulty with problems 4 and 5,

ask your teacher for help; then turn to Practice Tests 45 to 50,

pages 336 to 340, and find similar examples for further practice.



Exact and Approximate Measurements

No matter how carefully a measurement is made, the

result is never exact but only approximate. The picture shows

a micrometer which engineers, mechanics, scientists, and

others use to measure accurately. With it measurements can

be made accurate to .0001 of an inch. Such a high degree

of accuracy is not necessary in most measurements.

Dick Cunningham reseeded the lawn in the back yard.

To find how much seed he would need, he measured the yard

and found that it was about 26 ft. 3 in. by 40 ft. 10 in. He

called it 26 ft. by 41 ft., to the nearest foot. If it had been 26

ft. 6 in. wide, he would have called it 27 ft., as it is customary

to add 1 for a fraction equal to or greater than one half, and to

drop a fraction less than one half.

1. Measure the following to the nearest foot. To the

nearest inch.

The length and width of the blackboard

The height and width of the door

Your own height

2. Measure the following to the nearest half inch. To

the nearest eighth of an inch.

The length and width of this book

The length and width of a piece of paper

The length of this line of type

209



210 Common Errors in Measuring Distances

Agnes' class was studying measurement. Miss Smith

placed two dots on the blackboard and asked her pupils to

measure the distance between the dots. Eugene used a foot

rule. He placed the zero of the ruler on one dot and pointed

the other end in the direction of the second dot. He then

marked the position of the end with his thumb and slid the

ruler in the direction of the second point until the zero part

of the ruler was just below his thumb. Working in this way,

he found the distance between the dots was 8 ft. 7 in.

Miss Smith asked the class to criticize Eugene's method

of measuring the distance between the dots.

Agnes said Eugene had not measured the shortest distance

between the two dots, as he had not measured along a straight

line but had zigzagged back and forth. Albert suggested

that they cover a string with chalk and stretch it between

the two points. He pulled the center of the string away from

the board and let it snap back. This marked a straight line

on the board between the two points.

Harry suggested that Eugene mark the end of each foot

by a chalk mark on the board instead of using his thumb.

Why? Eugene measured again, following these suggestions,

and found that the distance was 8 ft. 2\ in.

1. Name two errors to be avoided in measuring the dis-

tance between two points. Tell how to avoid each.

2. Place two dots on the blackboard, at least 6 feet apart,

and measure the distance between them as accurately as you

can, Use a foot rule and string. Have at least two groups

from your class measure it independently, as a check.

3. Drive two small stakes in your playground about 20feet apart and measure the distance between them. Use a

yardstick and a string.

4. Measure the length and width of the schoolyard. Use

a yardstick. Use a string if necessary. Have at least two

groups make each measure, as a check.



Estimating Distances

Agnes Cunningham received a camera for her birthday.

Before taking a picture, the camera had to be adjusted accord-

ing to its distance fromthe object to be photographed. At

first Agnes had difficulty in estimating distance, but after a

little practice she did so quite accurately.

It is often useful to be able to estimate short distances

without having to measure them. The following exercises

will help you to develop this ability.

1. Study the distance between two successive inch marks

on your ruler. When you think you have a clear picture of it

in your mind, try to draw a line one inch long without using

your ruler. Measure it. Repeat this until you can draw,

without measuring, a line which is less than an eighth of an

inch longer or shorter than an inch.

2. Draw lines of the following lengths without measuring.

Check by measuring.

2 in. 5 in. 4 in. 7 in. 10 in. 9 in.

3. Can you estimate the length and width of the cover of

this book to the nearest half inch? Can you estimate the

thickness of this book to the nearest eighth of an inch?

4. Draw a line less than a foot long on the blackboard.

Ask someone to estimate its length. Check by measuring.

211



212 Estimating Distances

Scouts should be able to estimate short distances with an

error of not more than 25% of the correct distance. Thus, in

estimating a distance of 10 inches, the estimate should be

from 7^ to 12^ inches. Why? Between what limits should

the estimate be if the true length is 12 inches?

1. Find the per cent of error in each of the following:

Estimate True Length

1 ft. 8 in.

20 in. 2 ft.

2. Estimate the lengths of objects in the schoolroom.Check by measuring. Find what per cent the error is of the

correct length in each case. Practice until you can estimate

lengths less than a foot with an error of not more than 20%.

3. Study the length of your foot rule. When you think

you have a clear picture of it in your mind, try to draw a line

one foot long without using your ruler. Measure it. Practice

until you can draw a line which is not less than eleven andnot more than thirteen inches long.

4. What is the per cent of error in problem 3 if the line

drawn is 11 inches long? If it is 13 inches long?

5. Mark distances of the following lengths without measur-

ing. Check by measuring. Find the per cent of error.

3 ft. 2 ft. 6 ft. 12 ft. 8 ft. 41 ft.

6. Estimate different distances in or near the school.

Check by measuring. Find the per cent of error in each case.

Practice until you can estimate any distance up to 50 feet

with an error of not more than 25%.

7. In estimating a long distance, it often helps to lay it

off in shorter distances with the eye. Dick used this method

to estimate the distance from the boyscout cabin to the

road. He estimated the distance from the cabin to the large

oak tree as 50 ft., from the oak to a large rock as 40 ft., and

from the rock to the road as 30 ft. He estimated the whole

distance to be 120 ft. Use this method to estimate several

distances.



RECTANGLESRIGHT ANGLE

ACUTE ANGL]

OB i i -I. \-.<

;LE

Rectangles, Squares, and Angles

You have learned that the first two figures above are called

rectangles and that the second is also called a square. Asquare is a rectangle whose sides are equal. The corners of

squares and other rectangles are called right angles. Angles

less than right angles are called acute angles. Angles larger

than right angles are called obtuse angles.

1. Can you find any right angles in this linoleum pattern?

Any obtuse angles? Any acute

angles?

2. Turn to the map of

Smithville, on page 220. Can

you find any right angles?Any acute angles? Any ob-

tuse angles?

3. Fold a paper and crease

the fold sharply. Fold the crease

back on itself and crease again.

The four angles formed by the

two creases are all what kind of angles?

4. Is a square always a rectangle? Is a rectangle always

a square?

5. Using the paper you folded in problem 3 as a model,

draw a right angle. Draw an acute angle. Draw an obtuse

angle.

213



214 Finding the Area of a Rectangle

Mrs. Cunningham wanted a new carpet for her living room.

Dick helped her find the cost of the carpet.

Dick first found the amount of floor surface or area to be

covered. You have learned that in measuring lengths we use

some standard length as a unit. In the same way, in measur-

ing areas we use the area of a square, one unit on each side,

as a unit. The common units for measuring areas are the

square inch, square foot, square yard, and square mile. These

are the areas of squares 1 inch, 1 foot, 1 yard, and 1 mile on

a side, respectively.

The Cunningham's living

room was 7 yards long and

5 yards wide. Dick drew a

figure like the one at the

left. What does each square

represent? How many square

yards of carpet will it take?

Mr. Cunningham sug-

gested to Dick that he could

have found the number of square yards in the floor without

drawing a figure and counting them one by one. How many

square yards are there in the bottom row in the figure? How

many yards in the lengthof the rectangle? How many rows

are there? How many yards in the width of the rectangle?

How could you find the total number of square yards without

drawing a figure and counting the square yards?

•Can you state a rule for finding the area of a rectangle by

measuring the length and width? Mr. Cunningham gave Dick

the rule below. Do you think it is correct?

To find the number of square units in the surface of a

rectangle, multiply the number of units of measure in the

length by the number of like units in the width.



Problems* Finding 4jreas of Rectangles 215

i. In order to check the accuracy <»t Mr. Cunningham'

rule, draw the following rectangles and divide them into

square inches. Does the rule work in each ca

3 by 6 .V by 9 8j by 6 :>>\ by 2j*

2. In what unit will the area be if the length and width

are in inches? In feet? In yards? In miles?

3. Draw a figure to show the number of square feet in a

squard yard. How many square inches are there in a square

foot?

4. Find the area of a rectangle that is 1 foot long and 8

inches wide. Can you do this in two ways? Can you show

that the two answers are equal?

5. Is the following statement always true? The length

and width of a rectangle must be expressed in the same units

before multiplying to find the area.

6. Mrs. Cunningham considered two kinds of carpet for

her living room (page 214). One came in wide widths and

cost $3.50 a square yard. What would this carpet cost?

7. The other carpet came in a strip 27 inches wide and

cost $2.75 a running yard. How many strips would it take,

running the long way of the living room? How many yards

would it take? What would the carpet cost?

8. The Cunningham's dining room is 12 feet wide and

18 feet long. How many square yards are there in the surface

of the floor? Can you find this in two ways? How much

would a new carpet cost at $3 per square yard?

9. How many square yards of linoleum will it take for a

kitchen floor 9 ft. by 12 ft.?

10. How much would the linoleum for the kitchen in prob-

lem 9 cost at $1.50 a square yard?



216 Formulas for the Area of a Rectangle

You have already seen how the rule for finding interest

can be stated more briefly by the use of letters as abbrevia-

tions for words. Dick abbreviated the rule for finding the

area of a rectangle in the same way.

A=lXw, or A = lw

Dick read this, The area of a rectangle equals the length

times the width. He explained that the multiplication sign

is usually omitted between two letters, or a number and a

letter, in writing a formula.

The length and width of a rectangle are often called the

base and altitude. Either side of a rectangle may be used as

the base, but we usually call the side on which a figure rests

the base. The altitude measures the height of the figure.

Height and altitude mean the same thing. Notice that the

altitude forms a right angle with the base. The length and

width, or the base and altitude, of a rectangle are oftencalled

its dimensions.

1. Dick read in a newspaper that the altitude of Mount

McKinley, in Alaska, is 20,300 feet above sea level. What

does that mean?

2. Find the area of the following rectangles.

a. = 8 in., 6 = 12 in. c. 7=9 ft., w =Uft.

b. o = 13 ft., b = 17 ft. d. l = 6in.,w = 7m.

3. Find the number of square feet in a rectangle 7 ft. 5 in.

by 9 ft. 7 in. Do this in two ways.

4. Find the number of square feet in the floor of your

schoolroom.

A = lw A = ab

The area of a rectangle equals the length times the width,

or the altitude times the base.



Formula for the Area <>f a Square 217

Since a square is a rectangle with equal base and altitude,

we can write the formula, A=sXs, using s to represent the

length of the side of the square. This formula is usually

written A=s2. The small 2, written to the right and above

the s, shows that s is to be multiplied by itself, or used as a

factor two times, s2 is just a shorter way of writing sXs.

s2 is read s square, or the square of s.

1. Find the value of s2 , if s = 4 yards; 6 inches; 9 feet; 2 feet

8 inches.

2. Give the value of each of the following.

3 2 52 122 6.22

22 42 .52 (f)2

3. Find the areas of the following squares.

a. s = 7 inches d. s = 2.72 yards

b. s = 3 yards e. s - 8 feet 7 inches

c s = 17 feet f . s = 2 yards 9 inches

4. How many square feet are there in the square in problem

3b? How many square yards in the problem 3c?

5. The lot on which the Underwood Junior High School is

located is 250 feet square. The building is 120 feet by 80 feet.

How many square yards are there in the yard surrounding

the building?

6. How many square feet are there in a square 20 inches

on a side? Solve in two ways.

7. Can you find a square picture in this book? Find the

length of one side of the picture in inches. Find the number

of square inches in the area of the surface.

8.If

you canfind some large square surface, such as a

flower bed, measure it and find the area.



218 Finding the Perimeter. Order of Operations

The perimeter of a rectangle or a square is the distance

around it, or the sum of the four sides. Using p for the perim-

eter of a square, Dick wrote, p=s+s+s+s. Agnes wrote,

p = 4s. Which formula do you like better?

Dick wrote the formula for the perimeter of a rectangle,

p = a+b+a+b. Agnes wrote, p= 2a +2b. Which do you like

bettor?

They used Agnes' formula to find the perimeter of a rec-

tangle 8 feet by 7 feet. Their solutions are shown below.

Which is correct? Draw the rectangle and add the lengths

of the four sides, if you do not know.

Dick's Solution

p =2a+2b=2x8+2x7 = 2xl0x7 = 140 feet

Agnes' Solution

p=2a+2b=2x8+2x7

= 16

+14 = 30 feet

In a statement such as 2x8+2x7, which must be carried

out first, the multiplications or the additions? What does

5 X 8 _6 -=-2 equal? Study the rule below.

1. Find the value of each of the following.

3X6+8X9 d. 12-^2+4 g. 5x6+4

b. 5X7-3X4 e. 10-8-2 h. 8x9-26X8+7X9 f. 3x4-1 i. 9-3+5

2. Find the perimeters of the following.

a. s = 14 ft. c. a = 17 ft., & = 25ft.

b s =9yd #d. a = 49 in., b = 85 in.

a

c.

In a series of arithmetical operations, all multiplications

and divisions must be carried out before the additions and

subtractions.

5X8-6^2 = 40-3 = 37



Finding the Perimeter. Using Parentheses 219

The formula p=2 (a + b) is read, The perimeter Ol a rec-

tangle equals two times the sum of the altitude and bi

The parentheses,( ), tell you to perform the pro inside

the parentheses first.

Agnes used this formula to find the perimeter of a rec-

tangle 9 in. by 12 in. Her solution is shown below. Is it cor-

rect? Draw the rectangle and add the lengths of the four

sides, if you do not know.

p=2 (9+ 12) =2x21 -42 in.

1 Find the value of each of the following.

a. 2(8+9) d. 3x4+2 g. 4+8-2l>. 2x8+9 e. 3(4+2) h. (4+8) -2

c. 3(4-1) f. 7(8-5) i. (3 +2) x (4-1)

2. For what is the first formula at the bottom of the page

used? The second? State each of these formulas in words.

3. Agnes preferred the formula for the perimeter of a

rectangle given at the bottom of this page, to the one she

made (p = 2a+2b). Do you agree? Why?

4. Use the formulas at the bottom of this page to find

the perimeter of each of the following squares and rectangles.

a. s = 12 in. e. 0=9 ft., b = ll ft.

b. s = 125ft. f. a = 1.9 mi., b = 2.4 mi.

c. s = 3ft. 4 in. g. « = 1 ft. 9 in., b = 3 ft. 7 in.

d. s = 5yd. 2 ft. h. c? = 4 ft. 8 in., 6 = 6 ft.

5. Harry Smith helped his father build a fence around a

field 100 rods by 80 rods. Without allowing for a gate, how

many rods of fence did it take? How many feet?

In a series of arithmetical operations, the operations

inside the parentheses( ) must be carried out first.

3(15-7)=3X8= 24

p= 4s p= 2(a+ b)



220 Parallel Lines and Parallelograms

k



Parallel Lines an<l Parallelograms 221

1. Name two streets shown on the map of Smithville

that are parallel. Can you find two others? Two more?

2. Name two streets that are not parallel. I low many

other pairs of nonparallel streets can you name?

3. Are the two rails of a railroad track parallel? The two

sides of a street? The spokes of a wheel? Two opposite sides

of a rectangle? Two adjacent sides of a rectangle? Two sides

of a triangle?

4. See how many pairs of parallel lines you can find in

your schoolroom. Can you find ten pairs of lines that are

not parallel?

5. Name two streets, or roads, near your school that are

parallel. Two that are not parallel.

6. Are the top and bottom of this page parallel? The two

sides?

7. Is a rectangle always a parallelogram? Why? Is a

parallelogram always a rectangle? Why? Draw figures to

illustrate your answers.

8. Define a rectangle as a particular kind of parallelogram.

9. Is a square aparallelogram? Is it a rectangle?

10. What lots on the map of Smithville are parallelograms?

Which of these are rectangles? Are any of these squares?

11. How many parallelograms can you find in your school-

room? Can you find any that are not also rectangles?

12. Draw a parallelogram. Be sure to get the opposite sides

as nearly parallel as possible. It will help you if you rememberthat the opposite sides are equal.

13. Two railroads cross. One runs east and west, the other

northeast and southwest. What kind of figure is formed by

their tracks at the intersection? What kind of figure would

be formed if thev crossed at right angles?



The Area of a Parallelogram

Before Mr. Mullins built his house, he hauled in dirt to raise

the level of the lot three feet. To find out how much dirt he

would need he had to find the area of the lot. He drew a

picture like the first figure shown above and marked the

altitude and the base. Notice that the altitude, which meas-

ures the height of the parallelogram, makes right angles with

the base.

Mr. Mullins cut off the triangle on the left end of the figure

and fitted it on to the right end as shown in the second figure.

He thought, I now have a rectangle with the same area, base,

and altitude as the parallelogram. Since the area of the

rectangle equals the product of its altitude and base, the

area of the parallelogram must also equal the product of its

altitude and base. Hewrote the rule and formula at the

bottom of the page. In using them, the altitude and base

must be expressed in the same units.

1. How did he know that the parallelogram and rectangle

have the same altitude? The same base? The same area?

2. Was Mr. Mullins' reasoning correct?

3. How many square feet are there in the surface of Mr.

Mullins' lot? How much did it cost to fill the lot if the con-

tractor charged 35^ for each square yard of surface?

A= ab

The area of a parallelogram equals the altitude times the base.

222



Problems. Areas o£ Parallelograms 223

1. Find the area of the following parallelograms.

c = 2 ft., 6 = 9 in.

o = 3 ft. 2 in., 6 = 15 in.

a = 4 ft. 1 in., b = 2 ft. 3 in.

50

o = 7 in., 6 = 9 in.

a = ll ft., 6 = 3 ft.

= 2 yd., 6 = 5 ft.

2. Find the number of square feet in a rectangle 18 inches

by 15 inches. Work in two ways.

3. Are there any other lots on the map on page 220 that

have the same areas as Mr. Mullins' lot? How many can

you find? How do you know that they have the same area?

4. Agnes measured the length

of two adjacent sides of a

parallelogram, as shown in the

figure, and multiplied. She said

the area was 1750 sq. in. What

error did she make?

5. Cut out of paper a parallelogram that is not a rectangle.

Be sure to get the opposite sides as nearly parallel as possible.

Remember the opposite sides must be equal. Find the alti-

tude by folding through the center so that the parallel sides

are folded back on themselves. Measure the base and alti-

tude as accurately as you can and find the number of square

inches in the surface of the parallelogram.

6. How much of an error would you make on the area of

your parallelogram if you made the same error as Agnes

made in problem 3? Try it and see.

7. Each square in the figure represents one square inch.

Find the area of the parallelo-

gram ABCD by multiplying the

altitude by the base. Check the

result by counting the number of

square inches and halves of

square inches in the parallelo-

gram. Does the short rule give

the correct result?



Right Triangles

You have learned that all of the figures shown above are

called triangles. A triangle is a figure bounded by three

straight lines. If one of the angles of a triangle is a right

angle, the triangle is called a right triangle.

1. Which of the figures above is a right triangle?

2. Take a rectangular piece of paper and fold it along a

straight line passing through two opposite corners. Are the

two triangles formed right triangles? Cut the two triangles

apart. Can you put one on the other so they fit exactly?

Are they equal?

3. Cut out of paper a parallelogram that is not a rec-

tangle. Fold through the opposite corners. Are the triangles

formed right triangles? Are they equal?

4. Read and complete the following sentences.

a. A rectangle can be divided into two equal triangles.

A parallelogram can be divided into two triangles.

Turn to the map of Smithville on page 220. Can you

find any triangles? Are any of these right triangles?

6. Are there any triangles in your schoolroom? Are any

of these right triangles?

7. Using a model right angle, formed by folding a piece ot

paper twice (see page 213), draw a right triangle.

8. Draw a triangle that is not a right triangle.

b.

5.

224



The Area of Any Triangle

Mr. McMurray's farm is bounded by two roads and a rail-

road. To find the area he measured the base AB and the altitude

DC. Hethought, My farm is just half of a parallelogram. To

find the area of a parallelogram you multiply the base by the

altitude, so the area of my farm will be one half the product of

the altitude and base.

1. Was Mr. McMurray's reasoning correct? Explain.

2. Find the number of acres in Mr. McMurray's farm, if

the base measured 325 rods and the altitude 196 rods.

3. Each square in this figure

Find the area of triangle ABC

by counting the number of square

inches and halves of square inches

in the triangle. Find the area

by the short rule. Do your two

results agree?

represents inch.

A = |ab

The area of any triangle equals one half the product of the

altitude and base.

226



Problems. Finding Areas of Triangles 227

1. Find the areas of each of the following triangles.

a. a = 32 in., b= 13 in.

l>. a = 15 ft., b= 24 ft.

c. a =19 ft, 6 in., b = 27 ft. 9 in.

(1. a =121 yd., b = 200 yd.

c. a = 32rd., b= 58 rd.

2. Find the number of acres in the fourth triangle in problem 1.

In the fifth.

3. In a right triangle the two sides of the right angle may

be used as the base and the altitude. Why? Can any two sides

of a triangle that is not a right triangle be used as base and

altitude? Why?

4. Find the area of Lot 445 in Smithville (page 220). What

kind of triangle is this?

5. To find the area of the tri-c

angle shown, Mary measured the

sides AB and BC. She then found

the area as shown below. Was her

work correct? Why?

lift.

A=-|ab=|x/xll=44 sq. ft.

6. Find the correct area of the triangle in problem 5. Also find

the per cent of error in Mary's result.

7. Find the areas of each of the following triangles.

a. a = 8 in., b = 12 in.

b. a= 37 ft., b= 50 ft.

c. a = 9 yd., b = 13 yd.

d. a = 6yd., b = 7 yd.

e. a = 175 ft., b = 97 ft.

f. a= 87 in., b = 49 in.

g. a = 2ft, 7 in., b = l ft, Sin.

h. a= 3 yd. 9 in., b = 2 yd. 3 in.

8. Mr. Adams owns two fields

as shown at the right, Find the

number of acres in field B.



228 The Area of a Trapezoid

Jack's father owns

a lot of the shape

shownhere. Such a

figure is called a

trapezoid. A trapezoid

is a figure with four

4oFt. c no ft.

60 ft.

no ft. B 40ft.

sides, two that are

parallel and two that are not parallel. The two parallel sides

are called the bases.

To find the areaof his father's lot Jack measured the two

bases AB and DC, and the altitude EF. He then drew a plan,

like the one in the picture. Note that the altitude makes right

angles with the bases. Jack thought, The lot is half of a

parallelogram with an altitude of 60 ft. and a base of 150 ft.

Since the area of this parallelogram is 60X150, the area of the

lot is JX60X150 = 4500 sq. ft.

1. State a rule forfinding the area of a trapezoid. Compare

it to the one Jack made, at the bottom of the page. Do you agree

with Jack's rule?

2. What does a stand for in Jack's formula? What does

stand for? What does b' stand for? b' is read b prime.

3. Each square shown represents

1 square inch. How long is each

base of this trapezoid? How long is

the altitude? Find the area of the

trapezoid by Jack's rule and by

counting squares. Do the results

agree?

4. Find the number of acres in field A in problem 8 on page

227. As a check find the combined area of both fields in two ways.

A=ia(b+b')

The area of a trapezoid equals one half the product of the

altitude and the sum of the two bases.



Problems. Finding Areas of Trapezoids 229

1. Alice, Lucy, and Jane wished to find the area of :i trapezoid

with an altitude of 12 inches and bases of 20 inches and 32 inches.

Their solutions are shown below. Which one is incorrect? Why'

Alice's Solution

A = ia(b+b')=iXi2X20+32= 120+32= 152 sq. in.

Lucy's Solution

6

A = Ja(b+b')=iXl2X(20+32)=Jxi2X52= 312 sq. in.



230 Drawing the Altitude

1. Draw a triangle on a piece of paper. Cut out and fold

the longest side back on itself so the fold will go through the

Before Folding Folded Unfolded

corner opposite the longest side. Unfold. Is the crease an altitude

of the triangle? Why? Measure the base and altitude and find

the area.

2. Cut several triangles out of paper. Find the areas in the

same way as in problem 1.

3. To find the al-

titude of the triangle

ABC, Eugene took a

large piece of paper

DEFG, having a right

angle at E. He laid

the paper on the tri-

angle so that the edge

DE was on the side

AB and then he slid the paper along AB until the edge FE passed

through C. He then drew the line CE on the triangle with a

pencil. Is CE an altitude of the triangle ABC?

4. Draw a triangle on a piece of paper Draw an altitude as

described in problem 3. Measure the base and altitude and

find the area.

5. Find the areas of at least two other triangles by the method

used in problems 3 and 4.

6. Trace the trapezoid on page 229 on a thin sheet of paper.

Draw an altitude. Measure the bases and altitude and find the

area.



Drawing Circles 231

Take a cardboard about three inches Long and make a small

hole in one end. Stick a pin through the other end of the

cardboard, about two inches from the hole. Now lay a sheelof paper on a board and place the cardboard on the paper,

pushing the pin through the center of the paper into the

board. Put the point of a pencil through the hole in the card-

board and move the pencil and cardboard around the pin.

The pencil will draw a circle on the paper. The distance

around the circle is called the circumference. The pin is at

the center. Draw a straight line from the center to any pointon the circumference. This line is called a radius. Draw a

straight line through the center and ending on the circum-

ference at each end. This is called a diameter.

Compasses are convenient in drawing circles. If you do

not have a pair, you can use cardboard, as described above,

or string tied to a piece of chalk or a pencil for larger circles.

1. With a piece of cardboard or a pair of compasses drawa circle having a radius of 1^ inches. Draw a radius. A diam-

eter. How long is the diameter?

2. Draw a circle 5 inches in diameter. What radius will

you have to use?

3. Draw a circle that has a radius of § of an inch. What

is the length of the diameter?

4. All points on the circumference of a circle are the same

distance from the center. Why?

5. The plural of radius is radii. Are the radii of a circle all

equal in length? Why?

6. Are all diameters of a circle equal in length? Why?

7. Make a formula which tells you how to find the diameter

when you know the radius. The radius when you know the

diameter.

8. In answering problem 7, Dick made the formulas given

below. Are they correct?

d=2r r =U



ii

Relation between Circumference

and Diameter

Mrs. Cunninghamwished to buy a jardiniere for a large

fern. She tried to measure the diameter of the pot so she

would know what size of jardiniere to buy. Dick offered to

help. He measured the circumference of the top of the pot,

and from that he found the diameter. Do you know how

he did it?

1. Dick's class discovered the formula he used by measur-

ing several circular objects and makingthe table given below.

Copy it and complete the last column.

Object



Formulas. Diameter, Radius, ami

( Circumference

233

It is proved in higher mathematics

that the ratio of the circumference to

the diameter is the same for all circles,

and that it is equal to about 3.1416,

or approximately 31. It is customary

to represent this ratio by the Greek

letter tt, or pi. This is pronounced just

like the English word pie.

Dick's class, after making the table

given on page 232, wrote the formulas

at the bottom of the page. Explain

each formula in the first row. Do you

see how they obtained the first formula

in the second row? If you divide one side of an equation by

a number or quantity, what must you do to the other side

to keep the sides equal? What does ird^w equal? In whatother way can you write c -r- tt?

1. Explain how the class got each of the last two formulas

in the second row.

2. State a rule for finding the circumference of a circle

when the diameter is known. The circumference, when the

radius is known. The diameter, when the circumference is

known. The radius, when the circumference is known.

3. Dick Cunningham found that the circumference of the

flowerpot, page 232, was about 25 inches. He found the

diameter as shown below and decided it was about 8 inches.

Was his solution correct?

d = 25-3i =25^ =25X^ =W = 7 -95



234 Problems. Finding Diameter andCircumference

1. Harriet drew a circle on the blackboard with a radius of

1\ inches. She wished to know the circumference but could

not measure it accurately. Can you find the circumference?

Use t = ^-.

2. What is the circumference of a circular flower bed, 15

feet in diameter? How many bricks will it take to make an

edge around this bed if the bricks are laid end to end and

each brick takes 8| inches of the circumference? Use tt = 2^-.

3. Copy the following table andfill the blank spaces. Use

T _22TT--J-.

Radius Diameter Circumference

12 ft.

42 in.

57 in.

1 yd. 1 ft.

4. How far will an automobile wheel 32 inches in diameter

move forward in one revolution? Make a general statement

of how far any wheel moves forward in one revolution.

5. How many revolutions does a 30-inch wheel make in

going one mile?

6. How many revolutions a minute does a 32-inch wheel

make when the car is going 40 miles an hour?

7. How fast is the automobile going if it has 30-inch wheels

and they are making 300 revolutions a minute?

8. Change 3^ to a decimal fraction. Carry your result to

the nearest ten-thousandth. The correct value of t, to four

decimal places, is 3.1416. How many figures of the two values

of r are equal? Would 3^ be accurate enough to use for r

if you want two figures in your result? Three figures? Four

figures? Five figures?

9. Find the circumference, to the nearest foot, of a circle

800 feet in diameter. Estimate the answer. How many figures

will it contain? What value of ir should you use?



236 Problems. Finding the Area of a Circle

1 . Find the area of a circle having a 3 inch radius.

A=7rr2= -2

/x9=

H^= 28

f

scl- in -

2. The pool Mr. Sanders built was 60 ft. in diameter. Find

the' area of the bottom to the nearest square yard. Use

7T =3.1416. Find the cost of covering the bottom with con-

crete at $2 a square yard.

3 Mr. Sanders also built a curb around the pool to retain

the water. Find the length of the curb to the nearestyard.

Use tt= 3.1416. What did this cost at $1.50 a yard? What

was the total cost for building the pool?

4 Is it necessary to use 3.1416 for tt, instead of ^f in

problem 2? See problem 8, page 234. How much difference

would it make in the area of the pool, to the nearest yard, if

you used tt=-^?

5 A cement walk 8 ft. wide surrounds a circular flower bed

20 ft. in diameter. Find the number of square yards of cement

in the walk. Draw a figure. Use tt = 3.1416.

6. A horse is tied to a post by a rope 12 ft. long. Over how

many square feet can the horse graze? Use tt = 3.1416.

7 If the horse is tied to a corner of a barn by a rope 20

ft. long, over how many square feet can he graze if the barn

is' 20 ft', by 30 ft.? Draw a figure. Use tt = 3.1416.

8. Copy the following table and complete. Use tt=^-.

Radius Diameter Circumference Area

12 ft.

75 in.

5 in.

100 yd.

88 yd.

1000 ft.



Rectangular Solids and Cubes

You have been studying figures such as rectangles, squares,

triangles, and circles. All of these have only two dimensions

length and width. A solid is a figure having three dimensions

—length, width, and height. Finding the size of a solid is

called finding its volume.

If each face of a solid is a rectangle, as in the first figure

above, it is called a rectangular solid. If the length, width,

and height of a rectangular solid are all equal in length, as in

the second figure above, it is called a cube, and each dimen-

sion is called an edge, or side.

1. Is your schoolroom a rectangular solid? Your pencil?

A chalk box? A baseball? A football? A brick?

2. Is a cube always a rectangular solid? Is a rectangular

solid always a cube?

3. Name as many rectangular solids as you can. Which

of these are cubes?

4. What kind of figures are the faces of a rectangular

solid? The faces of a cube?

1. The faces of a rectangular solid are all rectangles.

2. The faces of a cube are all squares.

237



238 The Volume of a Rectangular Solid

When Mr. Cunningham bought his house, the coalbin

was partly rilled with coal. To find how much coal was in

the bin, Dickhad to know how to find the volume of a rec-

tangular solid.

In measuring areas, we use the area of

a square, one unit on each side, as a

unit. In the same way, in measuring

volumes we use the volume of a cube,

one unit on each edge, as a unit. The

common units for measuringvolumes are

the cubic inch, cubic foot, and cubic

yard. These are the volumes of a cube

1 inch, 1 foot, and 1 yard on a side,

respectively.

Dick took some small wooden cubes, one inch on a side,

and built a rectangular solid 3 inches long, 2 inches wide, and

4 inches high. He thought, This solid contains 4layers of

cubes, like the layers of a cake. Each cube is a cubic inch.

The top layer contains 2 rows of cubes with 3 cubes in each

row. There are 2x3 cubes in the top layer. Since there are

4 equal layers, there are 4x2x3 cubes in the solid. Dick

then wrote the formula, V = hwl. Can you state this formula

in words?

1. If possible, secure a supply of small wooden cubes, 1

inch on each side. With these build rectangular solids of

various sizes and verify Dick's rule.

2. Find the volume of each of the following.

a. h = 6 in., w = S in., Z = 10 in.

b. h = 12 ft., w=9 ft., 1 = 23 ft.

c. h=9 ft., m/ = 15 ft., 1 = 20 ft.

d. h = 3 ft. 6 in., w = 4 ft., 1 = 6 ft.

V=hwl

The volume of a rectangular solid is equal to the product

of its height, width, and length.



The Volume of a Cube 239

Dick also wrote a formula for the volume of a cube. Hethought, Since the length, width, and height of a cube are

all equal and each is called an edge of the cube, V=eX<= e\

Dick explained to his sister that the small 3, written to Hit-

right and above the e, shows that e is to be used as a factor

three times. ez

is just a short way of writing eXeXe. r is

read e cube, or the cube of e.

1. Read the following and find their value.

23

33

43

53

63

103

2. The squares and cubes of the numbers from 2 to 6, and

of 10, are used frequently. Make a table of these and then

memorize for future use.

3. How many cubic inches are there in a cubic foot? Howmany cubic feet in a cubic yard? Work these out, if you do

not know them, and then memorize for future use.

4. Find the volumes of the following.

a. e = 7in. c. e = 2ft. 5 in.

b. e=8ft. d. *=5ft., w=6ft., /= 10ft.

e. h = 2\ ft., w = 3h ft., l=4\ ft.

f. h = 2 ft. 3 in., w = 3 ft. 1 in., 1 = 4 ft. 7 in.

5. Mr. Mitchell has a bin 20 feet long, 8 feet wide, and

6 feet deep. How many bushels of wheat will the bin holdif one bushel occupies 1| cubic feet?

6. Mr. Mitchell's bin, problem 5, is now filled with wheat

to a depth of 4 feet. How many bushels are in the bin? Find

in two ways.

7. Mr. Mitchell wants to build a brick wall. The wall is

to be 40 ft. long, 18 in. thick, and 4 ft. high. How many bricks

will it take, allowing 22 bricks to a cubic foot? What will

the brick cost at $20 a thousand?



m&M*SK(B*^:

JProblems. Volumes of Rectangular Solids

Last winter Harry helped his father cut ice from their

pond They cut a rectangle of ice from the center of the pond,

65 feet long and 40 feet wide. The ice was 9 inches thick.

How many tons of ice did they get if 1 cubic foot of ice weighs

about 56j pounds?

2 Mrs. Cunningham bought 75 pounds of ice. Dick meas-

ured the piece and found it was 10 inches X 12 inches X 18

inches He figured the weight of the ice and told his mother

it was not full weight. How much underweight was the ice?

3 The coalbin in Mr. Cunningham's new house was 15 ft.

long and 6 ft. wide and was filled to a depth of 2 ft. Howmany cubic feet of coal were in the bin? If 1 cubic foot of coal

weighs 63 pounds, how many tons were in the bin? How much

did Mr. Cunningham have to pay the former owner of the

house for the coal if he paid $5.50 a ton?

4. The bin in problem 3 can be filled to a depth of 6 feet.

How many tons will it hold, when full?

5. A water tank is 12 ft. long, 2 ft. deep, and 3 ft. wide,

inside measurements. How many gallons of water will it

hold if 1 cubic foot equals l\ gallons?

6 A classroom is 36 ft. long, 28 ft. wide, and 11 ft. high

How many pupils will this room accommodate if each pupil

should have 231 cubic feet?

240



Problem Helps. Telling Hon to 241

Solve Problems

1. If you know the cos1 of an article and want to find

out how many of them you can buy for a certain amount,

what would you do?

2. How do you find the circumference of a circle if the

radius is known? Write a formula.

3. How do you find the radius of a circle if the diameter

is known? Write a formula.

4. How do you find a per cent of a number?

5. What measurements would you need to take to find

the capacity of a rectangular bin? How would you find the

volume? Write a formula.

, 6. What would you need to know and how would you

thWi proceed to find how many revolutions the wheels of an

automobile make in going a mile?

7. If you know the interest and the principal, how doyou find the amount? Write a formula.

8. If you know the principal and the amount of the inter-

est for one year, how do you find the rate of interest? Can

you write a formula?

9. Mr. McCoy is a dealer in used automobiles. He bought,

repaired, and sold a used Ford. What would you need to

know, and how would you proceed, to find what per cent

of the selling price he made as profit?

10. The length and width of a rectangle are known in

feet. How would you find the number of square inches in

the area? The number of square yards?

11. Harry wanted to find the area of a circular flower bed.

He measured the diameter. How did he find the area?

12. How would you find the number of cubic yards in a

bin if you knew the number of cubic feet?

13. How do you find the radius of a circle if the circumfer-

ence is known?



242 Problem Test

Solve each of the following problems and check.

1. Thomas is a member of his high school baseball team.

He made 12 safe hits last season and hit safely 2 times out

of every 5 times at bat. How often was he at bat?

2. Mr. Cunning-

ham's cistern is built

in the basement. It

is rectangular in shape

and is 10 feet long, 8

feet wide, and 6 feet

deep. How many bar-

rels of water will it

hold if 1 barrel equals

31| gallons?

3. One day Dick measured the water in the cistern. He

took apole and lowered it into the cistern until the end

touched the bottom. The pole was wet to a depth of 20

inches. How many gallons of water were in the cistern?

4. The Busy Bee Store held a July Clearance Sale during

which they offered a 20% discount on everything in the store.

During the sale Mrs. Conners bought the following: suit,

original price, $37.50; rug, original price, $125; curtains,

original price, $12.50; hat,original price, $7.50. Find the total

amount she paid.

5. Carpet is sold by the linear yard. Fractional parts of

a linear yard can be bought, but split widths cannot be pur-

chased. Find the cost of carpeting a room 14 ft. by 20 ft.,

with carpet 27 inches wide and costing $3.50 a linear yard,

running the carpet the long way of the room. How much

would it cost if the carpet is run the short way? Draw adia-

gram for each way.

6. Boy scouts should be able to estimate weights with an

error of not more than 25% of the true weight. Dick estimated

the weight of an iron bar to be 30 pounds. What was his

per cent of error if the true weight of the bar was 25 pounds?



a

Problem Test, Continued 243


1. A hollow brass tube is 12

inches long. The end is lj inches

square, outside measurements. The

opening is also square and 1 inch on

each side. What will the tube

weigh if 1 cubic inch of brass weighs

.3 lb.? Can you work this problem

in more than one way?

2. Mr. Peters paid $1.25 a cubic yard for an excavation

30 ft. wide, 45 ft. long, and A\ ft. deep. He sold the dirt for

45^ a cubic yard. How much did the excavation cost him

if the volume of dirt expanded 33J% in digging?

3. John Bowers & Company use gas in their factory. Last

month they burned 27,000 cubic feet. Find the cost accord-

ing to the schedule below if they took the discount.

Readiness-to-serve charge $1.00 per monthFirst 1,000 cu. ft. per month 1.20 per thousand



All over 20,000 cu. ft. per month 85 per thousand

Discount of 5^ per 1000 cubic feet for prompt payment.

4.

How muchdoes it cost to run an electric waffle iron for

1 hour if the iron uses 500 watts per hour and current costs

7^ per kilowatt hour?

5. Mr. Owens sells electric vacuum cleaners and receives

15% of the selling price as his commission. What must his

sales amount to in a week if he is to earn $45 a week?

6. Mr. Jones owned a cow that yielded 8950 pounds of

milk in one year, 3.6% of which was butterfat. His separator

removed about 96% of the butterfat from the milk. How

many pounds of butterfat did he obtain from this one cow in a

year? What was this worth if he received an average price

of 21 j£ a pound for the butterfat?



244 Diagnostic Tests in the FundamentalProcesses



without hurrying.

1.

2.

a.



Diagnostic Tests in IVrr«*iitaf;«* 245

LOCATING Vol K DIFFICULTIES

Work these ex

without hurrying.

1. a.

Write 23.7

as a per

cent.

e.

Write 850%

as a wholeor mixed

number.

imples and check. Work as f.t-t ae you can

2. a.

Write the

common

fraction

equivalentof 83§%.

e.

Write .0985

as a per

cent.

1>.

32 is what

% of 40?

f.

Find 73.4%

of $39.12.(Nearest

cent)

b.

8.7 is what

per cent of

9.3? (Nearest

tenth of aper cent)

f.

Write 212j%

as a wrhole or

mixed number.

c.

Write the%equivalent

off*if.

Change ^to a per

cent.

c.

Write 7.3%

as a

decimal.

f-r

-

d.

Writ,- L69%

as a decimal.

h.

Find the whole,

if

47.6%of it

equals S7.49.

(Nearest cent

d.

Find the whole,

if 300% of it

equals SI 74.

h.3

Find 6.27% Write If as a

of $92.18. percent.

(Nearest cent)




suggested below.

la — 40 le — 38 2a — 37 2e — 40

lb— 43 If — 42 2b— 43 2f — 38

lc — 37 lg— 41 2c — 39 2g— 42

Id— 39 lh— 44 2d— 44 2h— 44



your Score Card. Did you improve? Find the class average and

bring your graph up to date.



246 Chapter Test

1. Name one standard unit for each of the following: Length.

Area or surface. Volume. Capacity. Weight,

2. Name each of the figures given below.

3. State two facts about the opposite sides of a parallelogram.

4. Name each of the angles given below.

5. Write a formula for each of the following

a. Area of rectangle

b. Perimeter of rectangle

c. Area of square

d. Area of parallelogram

6. Write a formula for finding each of the following

e. Area of triangle

f. Area of trapezoid

g. Volume of rectangular solid

h. Volume of a cube

a. Area of a circle, when the radius is known.

b. Diameter, when the radius is known.c. Radius, when the diameter is known.

d. Circumference, when the diameter is known.

e. Circumference, when the radius is known.

f. Diameter, when the circumference is known.

g. Radius, when the circumference is known.



Chapter Test, Continued 247

7. Write true or false on your paper for each of the following

statements.

a. All rectangles are parallelogram -

1>. All parallelograms are rectangles.

c. AH rectangles are squares.

d. All squares are rectangles.

e. Alt rectangular solids are cubes.

f. A cube is a rectangular solid.

8. How many cubic inches arc there in a cubic foot? How

many cubic feet in acubic yard?

IsJZ. \9. Eugene wished to find the

\ area of the trapezoid shown a1 the

\ left. He measured the bases and one

5*\ of the sides. His solution is shown

\ below.

23 ft. \

6

A=JX12 (15+23) ^XJ^X 38 = 228 sq. ft.

Was Eugene's solution correct? Why?

10. The bottom of a clothes boiler has parallel sides and

semicircular ends (half a circle) as

shown in the figure. How many square

inches of material does it contain it

it is 21 inches long and 13 inches wide?Use7r =^.

11. A tin ring for the bottom of a cake pan is 8 inches in

diameter and has a hole in the center 2 inches in diameter.

Find the number of square inches of tin. (Find the number

of square inches of tin there would be if there were no hole in

the center. Then find the number of square inches of tin that were

cut out of the center to form the hole. Use iv = ^L

fL .)


a. 5X2+3=b. 5(2+3) =

c. 2X3+3X4 =



Roosevelt

McKinley

o Washingtono

ow

Adams

Wilson

Jefferson

| | | | | | |

T| | | | | | | | | | | | |

|



Z II A r I E R

8

Picturing lumbers. Qrapks

-ttES^asnjr *

You have probably heard the old saying that Seeing is

believing. Most people find it easier to understand numberfacts and relations when they are illustrated by pictures like

those on the opposite page. They are called bar graphs.

1. Mr. Anderson, superintendent of schools in Williams-

town, made the first graph, showing the number of pupils in

his elementary schools. Which school has the largest number

of pupils? Which has the smallest? How does the graph tell

this?

2. The numbers following the school names show that

there are approximately twice as many pupils in Adams School

as in the Jefferson. Does the graph show this? How?

3. Make similar comparisons for the McKinley and Adams

Schools. The Roosevelt and Jefferson. The McKinley and

Wilson. The McKinley and Jefferson.

4. How many pupils does one small square represent?

Five squares?

5. Agnes found the second graph in her geography. How

many millions of square miles are there in each continent?

6. Which is the largest continent? The smallest?

7. Asia is how many times as large as North America?

8. Compare the area of South America with that of Africa.

Europe with Africa. North America with Europe.

9. How many square miles does the height of a square

represent?

10. Bring to class as many examples of bar graphs as you

can find.

249



Directions for [Making liar Graphs

7. The unit in which the scale numbers arc expressed

should always be below or opposite the scale numbers. The

unitin the first

graph on page 250is

a square mile. Wh.n1

the unit in the second graph? In each graph on page 248?

a. The graph will be more useful if the number represented

by each bar is given. This may be placed within the bar as

shown in the second graph on page 250. On a horizontal bar

graph it is often placed to the left of the bar. On a vertical

bar graph it may be placed directly above the bar.

Problems. Making Bar Graphs

Make bar graphs to represent the following. You will find

it convenient to use graph paper which is ruled in squares.

1. On an arithmetic test Alice worked 8 examples cor-

rectly, Jane 9, Louise 6, Agnes 10, John 4, Harry 7, Robert 8,

Henry 10, and Tom 9. Let one square represent one example.

2. Louise Baker's final grades were: arithmetic 95, geogra-

phy 87, reading 72, spelling 80, history 65, manual training

97. Let one square represent a grade of 10. Why?

3. In 1875, Captain Webb of England swam the English

Channel in 21 hours 45 min-

utes. In 1926, Gertrude

Ederle of the United States

swam it in 14 hours 31 min-

utes. In 1927, Venceslas

Spacek of Bohemia swam it

in 10 hours 45 minutes.

4. Recent American men's

speed records for a mile are:

swimming, 21 min. 35.6 sec;walking, 6 min. 28 sec;

running, 4 min. 12 sec; skating, 2 min. 41.2 sec; riding a

bicycle, 2 min. 2 sec.

5. Make at least one original bar graph. Choose your own

subject and furnish the required information.



1,500,000

1,200,000

1,000,000

800,000

600,000

400,000

200,000

S2SJ35L

«

20Z281:

i I1911 1915 1919 1923 1927

GROWTH IN MEMBERSHIP OF BOY SCOUTS OF AMERICA

1943

Picture Graphs

Sometimes pictures are used instead of bars, to make the

graph more interesting. Two kinds are shown on this page.

1. In the graph at the top of the page, the heights of the

scouts show the relative sizes of the membership. Read from

the graph the total membership for each year shown.

2. The graph below shows that there were almost 14,500 aliens

naturalized in Chicago in 1932. How many individuals does each

figure represent? Explain the rest of the graph.

1932 A



SSIA

384,000,000 bu.)

ITED STATES

8,000,000 bu.)

NADA1,000,000 bu.)

)IA

1,000,000 bu.)

GENTINA

6,000,000 bu.)

\NCE

1,000,000 bu.)

STRALIA

3,000,000 bu.)



254 line Graphs

00 CO

6 r^00 00o> en

CNJ COCO CO

Years

ENROLLMENT IN ELEMENTARY

SCHOOLS OF WILLIAMSTOWN, 1930-41

Mr. Anderson wanted

to picture the changes in

the number of pupils in

his elementary schools

from 1930 to 1941. He

decided to draw a line

graph as shown at the

side.

1. The graph shows

that there were about660 pupils enrolled in the

elementary schools of Wil-

liamstown in 1930-31. Can

you estimate the enroll-

ment for each of the other

years shown?

1938 several rural districts2. During the summer of

decided to buy busses and transport their pupils to Williams-

town. What effect did this have on the enrollment?

3. A glance at the graph shows that the line, as a whole,

rises as it goes from left to right. What does this show about

the change in enrollment in the Williamstown schools?

4. If the enrollment had decreased, what would have been

true about the line? Can you find any years when there was

a decrease?

5. For which years was there no change in enrollment?

How does the graph show this?

6. A line graph is used to show changes. This graph pictures

the changes in enrollment from year to year. For which year

was the changethe greatest?

7. Which of the words, rising, descending, horizontal belongs

in each blank below?

a. A decrease is shown by a line.

b. No change is shown by a line.

c. An increase is shown by a line.



I sing Lino Graphs in School 255

1. Agnes made a score of ('>()'

joil

a spelling test, Monday. From the

graph read her scores on each of the

other days.

2. In general, did Agnes' score in-

crease or decrease during the week?

How much?

3. On which days did Agnes' score

improve over the preceding day?

Decrease? How can you tell this fromthe graph?

4. How much did her

prove on Friday?

100

90

80

70

60

50

40

30

20

10



90

80

60

40

20



1,391

59.3%



WHAT DO

24 HOURS

RACES OF PEOPLE

IN THE UNITED STATES

WHERE MY

MONEY GOES

B

23%

C

40%

D

21%

:.



GIRLS

16

59.3 %

BOYSli

40.7 %

DISTRIBUTION OF BOYS AND GIRLS IN THE

SEVENTH YEAR OF THE HOWARD SCHOOL

Making Rectangle Graphs

There were 16 girls and 11 boys in the seventh year of the

Howard School. Louise Henderson represented this by a

rectangle graph. She drew her rectangle 3 inches long. Since

the girls formed £f of the class, she took Jf of 3 inches and

found it was 1.78 inches approximately. Since this is just a

little more than If inches, she made the part of the rectangle

which represents the girls If inches long.

Draw rectangle graphs to represent the following facts.

1. Robert Sousa spent his allowance last week as follows:

moving pictures, 20^; candy, 10^;ice cream cone, 5£; saved,

15^. Make your rectangle 5 inches long.

2. In a recent, year about 46.3% of the farms in the United

States were operated by full owners, 10.4% by part owners,

.9% by managers, and 42.3% by tenants.

3. Make a rectangle graph showing the per cent of boys

and girls in your class.

4. Make a rectangle graph showing the distribution of the

enrollment in your school by grades.

5. Make an original rectangle graph. Choose your own

subject for this graph.

6. Find a rectangle graph in a newspaper or magazine.

262



The Meaning of ingle

Dick Cunningham road about angles in his Boy Seoul

Handbook. He had trouble understanding, so his Bcoutmaster

had Dick go through the exercises given below.

1. Face east. Turn to your left until you face north.

Through what part of a complete revolution have you turned?

One fourth of a complete revolution is called a right angle.

2. Face east. Turn to your left until you face between

east and north. Have you turned through more or less than

a right angle? What is such an angle called? (Page 213)

3. Face east. Turn to your left until you face west. Whatpart of a complete turn have you made? One half of a com-

plete revolution is called a straight angle.

4. Face east. Turn to your left until you face in a direc-

tion between north and west. Have you turned through more

or less than a right angle? More or less than a straight angle?

What is such an angle called? (Page 213)

5. The two lines marking the boundaries of the turning,

or angle, are called the sides. OX and OY are the sides of the

first angle below. Read the sides of the other angles.

6. The point about which the turning takes place is called

the vertex. Where is the vertex of each angle in problem 5?

7. The first angle in problem 5 is read Angle XOY, or Angle

YOX. The letter at the vertex is read in the middle. Read the

other angles.

8. Which angle in prob. 5 is a right angle? Acute? Obtuse?



D 90° 80°

£ ,o

<?

^B

°° A

Measuring Angles. A New Unit of Measure

You have learned that in measuring distances, areas, and

volumes, we must have a unit of measure which must be the

same kind of quantity as that being measured. The same is

true in measuring angles. The unit used in measuring angles

is a small angle called a degree. It is equal to fa of a right

angle. The sign ° is used for degree.

1. The figure shows a right angle divided into 90 equal

angles, each of which is 1°, or 1 degree. The sides of some of

the angles are not drawn clear to the vertex, as they would

be too close together. Find the following angles on the figure:

10° 30° 45° 60° 90° 57° 13° 1° 8° 79°

2. How many degrees are there in angle AOB in the figure?

AOC? AOD? BOC? BOD? COD?

3. How many degrees are there in a complete revolution?

In a straight angle? In an acute angle? An obtuse angle?

264



Using a Protractor to Measure Angles

In measuring distances it is convenient to have a straight

rule with units of distance marked along the edge. In measur-

ing angles it is convenient to have a circular rule with degrees

marked along the edge. Such a rule is called a protractor. It

may be a complete circle, a half circle, or a quarter circle. The

half circle

form shown aboveis

the commonest.To measure an angle, such as angle AOB above, place the

protractor with the center (marked by arrow) on the vertex

of the angle. Turn the protractor until one side of the angle

passes through 0°. Note where the other side of the angle

crosses the protractor. The angle AOB contains 35°.

1. Edward Thompson measured the angle AOB and said

it contained 145°. Do you see what mistake he made?2. Draw a number of angles on a piece of paper. Make

some less and some greater than a right angle. Measure them

with a protractor.

3. Using a protractor, draw an angle of 15°, 30°, 45°, 82°,

135°, 168°.

265



u*

jjBHf

WEIGHT OF PUPILS IN WASHINGTON SCHOOL

Making Circle Graphs

The pupils of the Washington School were given a physical

examination. Twenty per cent were underweight, 15% were

overweight, andthe rest were normal. Eugene Sanders made

a circle graph to show these facts. He proceeded as described

below. Study and explain,

a. He drew a circle.

_2o_X 360° = |X360° =72°

3^X360° = 23oX360° =54°

With his protractor he drew an angle equal to 72°, with

the vertex at the center of the circle.

Next to this he drew an angle equal to 54°.

Why did Eugene take-flft

of 360°? £& of 360°?

Draw a circle graph to illustrate the data given in prob-

lem 3, page 253.

3. Make circle graphs to show the following facts.

In 1930-31 the United States produced 1,932,000 bales

of cotton; India, 4,033,000; Egypt, 1,715,000; China,

2,250,000; Brazil, 455,000; Russia, 1,550,000.

Of the 12,049,000 radio sets in the United States in one

year, it is estimated that 1,371,000 were on farms.

Make an original circle graph. Choose your own subject.

b.

c.

d.

e.

1.

2.

a.

b.

4.

266



Choosing the Type <>f Graph 267

The kind of graph to be used depends on the nature of the

facts to be pictured. The graphs you have studied may t>e

used for many purposes, the most important being:

a. To show the changes in some quantity—how it increa e

and decreases. The line graph is best for this purpose,

but the bar and picture graphs are also used.

b. To show the relative sizes of two or more quantities.

The bar and picture graphs are best suited for this.

c. To show how the different parts of some whole compare

in size with the whole and with each other. Circle andrectangle graphs show this best.

1. The topics given below might be pictured graphically.

Which type of graph would you use to represent each? Could

you use more than one kind for any of them?

a. To show the changes in the number of airplanes manu-

factured in the United States each year for the last ten

years.

b. To compare the number of airplanes manufactured in

the United States last year with the number manufac-

tured in Great Britain.

c. To show what part of the total number of airplanes

manufactured in the world last year were made in the

United States, in Great Britain, and in the rest of the

world.

2. Name three sets of facts that might be pictured by a

circle graph. A line graph. A picture graph. A bar graph.

A rectangle graph.

3. Make a brief statement telling for what each type of

graph mentioned in the preceding question may be used.

Canany be used for more than one purpose?

4. According to a recent survey 25% of accidents to chil-

dren occur at home, 20% in school, 20% on school grounds.

8% on way to and from school, and 27% in other places.

Illustrate with a graph. What type might you use? Which

do you prefer? Give your reasons.



268 Making Graphs

IS THE WORLD GETTING SMALLER?

Draw a graph to illustrate each of the following. Choose the

type y°u think most suitable.

1. In 1519, Magellan sailed around the world in 1083 days.

Since then the following records have been made for circling

the globe, by different routes:

Charles Fitzmorris (1901) 60 £ days

J. W. Sayre (1903) 54i days

Henry Fredericks (1903) 54 £ days

Col. Brunlay-Campbell (1907) 40 1 days

Andre Jaeger-Smith (191 1) 39f days

J. H. Mears (1913) 36 days

E. S. Evans and L. Wells (1926) 28} days

Capt. Collyer and J, H. Mears (1928) 23* days

Graf Zeppelin, dirigible balloon (1929) 20£ days

Wiley Post, in monoplane Winnie Mae (1933) .... 7| days

Howard Hughes, in Lockheed monoplane (1938) . . 3| days

Capt. James W. Chapman, Jr., in Army plane (1941) 5 days

2. The first American ship to use steam in crossing the Atlanticwas the Savannah, a sailing vessel with auxiliary steam. She

crossed in 1819 in 26 days, during 18 of which she used steam.

The principal steamship records since then are:

Great Western (1838) 10} days

Persia (1856) 9 days

Scotia (1866) 8 days

Arizona (1880) 7\ days

Alaska (1882) 6| days

Etruria (1888) 6 days

Majestic (1891) 5f days

Lucania (1894) 5$ days

Lusitania (1909) 4} days

Rex (1933) 4} days

Queen Mary (1938) 3f days

3. Times for transatlantic airplane flights:

Lindbergh 33 hr. 30 min.

Chamberlin and Levine 42 hr. 31 min.Byrd, Neville, Acosta, Balden 46 hr. 6 min.

Williams and Yancey 30 hr. 30 min.

Amelia Earhart Putnam 14 hr. 56 min.

Mattern and Griffin 18 hr. 40 min.

Felix Wilkus 23 hr. 15 min.

Howard Hughes 16 hr. 38 min.

American-built bomber (1942) 6 hr. 40 min.



Making Graphs 269

CAUSES OF AUTOMOBILE ACCIDENTS

The number of persons injured and killed by automobiles greatly

increased in the years preceding World WarII.

Aknowledge of

the causes of accidents will help you to avoid them. Select facts

from the tables below to make three graph8.

1. Actions of Pedestrians Resulting in Deaths and Injuries.

Crossing at intersection:

With signal

Against signalNo signal

DiagonallyCrossing between intersections

Waiting for, getting on or off streetcar

Getting on or off other vehicle

Children playing in street

At work in road

Hiding or hitching on vehicle

Coming from behind parked car

Walking on rural highwayMiscellaneous

Total

Pedesl riane

Killed

190

7402,150

210

3,95040

140

650500

120

8902,270

650

12,500

Per< en1

1.5

5.917.2

1731.6

.3

1.1

5.L'

4.0

1.0

7.1

18.2

5.2

100.0

Pedeatriani

[njured

i:;.::mo

:il, >r.o

18,210

3,490

74,630

1,460

3,19034.S50

7,840

3,490

38,910

11,330

13,060

290,400

< '.lit

5 3

1 1

,916 B

i 2

25.7

.5

1.1

12.0

2.7

1.2

13.4

3.9

4.5

100.0

2. Actions of Drivers Resulting in Deaths



270 Problem Helps. Telling How to

Solve Problems

1. How do you find the area of a square? Write a formula.

2. How do you find the area of a triangle? Write aformula.

3. If you know the rate of interest, the principal, and

the time, how do you find the interest? Write a formula.

4. If you know the principal and the amount of interest

for one year, how do you find the rate of interest?

5. What is a trapezoid? How do you find its area? Write

a formula.

6. How do you find per cent of increase? Of decrease?

7. How do you find the diameter of a circle if the circum-

ference is known? Write a formula.

8. How do you find the diameter of a circle if the radius

is known? Write a formula.

9. What measurements would you need to take to find

the number of cubic inches in a box in the shape of a cube?

How would you find the volume? Write a formula.

10. If you know the length and width of a rectangular

field, in rods, how do you find the number of acres? If the

length and width are in feet, how would you proceed?

11. The drive wheels of

a railroad engine are mak-

ing 200 revolutions a min-

ute. What else would you

need to know, and how

would you proceed, to find

the speed of the train?

12. If you know whatan automobile cost and what it is now worth, how do you

find the per cent of depreciation?

13. If you know the number of games won by a ball team

and the number lost, how do you find what per cent of the

number played was won?



Problem Helps. Choosing the Facts

^ on No«hI

In each problem one or more facts arc given that .'ire not

needed in the solution of the problem. Decide which facts

you need and solve the problem. Check.

1. Mrs. Sanders used

two gas burners for 5 hours

in canning 12 quarts of

beans. Each burner con-

sumed 6 cubic feet of gas

per hour. How much did

the gas cost at $1 per thou-

sand cubic feet?

2. A $1700 note, due in

90 days with interest at

6%, is paid in 30 days. Find

the amount paid.

3. The McManns expect to spend 20% of their income for

rent, 25% for food, 20% for clothing, and 20% for all other

expenses. What should their income be to justify $50 a month

for rent?

4. During one month a farmer sent 8000 pounds of milk

to a creamery. The milk contained 304 lbs. butterfat andtested 3.8 per cent butterfat, on the average. How much

did he receive if he was paid 21^ a pound for the butterfat?

5. Mr. Smith sells automobiles. He receives a commission

of 10% on his sales. The car he sells has a 6 cylinder, 60

horsepower engine, and a 116 inch wheel base. It sells for

$950. How much does he receive on each car he sells?

6. A driveway 50 ft. long and 8 ft. wide is to be covered

with crushed stone to a depth of 10 inches. Find the cost

at $2 a square yard.

7. The bases of a trapezoid are 6 ft. and 12 ft. Each of

the other sides is 5 ft. The altitude is 4 ft. Find the area.



*

Problem Helps. Supplying the Missing Facts

In each problem below one or more needed facts are missing.

Look up the facts needed, or have your class agree on probable

figures, and solve the problems. Check.

1. In 1935, Campbell drove an automobile at Salt Lake,

Utah, at the average speed of 484.94 kilometers an hour.

This is equivalent to how many miles an hour?

2. The light in the lighthouse at Fire Island, N. Y., can

be seen at a distance of19 nautical miles in clear weather.

This is how many statute miles?

3. On June 30, the Brown's gas meter read 57,890 cubic

feet. What was their bill for June at $1.10 per thousand

cubic feet?

4. In 1933, Gar A. Wood drove his motorboat, the Miss

America X, at an average speed of 72.98 knots an hour. This

was how many statute miles an hour?

5. Mr. McAndrews owns a store. Last year his sales totaled

$12,239. The goods he sold cost him $7,813. Find his profit

for the year.

6. A note for $2500, with interest at 7%, is paid 30 days

before it falls due. Find the amount due.

272



Problem Helps. Making Problems 273

Enough data are given in each list below to enable you to

find one or more facts that arc not given. Make one

problem out of each list. Solve each of your problems andcheck.

1. The pupils of the Longfellow School held a sch<x)l fair

one Friday afternoon. The seventh grade pupils gave a play.

The fair lasted 3 hours and 15 minutes, and the play took 45

minutes. They could seat 50 pupils at each performance, and

they charged 5^ admission.

2. A farmer shipped five steers to Chicago. They weighed1280 lb., 1135 lb., 1076 lb., 1217 lb., and 1179 lb. when shipped.

The shrinkage in weight due to transportation was 3.2%.

The farmer sold them at 4j^ a pound.

3. Mr. Owen drove his automobile 79 miles in 2\ hours.

4. A farmer had a bin full of potatoes. The bin was 4 ft.

by 8 ft. by 5 ft. A bushel occupies about \\ cubic feet.

5. A train makes the trip from Buffalo to New York, a

distance of 398 miles, in 11 hours and 45 minutes. Of this

time 15 minutes is taken up in making stops.

6. Mr. Mason paid $1725 for an automobile and drove it

42,000 miles during the first year. He spent $85.50 for tires,

$57.40 for repairs, and $630.64 for gas and oil. At the end of

the year he sold the car for $650. The money invested in the

car could have been loaned at 6% interest.

7. A field is 125 rods long and 70 rods wide. A strip 4 feet

wide all around the field next to the fence is not cultivated.

8. The side of a square field is 100 rods long.

9. Five years ago Mr. West paid $4500 for a house and lot

in Smithville. Since then property in Smithville has increased

about

20%in value.

10. In a recent year there were approximately 45 million

automobiles registered in the entire world. Of these 30 million

were in the United States.

11. Mr. Simmons sold $1500 worth of goods in December.

His sales in January were 15% less than in December.



274 Problem Test

1. Draw a picture graph

to show the relative heights

of the following: EmpireState Building, New York,

1248 ft. ; Chrysler Building,

New York, 1046 ft.; Eiffel

Tower, Paris, 1000 ft.;

Great Pyramid, Egypt,

480 ft.

2. Last summer The

Busy Bee Store held a July

Clearance Sale during

which they offered a 25%

discount on everything in the store. Following are the original

prices of the things Mrs. Conners bought during the sale: suit,

$37.50; rug, $125; curtains, $12.50; hat, $7.50. Find the total

amount she paid.

3. Copy the following table and complete. Use x 227 '

Radius

12 ft.

Diameter Circumference Area

5 mi.

88 yd.

4. In 1920, there were 20,092,000 horses on farms in the United

States, while' in 1940, there were only 10,616,000. Find to the

nearest tenth of a per cent, the per cent of decrease. How do you

account for this decrease?

5. Mr. Norris sells furniture at such prices that 65% of the

selling price equals the cost. At what will he sell a chair that

costs him $8.25? Draw a diagram.

6. Mr. Simmons owns a store. The expenses of running the

store average 25% of the sales. At what must he sell an article

which costs him $8.50, in order to make a profit of 10% of the

selling price? Draw a diagram.



Diagnostic Tests in iln* FundamentalProcesses

275

LOCATING YOUR DIFFH IULTIES

Work as fasl as you canork these examples and check

without hurrying.

1. a.

2. a.

e.

3 v «






without hurryin



Chapter IVsi 27

Write the numbers 1 to 12 on a piece of paper. After the e

numbers write the words, or numbers, which belong in ) 1 it-


1. Bar and picture graphs are used to show the relative

of two or more quantities.

id

2. Circle and rectangle graphs show how different parts

of a whole compare in size to each other and to the(2)

3. Line graphs are used to show how a quantity(3)

How it or(4) (5)

4. The lengths of the great ocean steamships of recent

years were: Majestic, 915 feet; Berengaria, 884 feet; Bremen,

899 feet; Rex, 880 feet; Europa, 890 feet; Leviathan, 908

feet; Olympic, 852 feet; Aquitania, 869 feet; Normandie, 981

feet; Queen Mary, 975 feet. To illustrate these facts, I would

draw either a graph, or a graph.(6) (7)

5. For safe driving you should know how quickly you can

stop your car. In the graph below, each black car represents

one car length you travel after you decide to stop and before

you place your foot on the brake. Each white car represents

one car length you travel after you apply the brakes. The

graph shows that when driving 30 miles an hour, you travel

car lengths while thinking and more while applying(8) (9,

the brakes, or car lengths before you stop.

(10)

•20 4»p 4Cp C=>

30 mm 0K+ C^ C^> «>

50^^ ** «* *2p C=> C^> C^> C^? C^? C=^ £^p C=S

'Milesper hour

DISTANCE REQUIRED FOR STOPPING AN AUTOMOBILE

6. The graph above is called a graph.(id

7. Every graph should have a title. The title of the graph

above is .

(12)



c ii a p r E it

Using geometry

One afternoon Dick Cunningham stopped to watch a sur-

veyor making a map of a city lot. To locate a large tree in

the proper place on his map, the surveyor measured the dis-

tance and direction of the tree from the corner of the lot.

In measuring direction, he used an instrument called a transit.

This is a telescope mounted over a circular protractor on a

tripod. A transit is used to measure angles in surveying.

The angle is measured on the protractor, and the telescope

enables the surveyor to see a greater distance.

The surveyor first placed the transit over the corner of the

lot A, as shown above, and sighted through the telescope at

the tree B. Next he turned the telescope and sighted along

the edge of the lot at a pole C. He then read on the protractor

the angle through which the telescope had turned. This

angle measured the difference in the directions of B and C

from A. Whenever it is necessary to determine direction or

differences in direction, we must measure some angle. Most

work in surveying is based on the measurement of angles and

distances.

Angles and other geometrical figures such as squares,

rectangles, parallelograms, triangles, and circles are used in

architecture, landscape gardening, engineering, art, and in-

dustry. In this chapter you will learn how to construct these

figures and some important facts about them. You will find

these facts useful in many ways.



JUUDp DDpr

LJLJUUL|U

JoanDODO~1DI

aJDDOI

LCioriripinr-inr

Making Geometrical Designs

Thousands of years ago the Babylonians, Egyptians, and

others used geometric forms and patterns in decorating floors,

walls, vases, cloth, and other articles. Today geometric forms

are even more widely used for decorative purposes.

Many designs can be made by the use of paper ruled in

squares. Secure several sheets ruled in squares about an

eighth of an inch on a side.

1. The first design illustrated above shows a popular old

quilt pattern, called the Double Irish Chain. Find examples

of other old quilt patterns using geometric figures.

2. Can you find any other examples of designs using

squares, rectangles, and triangles? Examine wallpapers, lino-

leums, tile floors, towels, tablecloths, luncheon sets, lace

embroidery, bedspreads, curtains, carpets, baskets, book

covers, and so on. You will also find many illustrations in

magazines. Bring as many examples as you can to school.

Copy on squared paper some of the designs you find.

3. Make at least one original design using squares, rec-

tangles, and triangles. Make your design for some definite

purpose, such as a border for a book, or a linoleum pattern.

280



I.

^

\

Making Geometrical Designs

Simple geometric designs can be made with squared paper; hut

before you can make one like the cathedral window above, you

must know how to make geometrical const ructions with drawing

instruments. You will need:

A straightedge and rule to be used in drawing straight lines and

measuring distances.

A pair of compasses to be used in drawing circles.

A protractor to be used in measuring angles.

A draftsman's triangle to be used in drawing perpendiculars.

1

1

)

1

1

1

1

1

1 1

11

1

1

1 1

1

1

1

1

I

1



282 Vertical Angles

AN IMPORTANT GEOMETRICAL FACT

Study this page and the next and you will discover two im-

portant facts about angles that are often used by carpenters,

engineers, and many other people. If you have forgotten about

angles, review pages 213 and 263.

1. Draw three straight lines through the same point, as

shown in the figure. Meas-

ure each of the six angles

formed. Add as a check

to your measurements. What

should the sum equal? Did

it come out exactly correct?

Why?

2. Draw a straight line as AB. Draw three other lines

meeting it at some point as

C. Measure each of the

four angles formed. Add as

a cheek on your measure-

ments. What should the sum

equal? Why?

3. Draw two intersecting straight lines as in the figure.

Measure angles aand b.

What do you find? Meas-

ure angles c and d. What

do you find about these

angles?

4. Angles such as a and b, or c and d, in problem 3, are

called vertical angles. Make a statement about the sizes of

two vertical angles.

5. Verify your statement by drawing another pair of inter-

secting lines and measuring the vertical angles.

Vertical angles are equal.



The Sum of the Angles of a Triangle 283

One of the most important facts about triangles ha to do

with the sum of the three angles. You ran discover wli.n

the sum is by studying this pa.u*

Draw a large triangle on paper. Measure each of the three

angles and add. Draw at least two other triangles of diffi

sizes and shapes and find the sum of the three angles of each.

What do you find about the sum?

If your triangles were carefully drawn and your measure-

ments accurate, the sum of the three angles of each triangle

was about 180°. It is proved in geometry that this sum is

exactly 180°. This is one of the most important facts in

geometry, being frequently used by architects, surveyors,

carpenters, and other practical men.

1. A straight line is what part of a complete revolution?

How many right angles equal a straight angle? How many

degrees?

2. Draw a large triangle on paper and cut it out. Tear

off the three corners and put the three angles together so as

to get their sum. What kind of angle does the sum seem to

be? Test it with the edge of your ruler. Draw another triangle

of a different size and shape and repeat the experiment. Is

the sum always the same?

3. Make a general statement concerning the sum of three

angles of a triangle. Compare your statement to the one at

the bottom of the page.

4. Two angles of a triangle are as given below. Find the

third angle in each triangle.

a. 112°, 51°. b. 75°, 27°. c. 60°, 60°. d. 50°, 50°.

e. 30°, 60°. f. 45°, 45°. g. 37°, 81°. h. 128°, 23°.

i. 40°, 50°. j. 30°, 30°. k. 110°, 40°. 1. 70°, 50°.

The sum of the three angles of any triangle is 180°, or a

straight angle.



284 Problems. Using the Angle Sum

1„ Dick Cunningham watched a surveyor measuring the

three angles of a triangular field. After he had measured

angle A and angle B, Dick said he

knew how to find angle C without

measuring. He said that angle C was

46°. Was he right? How did he get

\£ this result?

2. The surveyor told Dick that he always measured all

three angles in order to check on his measurements. What

was his method of checking?

3. Draw a triangle and measure the angles. Check by

adding. Can you expect the sum to be exactly 180°? Why?

4. If one angle of a triangle is a right angle, the triangle

is a right triangle. How many degrees are there in the sum

of the other two angles? What kind of angles are they?

5. Could a triangle have angles of the following sizes?

Why?

48°, 63°, 69c

b. 78°, 81°, 32c

c. 57°, 95°, 20°

6. How many right angles can there be in a triangle?

How many obtuse angles? Why?

7.

Twolines intersect, as shown below. Angle a is equal

to 37°. What is the sum of angle a and

angle b? Without measuring, find angle

b. What is the sum of angle b and angle

c? Without measuring, find angle c.

Find angle d in the same way. How

can you check your results? What

ought to be true about angle a and

angle c? Angle b and angle d?

8. How many degrees are there in

angle a, in the figure at the left? In

angle c? Why? In angle b? Why? In

angle d? Why? Check.



Bisecting a I me 285

Tc

In the construction of many designs u is necessary to find

the mid-point of a line or to divide it into two equal p

This is called bisectingthe

line.

Method 1. Measure the line and divide the length by '1.

Measure this distance from either end. If the line is 5 in.'

the mid-point will be 2\ in. from either end.

Method 2. To bisect the v . >

line AB draw an arc, or part

of a circle, with A as a center

and a radius longer than half

of AB. Which arc is this in

the figure? With B as center

and the same radius, draw the A

arc of a second circle. This

will cut the first arc at C and

D. Draw a straight line through

C and D. The point E, wherethis line meets the line AB,

is the mid-point of AB.

I. Draw a straight line 9 inches long. Bisect, using Method

1. Check with your compass. To do this, set one point of

the compass on one end of the line and the other on the center.

Move compass to the other end of the line. Are the two

parts equal?

2. To construct this design accurately,

you must first bisect the radius of the

circle. Do you see how the design was

constructed? Draw a circle and construct a

similar design about twice as large.

3. Edgar Jones wanted to bisect the side ofa square. Hefound that the side was a little more than 3<r inches long.

Could he bisect it accurately by the first method?

4. Draw three lines of different lengths without measuring.

Bisect each by the second method. Check by using your

compass to see if the two parts are equal.



4- ^-- %k«~: & ^—°§£

-:^:;

.-i

\

OfN/NC/ HALL

Scale: ^ inch ^ too feet

i

\ 7

\

Reading Scale Drawings

Dick spent two weeks in a boy scout camp last summer.

From an illustrated map of the camp he found the distance

of his tent from the flagpole. Do you know how he did this?

A map is a scale drawing. The real distance is represented

by a shorter distance on the map. What real distance does

J inch on the map ofthe

camprepresent? What distance

would | inch represent? 1 inch? lj inches?

1. Measure the dotted line from Dick's tent to the flag-

pole on the map. What was the real distance?

2. Find the distance from the flagpole to each of the other

points given on the map.

3. How far was Dick's tent from the dining hall? From

the spring? From the gate?

4. On an automobile map of Michigan, \ inch represents

10 miles. How far is it from Detroit to Lansing, the capitol

of the state, if the distance on the map is 8j inches?

5. On a large map of Ohio, 1 inch represents 13 miles. Find

the actual distances between cities if the distances on the

mapare: Cleveland to Toledo, 7 in.; Columbus to Toledo,

8J-in.; Columbus to Cincinnati, 7f in.; Toledo to Cincinnati,

14 in.; Cleveland to Columbus, 9j in.

6. Get a map of your state. Find the scale. This is usually

given on the map. Measure the distances between the prin-

cipal cities and find these distances in miles.

28G



J

Making Scale Drawings

Plans of rooms and buildings, as well as maps, are scale

drawings. On such plans and maps, shapes and directions

are the same as the real object pictured, but distances are

reduced in size according to some scale.

1. In drawing a map, using \ inch to represent 100 miles,

how long would you make the distance between two points

on the map if the actual distance is 200 miles? If it is 500

miles? 750 miles?

2. Dick made a scale drawing of his schoolroom, using

\ inch to represent 5 feet. The room was a rectangle 20 feet

by 30 feet. How long was his drawing? How wide?

3. Make a scale drawing of Dick's schoolroom. Choose

your own scale. Use your protractor or a draftsman's triangle

to make the right angles.

4. Measure your schoolroom and make a scale drawing.

Use a scale so that the finished drawing will be fairly large

(about 10 inches long). Draw the principal articles in the

room on your map, to scale and in the proper places.

5. A surveyor measured two

sides and an angle of a triangular

field and made the rough sketch

shown at the right. Make a

scale drawing, using \ inch to

represent 100 feet. From your

drawing, find the length of the

third side of the field.

287



Locating Points

Dick read a book about a hunt for buried treasure. The

boy in the story found an old scrap of paper giving directions

for finding gold hidden by pirates. The directions read:

Stand at the foot of the large elm tree and face the high rock.

Turn 40° to the left and walk straight ahead. Stand at the

foot of the rock and face the elm. Turn 60° to the right and

walk straight ahead. Dig where the two paths meet. It is 500

feet from the elm to the rock.

1. Dick made a scale drawing showing the location of the

treasure. Can you make such a drawing?

2. Dick andhis friends played a game which they called

Buried Treasure. They chose one of their group to be the

pirate who buried the treasure and wrote a set of directions

for finding it similar to those given above. The first to find

the treasure became the pirate. Play this game at home or

on the school grounds. Estimate the angles as accurately

as you can.

288



y&

Locating Points on Maps and Plans

In locating objects in map making, surveyors often use

the method described on page 288. Make scale drawings or

maps of the following fields. Choose your own scale.

1. A rectangular field, 40 rods by 85 rods, the long sides

running north and south. North on your map should be at

the top. From the southwest corner of the field the lines of

sight to the various objects in the field make the following

angles with the south side of the field: to walnut tree, 85°;

to maple tree, 68°; to large stone, 60°; to spring, 43°. From

the southeast corner the lines to the same objects make the

following angles with the south side of the field: to walnut

tree, 45°; to maple tree, 82°; to stone, 25°; to spring, 76°.

2. A lot is shaped like the figure at the right.

AB is 170 feet and BC is 60 feet. Angle A is

75° and Angle B and Angle C are each right

angles. From A the lines of sight of the different

trees on the lot make the following angles with

AD: to oak tree, 70°; to elm tree, 65°; to maple

tree, 40°. From D the lines of sight make the

following angles with AD: to oak tree, 20°; to

elm tree, 75°; to maple tree, 85°.

3. If possible, make a scale drawing, or

map, of some field or lot near your school.

289



Dividing the Circle into Six Equal Parts

.J

Many beautiful designs can be

made by dividing a circle into six

equal parts. The method is illustrated

in the figure at the left.

A is any point on the circumfer-

ence. The arc cutting the circle at

B is drawn with A as a center and a

radius equal to AO, the radius of the

circle. Arcs at C, D, E, F, and A

are drawn in the same way. If your

work is well done, the last arc will

cut the circle at A.

1. The six-sided figure, ABCDEF,

in the figure above, is called a regular hexagon. Construct a

regular hexagon in a circle with a radius of lj inches.

2. The figures at the

left show the method of

constructing a six-pointed

star. Construct such a

star, starting with a cir-

cle 4 inches in diameter.

3. Use your compasses and construct a design similar to

the church window at the top of the page.



Dividing a Circle into Three Equal Tarts 291

Can you divide a circle into three equal parts? Agnes did it

by first dividing the circle into six equal parts and takingevery other

point of division.

1. Can you make a design similar to

the one at the left? Start by drawing a

large circle and dividing it into three equal

parts. When you have finished, erase all

but the heavy lines in the drawing youhave made.

2. Copy this design for a circular win-

dow. When you have finished, erase all

but the heavy lines, as you did in exercise

1. Make your drawing about twice as

large as the one in the book.

Drawing an Equilateral Triangle

A triangle whose sides are all equal

in length is called an equilateral tri-

angle. The method of constructing

such a triangle is shown at the right.

The two arcs through C are drawn with

a radius equal to AB and with A and

Bas centers.

1. Draw equilateral triangles whose

sides are each lj inches, 2 inches, 2\

inches.

2. Can you copy the designs at the

right? Make them about twice as

large as those in the book.

3. The three angles of an equi-

lateral triangle are all equal. What is

the sum of the three angles? Howmany degrees are there in each angle?

Check with your protractor.



292 Drawing Perpendiculars

STRAIGHT EDGE

Two lines that meet at right angles are said to be perpen-

dicular to each other.

Method1. To draw a

perpendicular to the line AB

at C, place a straightedge

along AB. Place a drafts-

man's triangle as shown in

-^.the figure. Draw CD. If

_Jyou do not have a drafts-

man's triangle, you can eas-

ily make one by following

the directions given in prob-

lem 1 below.

Method 2. Lay a pro-

tractor on AB as shown in

Bthe figure. Make a mark on

thepaper at D opposite

90°. Remove the protractor

and draw a line through C and D.

1 Draw a straight line on a piece of cardboard. At one

end draw a second line perpendicular to the first. Use your

protractor. Locate a point on each side of the right angle

and 4 inches from the vertex. Draw a straight line through

these points. Cut out the right triangle formedIf you do

not have a draftsman's triangle, you may use this, this is

called a 45° right triangle. Why?

2. Draw a square 3 inches on each side,

triangle to draw the angles.

3. Draw a rectangle 5 inches by 2^

inches.

4. Make a scale drawing showing the

floor plan of a room 12 feet by 16 ft. with

a rug 9 ft. by 12 ft.

5. Draw a square about 4 inches on

each side and copy the design at the right.

Use a draftsman's



I>ri»w in^ Perpendiculars

USING COMPASSES AND STRAIGHTEDGE

Julia is studying geometry in the senior high school. Shelearned how to draw a perpendicular by using her com]

Her method is given below.

With C as a center and any

convenient radius, draw two arcs

cutting AB at E and F. With

E and F as centers, and a radius

greater than EC, draw two arcs

intersecting at D. Draw a line

through C and D.C

E F

drawing perpendiculars,

by using a draftsman's

1. Practice on this method of

Check the accuracy of your work

triangle or a protractor.

2. Julia's brother Ned wished

to measure the distance from the

point P to the line AB. He

measured the distance PC. Was

this right? Why?

3. Julia told Ned that the dis-

tance from a point to a line al- a ' B

ways means the shortest distance,and must be measured on the perpendicular, PD. Measure

PC and the length of the perpendicular. Which is longer?

4. Draw a straight line, AB. At D, any point on AB, draw

a perpendicular to AB. Mark any point, P, on the perpen-

dicular. Draw several other lines from P to AB. Measure

the distance from P to AB, along the perpendicular. Along

each of the other lines. What do you find?

The shortest distance from a point to a line is the perpen-

dicular distance.



294 Bisecting an Angle

MAKING A 30°-60° RIGHT ANGLE

The construction of many beautiful designs, and many

other practical problems depends on dividing an angle into

two equal parts. This is called bisecting the angle. Two

methods are given below.

Method 1. Measure the angle

with your protractor. Divide

by 2. Make a mark on the

paper opposite this division on

the protractor (D in figure).

Remove protractor and draw a

line through this point and the

vertex (BD in figure).

Method 2. With B as a

center and any convenient radius

draw an arc cutting the sides

of the angle at M and N. WithM and N as centers, and a

radius greater than half the dis-

tance between M and N, draw

arcs intersecting at D. Draw a

line through B and D.

1. Draw several angles. Bisect each. Use the first method

on about half of them and the second on the rest.

2. Construct an angle of 45° by bisecting a right angle.

3. On a piece of cardboard B

draw an equilateral triangle 5

inches on each side. How many

degrees are there in each angle?

Bisect one of the angles, as angle

BAC in the figure.

4. Cut out triangles ADC and

ADB. How many degrees are inA

angle a?. Why? In angle b? Why? In angle c? Why? Save

these triangles and use them as draftsman's triangles. These

are called 30°-60° right triangles.



Dividing a Circle into 4, 8, and 12 295

Equal Parts

Many beautiful designs are made by dividing a circle into

4, 8, and 12 equal parts. To divide a circle into 4 equal parts,

draw any diameter and then draw a line perpendicular to it

through the center.

How would you divide a circle into 8 equal parts? Into 12

equal parts?

1. Copy the following designs. Erase the dotted lines

when you have finished.

Dividing a Circle into Five Equal Parts

THE FIVE-POINTED STAR OF OLD GLORY

The stars on the United States flag are five-pointed. To

draw such stars, you must divide a circle into five equal parts.

This cannot be done easily with compasses but can be done

approximately by using a protractor. A

If a circle is divided into five equal

parts, each angle at the center will be

i of 360°, or 72°. Draw any radius as

OA. With your protractor draw angle

AOB equal to 72°. Draw angle BOCequal to 72°, and so on.

1. Divide a circle into five equal

parts. Draw lines to join alternate

points, making a five-pointed star, as in

the figure.



296 Drawing Parallel Lines

You have learned that two lines which are everywhere the

same distance apart are called parallel lines. Many designs

and constructions depend upon knowing howto draw parallel

lines.I—|c d

Method 1. Parallel

lines can be drawn

with a straightedge

and draftsman's tri-

angle as shown in the

first figure.

Method 2. Dick

drew parallel lines by

using two draftsman's

triangles as shown in

the second figure. The

distance between the

parallel lines can be

changed by sliding one

triangle along the

other.

Method 3. To

draw a line through P

(third figure) parallel

to AB, draw any line

through P crossing

AB at C. Measure the

angle BCP (40° in the

figure). Move the cen-

ter of the protractor to P, and locate point D so angle DPti

is equal to angle BCP. Removeprotractor and draw a line

through D and P.

1. Reproduce the border designs shown below. Make an

original border design, using parallel lines.



Important Farts about Parallel Lines 29:

Agnes and Dick made a list of all of the importanl fad

they knew about parallel lines. They discovered the second

fact given below from the first method of drawing parallel

lines given on page 296.

Dick's teacher told

him that pairs of angles

such as angle a and

angle b, and angle c

and angle d, are called

alternate interior angles.Do you see why they

are called alternate

angles? Why they are

called interior angles?

Dick's teacher said that line EF is called a transversal.

Why? Study the second and third methods of drawing parallel

lines. What can you discover about alternate interior anglesformed by a transversal cutting two parallel lines?

1. Draw two parallel lines by Method 1. Draw any trans-

versal and measure the alternate interior angles. Are they

equal?

2. Use the third method to draw a line through a given

point parallel to a given line.

3. Construct a parallelogram with sides 2 inches and 3

inches long and one angle equal to 50 degrees.

4. If angle a at the top of this page is 30°, how large is

angle b? Why? Angle c? Why? Angle d? Why? Can you

find the size of each of the other angles? How can you check

your results?

a. Parallel lines are the same distance apart at all points.

b. If two lines are perpendicular to the same line, they

are parallel.

c. If two parallel lines are cut by a transversal, the alter-

nate interior angles are equal.



Problem Helps. Choosing the Facta

^ on Need299

In each problem one or more facts are given 1h.it are not

needed to solve the problem. Decide which facts you need andsolve the problems. Check. >

1. A street a mile long

and 40 feet wide is to be

paved with concrete 12 in.

thick, at a cost of $2.75 a

square yard. There is to be

a curb on each side, 8 in.

high, costing 60^ a foot.

Find the total cost.

2. Mr. Smith, Mr. Jones,

and Mr. Brown shared the

cost of an automobile trip.

Mr. Smith furnished the car

and paid only half as much

as each of the others, who

paid equal amounts. How much did each have to pay if they

drove 475 miles and used 25 gal. gasoline at 17^ a gallon and

3 qt. oil at 3Q£ a quart?

3. Mr. Cox bought a car for $977. At the end of one year

he sold it for $425. He received $200 cash and took a note

for 90 days, with 6% interest, for the balance. The deprecia-tion was what per cent of the original price?

4. A man starts at 8 A.M. on an automobile trip of 210

miles and plans to reach his destination in 6 hours. He drives

110 miles the first 3 hours; then stops half an hour for lunch.

How many miles must he average an hour for the rest of the

trip in order to reach his destination on time?

5. Mary took a trip with her father. They drove 312 miles

in 8 hours the first day; 240.2 mi. in 7 hours the second day;

189.7 mi. in 6 hours the third; 203.4 mi. in 4J hours the fourth;

125.8 mi. in 3 hours the fifth; and 323.6 mi. in 8 hours the

last day. Find the average number of miles they drove a day.

to the nearest tenth of a mile.



300 Problem Helps. Supplying the Missing

Facts

In each of the following problems one or more facts are

missing that are needed in solving the problem. Look up

the facts needed, or have your class agree on probably figures,

and solve the problems. Check.

1. Mary gained 7 lb. in weight last year. Find the per

cent of increase.

2. John worked 12 problems correctly on an arithmetic

test. What was his mark expressed as a per cent?

3. Mrs. Monroe's refrigerator will hold a cake of ice 12

inches by 12 inches by 18 inches. How many pounds will it

hold?

4. Find the cost of running an electric washing machine

for one hour at 8j£ a kilowatt hour.

5. A train makes the trip from Washington to Boston in

11 hours 25 minutes. Find the average number of miles per

hour to the nearest tenth of a mile.

6. An agent sold a used automobile for Mr. Adams and

mailed him a check for $245 in payment. For how much did

he sell the car?

7. On April 1, the Atkins' gas meterread

91,270cubic

feet. On May 1, it read 95,100 cubic feet. Find the amount

of their bill for April.

8. A grocer bought 150 bushels of potatoes at $1.00 a

bushel. He sold them at 3^ a pound. He much did he make

or lose?

9. It is estimated that the people of the United States

spend $1,000,000,000 a year at the movies. How much a week

is this for a family of five persons?

10. Mr. Cummings sells groceries on commission. Last

year his sales totaled $27,840. What was the amount of his

commissions for the year?



Problem Helps. Making Problems 301

Enough information is given m each list of data below to

enable you to find one or more tacts that arc not given. Make

one good problem out of each list. Solve each of your prob-

lems and check.

1. Helen read this advertisement: Make extra money

by taking subscriptions for our magazine at SI a year. Wepay 12% commission. Last

month, by working in her

spare time, Mrs. Barnes

earned $24. Helen made

a problem out of this. Can

you?

2. A farmer sent 8300

lb. milk to a creamery dur-

ing one month. It tested

3.9% butterfat, on the aver-

erage. He was paid 27^ a

pound for the butterfat.

3. The table at the right gives the

average number of pounds of tea and

coffee consumed by each person in the

United States in 1910, 1920, and 1930.

4. A train goes 317 miles from Philadelphia to Pittsburgh

at the average speed of 40 miles an hour while moving. The

total time lost on stops amounts to 39 minutes.

5. An old quarry 75 ft. long, 50 ft. wide, and 20 ft. deep is

to be filled. Dirt is bought at 45^ a cubic yard. It is found

that the dirt in settling shrinks \ of its volume.

6. A train is scheduled to make the trip from Boston to

Cleveland, a distance of 668 miles, in 17 hours and 12 minutes.

A total of 32 minutes is lost in making stops.

7. A doctor travels about 20 miles a day visiting patients.

He used to drive a horse and average 8 miles an hour. He

now drives an automobile and averages 30 miles an hour.

Year



302 Problem Test

Solve each of the following and check.

1. Mrs. Vaughn is planning a trip to Europe. On the map

the distance from Paris to Rome measures 2\ inches. Thescale on the map states that \ inch represents 100 miles.

How far is it from Paris to Rome according to her map?

2. Mr. Grimes is using a map which does not give the

scale to which it is drawn. In order to determine this, he

measures the distance between Louisville and Kansas City,

which he knows is about 550 miles, and finds it is about 2f

inches. To what scale is the map drawn?

3. Margaret drew a map, using \ inch to represent 100

miles. How far apart should she place two points if the dis-

tance between them is known to be 375 miles?

4. A magazine took a vote among its readers as to their

likes and dislikes in radio programs. There were 78 who

voted that they liked tenors and 52 who voted that theydid not. What per cent of those voting on this question liked

tenors? Illustrate graphically.

5. A walk 4 feet wide is to be built along the border of a

rectangular grass plot that is 80 feet by 110 feet. The walk

is to be built on the grass plot. How many square feet of con-

crete will there be in the walk? Draw a figure. Check.

6. Common brick are usually 8 x4 x2 . In estimating

how many bricks it will take to build a wall, it is customary to

allow 22 bricks for each cubic foot. Show that this is approxi-

mately correct. Add \ inch to each dimension of the brick to

allow for the mortar.

7. Allowing 22 bricks per cubic foot, how many bricks are

needed for a wall 30 ft. long, 6 ft. high, and 1 ft. thick?

8. How many bricks are needed for a foundation 16 inches

wide and 7 feet high for a building 30 feet by 40 feet? Allow

22 bricks to the cubic foot. Count the corners only once.

9. How many packages of seed will it take to plant a cir-

cular bed, 8 feet in diameter, if one package plants 10 sq. ft.?



Problem Test 303


1. In one year there were 598 students participating in

athletics at Harvard University. The next year there were937. Find the per cent of increase. Illustrate graphically

2. The size of sheets of tin is 20 inches by 28 inches. Howmany circular discs for can bottoms, each 4 inches in diameter,

can be cut from a sheet? What per cent of the tin is \va

Draw a figure before trying to solve the problem.

3. Mrs. Norris bought an electric washing machine. Howmuch did it increase her electric bill if she ran it four days

a month, three hours on each day? The machine used 200

watts an hour, and the current cost 7^ a kilowatt hour.

4. Mrs. Norris also bought an electric iron. Find the

increase in her bill, due to the iron, if she used it 4 days a

week, 2\ hours each day. The iron used 600 watts an houi

and the current cost 7^ a kilowatt hour.

5. Mrs. Norris bought A\ yards dress goods at $2.50 a yard,

f yard trimming at $3.75 a yard, a pattern for $1.50, and

thread, buttons, and a buckle for 60^. It took her 12 hours to

make the dress. How much did she earn an hour if she could

have bought a similar dress ready-made for $20?

6. An excavation 12 inches deep is being made for a street

850 ft. long and 30 ft. wide. How many cubic yards of earth

must be removed?

7. The dirt in problem 6 is to be hauled away in trucks

each of which holds 10 cubic yards. How many truck loads

must be hauled if the dirt expands § in digging?

8. Grindstones when turned at too high a speed tend to fly

apart or burst. Experiments show that it is not safe to turna grindstone, made of a certain kind of stone, so that a point

on the circumference moves more than 3600 feet per minute.

What would be the highest number of revolutions per minute

that it would be safe to turn a grindstone 4 feet in diameter

and made of this material?



.04 Diagnostic Tests in the Fundamental

Processes



without hurrying.

1.



Diagnostic Tests in Percentage

LOCATING Vol i; DIFFICULTIES

Work these examples and check. Work as fast BS you CftX)

without hurrying.

1.

2.

a.

Write .1982

as a per

cent.

e.

Find 37i%of 92.

a.

Find 18.6%

of $2.87.

(Nearest

cent)

e.

Write 13.97

1>.

875 is what '

,

of 1027

(Nearest tenth

of a per cent)

f.

Find the wholeif 85.1% of it

equals $74.39.

(Nearest cent)

b.

Write 166f%as a whole or

mixed number.

f.

Changemas a per

cent.

Write the

per cent

equivalent

off.

Change137 to fl

2to ^

per cent.

c.

Find the

whole if

33%% of it

equals 89.7.

g-

Write .9%

as a

decimal.

d.

Write 7.:.'; as

a decimal.

h.

Write 1-200%as a whole

or mixed

number.

d.

Write the

common frac-

tion equiva-lent of 90%.

h.

$2.50 is what

per cent ofto nearest

hundredth of

a per cent.



turn to the Practice Tests (pages 307 to 840) for further practu

suggested below.

la — 40 le— 42 2a — 42 2e — 40

lb— 43 If— 44 2b— 38 2f — 41

lc — 37 lg— 41 2c — 44 2g — 39

Id— 39 lh— 38 2d— 37 2h— 43



your Score Card. Did you improve? Find the class average




306 Chapter Test

Write the numbers from 1 to 16 on a piece of paper. After these

numbers write the words, or numbers, which belong in the cor-

responding blanks below.

1 . A rotation, or turning, is called an

2. The unit used in measuring angles is called a

3. A degree is one of a right angle.

4. On a certain map one-half inch represents 100 miles.

Two cities are 3 inches apart on this map. These cities are

really miles apart.(4)

5. Harry draws a map so that one-fourth inch on the map

represents 100 miles. City B is 150 miles from City A. Harry

makes them . inches apart on the map.(5)

6. When a line, or angle is divided into two equal parts,

it is said to be(6)

7.

Atriangle whose sides are all equal is called an

triangle.

8. Two lines that make right angles with each other are

said to be to each other.

(8)

9. Vertical angles are(9)

10. The sum of the three angles of any triangle is always

equal to. degrees.(10)

11. Angle 1 = 80°.

Angle 2 = °

(in

Angle 3 =°

(12)

Angle 4 =

12. If two lines are perpendicular to the same line, they are

13. Angle a = °.

(15)

14. Angle a and angle b are called

angles.(16)



PRACTICE TESTS



Practice IVsi Number 1 309

THE 1(H) BASIC ADDITION FACTS

Write the results on a folded paper. Also practice giving the

results mentally. Note the trouble Bpota for extra practice.

2



310 Practice Test Number 2

HIGHER DECADE ADDITION FACTS

Write the sums on a folded paper. Also practice giving them

mentally. Think the left figure of the result first. Note the trouble

spots and make similar examples for extra practice.

14



Practice Teal Number 4 :,ii

CARRYING

Write the results on a folded paper. Check. Watch the tens

when the sum of a column is more than 9. Make similar examples

for further practice, if needed.

37




THE 100 SUBTRACTION FACTS

Write the results on a folded paper. Also practice giving them

mentally. Note the trouble spots for extra practice.

476759 12 9 10 3

®_JLJLAJL-L — — — -_

13 11 13 2 12 8 12 11 5 6

_6__L_£_°_Z — — — — --

95787616

8646 AJLJLJL-Q-JL — — —

10 10 12 2 11 15 13 12 7 6

_8j__4_l__9_8_7__3J7_5

37993411 8 10 7

J_J^_9_5_0_£_2J__5_114 15 11 9 10 17 18 10 1 5

A JL JL — -1 J*. — — — —

8 8 11 9 10 11 7 15 6

_2._9_Ji-ii-J._H _- -1 — —

15 10 14 9 14 14 17 12 3 9

_i__.__-i_-__-- — — —

485516 13 8 13 9

____J-J>_-___- -i- — —

13 10 11 2 12 16 16 14 4 8

8642897713Practice makes perfect only if it is correct practice.



Practice Tesl Number 7

MINUEND I 1<

:l RE SMALLEB THAN 81 BTB Ml

Write the results on a folded paper. Check. Make

examples for further practice, if needed.

71

28

825619

257382

735192

351899

825

273

753

:;sd

875649

281952

591386

278197

725

639

876

382

761

287

721571

365793

218978

159183

628751

395214

968325

283187

313

;.\D

similar

938

ST:.

592876

27819S

417288

385529

Practice Test Number 8

ZEROS IN THE MINUENDWrite the results on a folded paper. Check. Make similar


90




FEWER FIGURES IN THE SUBTRAHEND

a. Write the results on a folded paper. Check. Make similar


27



\.

Practice Teal Number II 315

THE 100 MULTIPLICATION I \« rS

Write the resultson a folded paper. Also practice giving

thom mentally. Notice the trouble spots for extra practice.

4 3 1 6

3 5 2

6




ADDING TO THE PRODUCT

a. Write the results on a foldedpaper. Also practice giving

them mentally. Make similar examples for further practice,

if needed. -^

6X9+1



Practice Test Number L3 .U7

PLACING PARTIAL PRODUCTS

Do not copy the examples. Work on folded paper. Checkeach step. Be sure each partial product is in the right place.

Make similar examples for further practice, if needed.

234




THE 90 DIVISION FACTS

Writethe results on a folded paper. Also practice giving them

mentally. Note the trouble spots for extra practice.

1)0



Practice Test Number 16 Blfl

DIVISION FACTS WITH REMAINDERS

Do not copy. Write the (plot nut- and remaindere on B folded

paper. Also practice giving 1 1nn) mentally. Make -mnl:,

amples for further practice, if needed.

2)9




CORRECTING THE TRIAL QUOTIENT

The trial quotients are given. Multiply and write the product

on a folded paper. Subtract mentally. Write C on your paper

if the trial quotient is correct, S if it is too small, and L if it is

too large. Note the trouble spots for extra study.

3



Practice Tesl Number 20 32]

MIXED PRACTICE IX DIVISION

Copy these examples and divide, ('heck eaeh step. M.ik<-

similar examples for extra practice, if aeeded.

a. Short division:

8)7569 6)8198 7)3557 '.'.-,7870 7)9685

b. Long division

38)36905 67)39564 73)58997 47)47950 71)4900

28)5725 89)62907 59)17807 37)29699 43)42876


FINDING THE LEAST COMMON DENOMINATORFind the L.C.D. Do not work the examples. Write the L.C.D.

on a folded paper. Check. Note the trouble spots for extra study.25.35^^5.4. 33686 10 678. 2. 1 l 3. _2_ 5 ___2 3 4 9 4 15 8 15

5. 2 5 3 3. l 7 2

8 3 6 10 8 20 15 9i 3. 3 1 1 1 3 7 .

2 4 4 8 6 1 5 10 1 5

7 2 3 5 3. 1 3 4

12 15 14 9 4 7 10 152.1_3_i2__5___5 10 6 5 6 6 12

3 JL5.335._7_102 64 8 8 12 2

13.l

15.13.2412

262451 2 2 _5_ 2 1 1

5 6 3 3 12 3 8 _$_

Work as fast as you can with comfort. Haste makes waste.




CHANGING THE DENOMINATOR

Write the numbers that belong in the blanks on a folded paper.

Check. Make similar examples for extra practice,if

needed.

1

2



Practicer

IVst Number 21 323

CHANGING IMPROPER FRACTIONS TO WHOLE ORMIXED NUMBERS

Write the results on a folded paper. Be sure the fractions in

the results are in lowest terms. Check. Make similar examples


4-5




SUBTRACTING FRACTIONS AND MIXED NUMBERS

Copy the examples on a large sheet of paper and subtract.

Check. Make similar examples for further practice,if needed.

a.




CANCELLATION

Copy the examples and carry out the indicated operations.

Watch for opportunities to cancel. Check. Make similar examples


3 v 2 5 . 3 4 v9n 2. 3 3 v 3

4 X 3 8~4 5* /u 3~2 5 X 5

n • 3. 5\/4 3.. 3. 3 v in 4.2y— 4 -jX^- 4— 5 4XI/ 5 — 3

Sj^S- 19^.6 4

V3 4^.2. 2-Vfl

F~5 Hi-f 9X2 3--f 3XO

4X 4 8. 6 12 •? 4 • <* 2-4

8-7-5 3X2 5^8 g 1 ^ 8X2

3.3. « • 2 5 v^ 6.2 2.44-7-4 O—3 T6 X 5 T~T 5~5


CHANGING MIXED NUMBERS TO IMPROPERFRACTIONS

Write the results on a folded paper. Also practice giving them

mentally. Check. Make similar examples for further practice, if

needed.

34




MULTIPLICATION OF FRACTIONS AND MIXED

NUMBERS

Copy and multiply. Be sure to use the shortest method. Check.


|Xf 4X32J 4X4J 1|X| 5JX135

3§X4| 3X£ 12^X8 2jx3i 2§X84

4X27| 8X5i fXT7

5 3JX1J9X18^

8JX237 12X57J 6§X75 8Xf 57X8J

3|xli 36§X7 9X2f 8^X56 7X§

5X132§ 439X2J ljxl2 lfX85 5jx8j


DIVISION OF FRACTIONS AND MIXED NUMBERS

Copy and divide. Check. Make similar examples for further

practice,



Practice Test Number .'11 327

ADDITION OF DECIMALS

a. Add. Write the results on ;t folded paper.

similar examples for further practice, it needed.

32.7 .384 327.8

8.3 .756 564.1

19.8 .829 380.5

32.9 .568 761.7

b. Copy and add. Watch the decimal points,

similar examples for further practice, if needed.

1. 238.4 and 25.9 7.

2. .5, .7 and .9 8.

3. 5.8 and .39 9.

4. .001, .005 and .002 10.

5. 2.37, 85 and 7.2 11.

6. .726, .8 and .07 12.

Check. Make

.973

.85

.8761

.9

Check. Make

239.2, 78 and 96.7

27,834.2, 84.86 and 236.07

.029, 37.2, 8.97 and 91.789

.32, .7, .938 and .2857

.81, .756, 3.8 and 7.894

87.3, 9.57, .47 and 156.8


SUBTRACTION OF DECIMALS

a. Subtract. Write results on a folded paper. Check. Make

similar examples for further practice, if needed.

.3897 12.75 7.00 .3504

.2958 8.5 .28 .0179

Watch the decimal points. Copy and subtract


1. 8.72 from 32.45 8. 18.2 from 39.05

2. .27 from .29 9.

3. .003 from .009 10.

4. 2.85 from 7 11.

5. 32 from75.9 12.

6. 98.2 from 100 13.

7. .389 from 7.1 14.

279.2

181.7

Check.

.389 from 2

28.756 from 75.027

.986 from 2.3

$.16 from $1

$3 from $5.28

$1.96 from S10

Safety first. Be sure you are right, then go ahead.




MULTIPLICATION OF DECIMALS

Copy and multiply. Watch the decimal point. Be sure to

supply zeros when needed. Check each step. Make similar


.09

.07




FINDING APPROXIMATE RESULTS WITH DECIMALS

a. Write the approximate value of each of the following as

indicated. Check by going over your result a second time. Makesimilar examples for further practice, if needed.

1. To nearest cent.

$4,754 $.832 $2,875 $17,296

2. To nearest tenth.

8.723 .89 6.35 5.97 .06

3. To nearest thousandth.

.8971 7.0096 .39752 1.0003

b. Copy and carry out the indicated operation. Check by

going over each step and by estimating the result. Make similar


1. Give the results to the nearest hundredth.

$18,257 719)839 6.82)398.7

39.848

8.694 7.89 872.6

+ 5.735 X8.37 -897

2. Find the following



Practice Test Number 37 33]

PER CENT AND COMMON FRACTION EQUIVALENTS

Write the results on a folded paper. Also practice giving

them mentally. Note the trouble spots lor further study.

a. Give the common fraction equivalents.

33§% 16§% 80% 00% 86§%

90% 75% 25% 12j% 40%

8j% 70% 10% 50% 37i' ;

83j% 62j% 87^% 20% 30%

b. Give the per cent equivalents of each of the following

common fractions.

2 3 113TO 3 4 8

2 17 9 13 8 8 TO 5

JL. 1 i4 7

10 2 6 5 10

3 3 i 5 55 4 T2 6 8

c. Give the common fraction or per cent equivalents.

66 §% | 10% i 40%

f 62 1 % i 80% |

90% f 25% % 12J%§ 70% J 20% A

33 J % A 87 J % ^ 50%

^ 8£% i 10f%O ° 3 /O 8 iU 3 /O 4

>% f 30% i 37 1 %

f 60% £> 83 \% %

Work as fast as you can with comfort Haste makes waste.




WRITING LARGE PER CENTS AS WHOLE OR MIXEDNUMBERS

Write the results on a folded paper. Also practice giving themmentally. Check. Make similar examples for further practice,

if needed.



4




FINDING WHAT PER CENT ONE NUMBER IS OF

ANOTHER

Use a folded paper. Make similar examples for further practice,

if needed.

a. Write on your paper the per cent that belongs in each blank.

For the first one think, 8 is -&, or J,or 50% of 16. Then

write 50% on your paper.

8 is % of 16 10 is—% of 5

9 is

%of 12 20 is % of 4

16 is % of 20 12 is % of 8

12 is % of 36 9 is % of 9

2 is % of 12 18 is % of 24

6 is % of 9 24 is % of 18

15 is % of 20 5 is % of 2

b. Find the following mentally.

3 is what % of 24?

7 is what % of 56?

19 is what % of 19?

25 is what % of 5?

12 is what % of 48?

15 is what % of 45?

Write the results only. Check.

What % of 36 is 6?

What % of 27 is 54?

What % of 8 is 20?

What % of 18 is 12?

What % of 5 is 15?

What % of 6 is 9?

c. Estimate the following and write your estimate on the

paper. Then find the result to the nearest tenth of a per cent.

Check.







Practice makes perfect only if it is correct practice.



Praotirr IVst NiihiImt II 335

FINDING THE WHOLE WHEN A PEB CENT OF IT

IS KNOWN

Use a folded paper. Make similar examples for further practice,

if needed.

a. Find the whole number in each of the following mentally.

Cheek. Write the results only.

2\% of a number = 12 16§% of a number = 7

300% of a number = 37. ) 50% of a number = 25

33^ % of a number = 45 37^% of a number = 18

700% of a number = 82 40% of a number = 80

75% of a number = 42 250% of a number = 35

1000% of a number = 375 66f% of a number = 24

b. First write on the paper your estimate of the whole. Then

find it to the nearest cent. Check.

17% of a number = $3.25

9.7% of a number = $7.15

32.8% of a number = $2.50

73.7% of a number = $9.60

212%, of a number = $12.50

c.

Findthe whole number, N.

results to nearest cent.

75% of N = $12.50

66§%of N= $8.75

200% of N= $9.27

37j%of N = $10

300% of N = $25

80% of N = $9.75

16f% of N= $22.50

.7%) of a number = $2.85

85.2% of a number = $6.25

8.5% of a number= $7.10

145% of a number =$8.36

65.7%c of a number = $15.95

in each of the following. Give

72.3% of N = $75

7.9% of N = $5.25

217% of N = $80

.4% of N = $5.10

62% of N = $17.50

57.8% of N = $6.35

19% of N= $8.49

Safety first. Be sure you are right; then go ahead.




CHANGING THE UNIT IN DENOMINATE NUMBERS

Write the results on a folded paper. Check. Make similar ex-

amples for further practice, if needed.

a. Changing to smaller units.

3 hr. = min.

5 lb. = oz.

7 ft. = in.

4T.= lb.

7 bu.= pk.

b. Changing to larger units.

(Express results as common

fractions or mixed numbers)

90in.=_

28ft.=_

12oz. =_40 min. =

39pt.=_

ft,

yd.

lb.

_hr.

qt.



Practice Tesl Number 17 337

CHANGING DENOMINATE NUMBERS PROM TWOUNITS TO ONE

Write the results on a folded paper. Check. Make similar

examples for further practice, if Deeded.

a. Changing to smaller unit.




DIVISION OF DENOMINATE NUMBERS

Work on a folded paper. Check. Make similar examples for

further practice, if needed.

a. Use two units.

3)9 lb. 12 oz. 3)5 ft. 8 in. 5)12 hr. 15 min. 3)14 gal. 2 qt.

2)3 yd. 10 in. 3)7 qt. 1 pt. 2)3 yr. 117 da. 5)8 min. 10 sec.

b. Change tosmaller unit before dividing. Change result

back to two units.

2)5 bu. 2 pk. 5)7 qt. 1 pt. 6)7 hr. 12 min. 10)8 ft. 10 in.

c. Change to larger unit before dividing. Change result

back to two units.

3)7 qt. 1 pt. 10)12 min. 40 sec. 8)10 yd. 2 ft. 4)2 lb. 8 oz.

d. Change to smaller unit before dividing.

1. 8 ft. 9in.-J-l ft. 9 in.

2. 3 lb. 4oz.-s-l lb. 10 oz.

3. 1 min. 15 sec. -s- 30 sec.

4. 5bu. 2pk.^-2pk.

5. 2 hr. 15 min.-r-5 min.

e. Change to larger unit before dividing.

1. 4 qt. 1 pt.-^l qt. 1 pt.

2. 6 yd. 2 ft.^-1 yd. 2 ft.

3. 5 hr. 30 min. ^2 hr. 45 min.

4. 7 lb. 8 oz.-i-12oz.

f. Choose your own method.

1. 5 yd. 2 ft. -5- 3. 3 1b. 4oz.-^5

2. 8 ft. 8 in.-^4 4. 4 qt. 1 pt.-^l qt. 1 pt.

Practice makes perfect only if it is correct practice.



Score Card 341

IMPROVEMENT TEST NUMBER ONEWHOLE NUMBERS

Trial



342 Tables of Reference

LENGTH

Common or English System

12 inches (in.) = 1 foot (ft.)

3 feet=1 yard (yd.)

5J yards = 1 rod (rd.)

16§feet=l rod

320 rods= 1 mile (mi.)

1760 yards=1 mile

5280 feet=1 mile

Mariner's

6 feet= 1 fathom

120 fathoms=1 cable length

5280 feet = l statute mile

6080 feet = l nautical mile

3 nauticalmiles= 1 league

1 knot= a speed of

1 nautical mile per hour

Conversion Table

1 nautical mile=

1.1515 statute miles*

1 statute mile=0.8684 nautical miles*

CAPACITY

Liquid

2 pints (pt.) = 1 quart (qt.)

4 quarts=1 gallon (gal.)

8 pints= 1 gallon

231 cu. in. = 1 gallon

1 cu. ft. = l\ gallons*

Dry

2 pints (pt.) = 1 quart (qt.)

8 quarts= 1 peck (pk.)

4 pecks= 1 bushel (bu.)

32 quarts= 1 bushel

64 pints= 1 bushel

2150.42 cu. in. = 1 bushel

\\ cu. ft. = 1 bushel*

Miscellaneous

35 cu. ft. = 1 ton of hard coal*

500 cu. ft. = 1 ton of hay*

128 cu. ft. = 1 cord of wood

3 teaspoons (t.) = j

1 tablespoon (T.)f All

16 tablespoons= ^measures

1 cup (c.) I level

2 cups= l pint )

TIME

60 seconds (sec.) =

1 minute (min.)

60 minutes=1 hour (hr.)

24 hours= 1 day (da.)

7 days= l week (wk.)

30 days= l month (mo.)*

*Approximate.

12 months=1 year (yr.)

365 days=l commonyear

366 days=l leap year

360 days= l year in computing

common interest.

100 years= 1 century



Table of References 343

WEIGHT

Avoirdupois

16 ounces (oz.) = l pound (lb.)

100 pounds

1 hundredweight (cwt.)

2000 pounds = 1 ton (T.)

2240 pounds=l long ton

Miscellaneous

1 pt. water weighs 1 lb.*

1 gal. water

8j lb.*

leu. ft. water 62\ lb.*

1 cu. ft. ice

1 bbl. flour

1 bale cotton

57J lb.*

196 lb.*

500 lb.*

Weights of a Bushel

(Most States)

Barley 48 lb.

Com (shelled) 56 lb.

Corn (on cob) 70 lb.

Oats 32 lb.

Rye 56 lb.

Wheat 60 lb.

Apples 48 lb.

Potatoes 60 lb.

Weights of a Cup

(In Pounds)

Sugar J*Flour J*Lardutter J*

COUNTING

12 units=1 dozen (doz.)

12 dozen = 1 gross (gro.)

20 units = 1 score

*2

*Approximate

AREA OB SI i:i \< E

( ommom ob English System

I 1 1 square inches (sq. in.) =

1 square foot (sq. ft.)

9 square feet =

1 square yard (sq. yd.)

3O5 square yards =

1 square rod (sq. rd.)

160 square rods= 1 acre (A.)

43,560 square feet = 1 acre

640 acres = 1 square mile

(sq. mi.)

LAND MEASURE

5j yd. = 164 ft-

= l rod (rd -)

320 rd. = 1 mile (mi.)

160 sq. rd. = 1 acre (A.)

640 A. = 1 sq. mi.

VOLUME

Common or English System

1728 cubic inches (cu. in.) =

1 cubic foot (cu. ft.)

27 cubic feet = 1 cubic yard(cu. yd.)

ELECTRICAL

1000 watts =1 kilowatt

746 watts = 1 horse power

ANGLES AND AR< >

60 seconds ( ) = 1 minute (')

60 minutes = 1 degree (°)

90 degrees = 1 right angle

180 degrees = 1 straight angle

360 degrees

1 complete revolution



INDEX

Addition, how used, 8; avoiding

errors in, 9, 10; checking, 11;

test in, 12: totals from tables,

13; tricks of, 19; 100 basic facts,

309.

Accounts, personal, 128, 130; fam-

ily, 129, 130; savings, 161;

postal savings, 162.

Altitude, drawing, 230; of tri-

angle, 230.

Angle, bisecting, 294; measuring,264.

Angles, recognizing, 213; meaning

of, 263; protractor, 265; verti-

cal, 282; sum of angles of tri-

angles, 283; using angle sum,

284.

Approximate results, finding, 73.

Areas, rectangles, 214, 215, 216;

squares, 217; parallelograms,

222, 223; triangles, 225, 226;

trapezoids, 228, 229; circles,

235, 236.

Arithmetic in the home, personal

accounts, 128, 130; family ac-

counts, 129, 130; personal bud-

get, 131; family budgets, 132,

133, 134; how banks serve us,

134; doing business with a bank,

135; paying bills by check, 136;

writing and cashing checks, 137;

bills and statements, 138; re-

ceipts, 139; postal money orders,

140; bank drafts, express and

telegraph money orders, 141;

buying, 142, 143; buying at

sales, 144, 145; gas used in home,

146, 147, 148, 149; electricity

used in, 150, 151, 152, 153;

water used in, 154, 155.

Averages, finding, 33.

Automobile, cost of running, 170.

Bank, how serves us, 134; doing

business with, 135; drafts, 141;

savings accounts, 160.

Bills, and statements, 138.

Bisecting, a line, 285; a circle, 290,

291, 295; an angle, 294.

British, unit of measure, 207.

Buying, in larger quantities, 142,

143; at sales, 144, 145; war

bonds, 163; a home, 164, 165;

installment, 167, 168.

Budgets, personal, 131; family,

132, 133, 134.

Cancellation, time saved by, 53.

Chapter Tests, 1: 37; 2: 57; 3:

85; 4: 125; 5: 177; 6: 201; 7:

246-247; 8: 277; 9: 306.Checking, two ways of, 187; a

new method of, 188; by starting

with answers, 189; problems,

190, 191, 192, 193.

Checks, paying bills by, 136;

writing and cashing, 137; travel-

ers', 172.

Circles, drawing, 231; circum-

ference and diameter of, 232;

formulas for circumference and

diameter of, 233; finding cir-

cumference and diameter of,

234; area of, 235; finding area

of, 236; graph, 260; reading

graph, 261; making graph, 266;

dividing, 290, 291, 295.

Circumference, 231; and diameter,

relation between, 232.

Commissions, meaning of, 93.

Compound numbers, avoiding er-

rors in, 80, 81; test on, 81; using

in problems, 82.

Cubes, 237; volume of, 239.

Decimal, system of numbers, 60,

61, 62.

Decimals, meaning of, 59; frac-

tions, 60; reading, 61, 63, 108;

writing, 62; avoiding errors in

addition and subtraction of, 65;

avoiding errors in dividing, 66;

checking, 67;test in,

68;to

common fractions, 70; fractions

to decimals, 71; common and

decimal fractions, 72, 74, 75;

ratios, 76; in problems, 79;

from per cents, 105; using, 106;

as per cents, 109; placing deci-

mal point, 114.

344



Denominate numbers, 80, .si 82:

teal in, 81, 208.

Denominator, Least Common, 44,

46, 47.

Diameter, 231.

Diagnostic Tests, in addition, L2;

in subtraction, 17; in wholenumbers, .

r>.r); in multiplication,

24; in division, 32; in commonfractions, 49, 50, 56, 122; in

decimals, 68; in fundamentalprocesses, 84, 121, 175, 199,

244, 275, 304; in percentages,

123, 176, 200, 245, 276, 305; in

denominate numbers, 208.

Diagrams, using in problem solv-

ing, 183; using in problems, 184,

185, 186.Distances, errors in measuring,

210; estimating, 211, 212.

Division, how used, 28; avoid-ing errors in, 29, 30; checking,

31; test in, 32; by 10, 100, 1000,

69; 90 division facts, 318.

Drawing instruments, 281, 296.

Electricity, how measured, 150;cost of, 151, 152; used in home,

153.Estimating, distances, 211, 212.

Express, money orders, 141.

Formula, circumference and diam-eter, 233; area of rectangle, 216;area of square, 217; area oftrapezoid, 228; interest, 158,159.

Fractions, history of, 39; twomeanings of, 40; third meaningof, 41; comparing, 42; six prin-ciples of, 43; checking, 44; avoid-ing errors in adding and sub-tracting, 46; avoiding errors in

adding and subtracting mixednumbers, 47; avoiding errors inmultiplying and dividing, 48;tests in, 49, 50, 56; in problems,51, 52; decimals to fractions, 70;fractions to decimals, 71; com-

mon and decimal, 72; ratios, 76;types of problems in, 78; chang-ing to per cents, 110, 111.

Gas and electric bills, fifirurine

149, 151.

Geometrical designs, 280, 281, 290.Geometry, using, 279; making

geometrical designs, 280-281.

Graphs, lor impro\ emenl le.sts. fi,

bar, 249; directions lor makingbar, 250-251; making bar, 251;picture, 252, 2.

r>.'{; line,

using line, 265; reading line,

256-257; directions lor makingline,

258, 259; making line, 259;circle, 260; making circle, 266;rectangle, 260; making redangle, 262; reading circle andrectangle, 261; choosing, 267;making, 268-269.

Hexagon, regular in circle, 290.

Home, methods of paying for, 164,165.

Improvement Tests, directions for

taking, 5; 1: Whole Numbers,7; 2: Common Fractions, 45;3: Decimal Fractions, 64; 4:

Percentage, 124.

Installment plan, buying on, 167,168, 169.

Interest, how to figure, 156; for-

mula, 158, 159; on savings ac-

counts, 161; on money loaned,166; finding rate of, 167.

Least Common Denominator, 44,

46, 47.

Map making, scale drawings, 286,287, 288.

Meaning of tv, 233.

Measure of capacity, British units,207.

Measurements, practical, 203; us-ing, 204; units of, 205; span,thumb width, pace, 206; stand-ard units, 207; exact and ap-proximate, 209; errors in, 210.

Meters, how gas is measured, 146;reading, 147; checking gas bill,

148; figuring gas bill, 149; howelectricity is

measured, 150;figuring cost of, 151; cost of

lights, 152; in the home, 153;water bill, checking a, 154;cost of, 155.

Money orders, postal, 140; bankdrafts, 141; express, 141; tele-

graph, 141.

Mortgage, giving, 164,

345



Multiplication, how used, 20;

avoiding errors in, 21; checking,

22, 23; test in, 24; short methods

of, 25, 26; using, 27; cancella-

tion, 53; by, 10, 100, 1000, 69;

100 basic facts, 315.

Numbers, reading and writing

large, 2, 3.

Parallel lines, 220, 221; drawing,

296; facts about, 297.

Parallelogram, area of, 222, 223.

Per cent, working with, 87; writing

hundredths, 88; of numbers us-

ing decimals, 89; as commonfractions, 90; of numbers using

common fractions, 91; meaningof, 92; in commissions, 93;

ratios, 94, 112; problems in, 95,

108, 113; increase and decrease,

96, 97; large, 98, 99; expenses

and profits, 100; in problems,

101; finding the whole, 102, 103,

107; finding the selling price,

104; as decimals, 105, 106; frac-

tions to, 110, 111; estimating,

114, 115.Perpendiculars, drawing, 292; us-

ing a compass and straightedge,

293.

Postal money orders, 140.

Postal savings accounts, 162.

Practice Tests, directions for using,

308; addition of whole numbers,

309-311; subtraction of whole

numbers, 312-314; multiplica-

tion of whole numbers, 315-317;

division of whole numbers, 319-

321; Least Common Denomina-

tor, 322; fractions, 322-326; deci-

mals, 327-330; per cents, 331-

335; denominate numbers, 336-

340.

Problem Helps, how to solve prob-

lems, 270, 298; choosing facts,

271, 299; supplying missing

facts, 272, 300; making prob-

lems 273, 301.Problems, steps in solving, 80;

facts in, 181; help for the solu-

tion of, 182; using diagrams in,

183; checking, 187, 188, 189,

190, 191, 192, 193; how to solve,

194; missing facts in, 195;

choosing the facts, 196; making,

197.

Problem Tests, 36, 54, 83, 120,

173, 174, 198, 242-243, 274, 302,

303.

Protractor, 265.

Promissory notes, how used, 157.

Radius, 231.

Ratios, finding, 76; in per cents,

94, 112.

Receipts, 139.

Rectangles, recognizing, 213; find-

ing area of, 214, 215; formula

for area of, 216; perimeter of,

218, 219; graph, 260; reading

graph, 261; making graph, 262.

Rectangular solids, meaning, 237;

volume of, 238.

Round numbers, 116; in multipli-

cation, 117; in division, 118; in

problems, 119.

Savings accounts, at bank, 160;

postal, 162.

Scale drawings, reading, 286, 288;

making, 287; locating on, 289.

Score Card, 341.

Selling price, 182.

Solids, volume of, 237, 238, 240.

Solving problems, process to use,34; telling how, 35, 241.

Square, recognizing, 213; formula

for area of, 217; perimeter of,

218.

Standard units, 207.

Statements, and bills, 138.

Subtraction, how used, 14; avoid-

ing errors in, 15, 16; test in, 17;

comparing numbers by, 18; 100

basic facts, 312.

Tables, 13.

Table of References, 342-343.

Trapezoid, area of, 228, 229;

formula for area, 228.

Travelers' checks, 172.

Triangle, right, 224; area of, 225;

area of any, 226; finding area of,

227; altitude, 230; sum of angles

of, 283; using angle sums, 284;

equilateral, 291.

Vacation, figuring cost of, 171.

Volumes, of rectangular solids,

237, 238, 240; of a cube, 239.

Whole numbers, review, 1-3; test

in, 55; finding value of, 77; in

per cents, 102, 103, 107.

346

arithmetic for you

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