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Chapter 1 Contents1.1 Exponents and Roots

1.2 Order of Operations

1.3 Geometry

1.4 Statistics

1.5 Translation of Words into Expressions

1.6 Application Problems

1.7 Bar Graphs and Line Graphs

Image from Microsoft Office Clip Art

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Complex Numbers

Real Numbers

Irrational Numbers Rational Numbers

Integers

Whole Numbers

Counting Numbers

ImaginaryNumbers (i)

You are here!

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Exponents

In mathematics, when we use the symbol , we mean 4 x 4 x 4, which equals 64. The 3 in is

called the exponent and the 4 is called thebase. We say or read asfour raised to the third poweror

four cubed. Likewise, = 9 x 9 = 81, and we say or read as nine to the second poweror nine

squared.

By definition, any number (except 0 itself) raised to the power 0 is 1; thus, , , and

. We can even raise a number to very high powers, and this is where exponents are particularly

useful in applications to higher mathematics and physics. You may have heard of the number ,

known as a googol, which is written as a 1 followed by one hundred zeros. The popular search engineGoogle changed the words spelling but is named for .

Example 1: Write the following in exponential form.

a. 3 3 3 3 = 34

(3 is multiplied with itself 4 times)

b. 7 7 7 7 7 7 = 76

(7 is multiplied with itself 6 times)

Example 2: Write in expanded form and simplify.

a. 53 = 5 5 5 = 125

b. 25 = 2 2 2 2 2 = 32

c. 70 = 1 (bydefinition)

Example 3: Write the following in exponential form and simplify:

a. 2 2 2 3 3 = 23

32

= 8 9 = 72

b. 4 4 5 5 = 4

2

5

2

= 16 25 = 400

c. 3 3 6 6 = 32

62

= 9 36 = 324

d. 1 1 1 5 5 5 = 135

3= 1 125 = 125

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Roots

An operation related to exponents in mathematics is taking thesquare root orcube root of a number.

We call these sorts of expressions radicals because they involve the symbol , which is called aradical. When we write 2 16 , we mean the number that when raised to the second power gives 16. In

other words, to calculate 2 16 , also known as the square root of 16, we ask the question, What number

do we have to raise to the second power to get 16? The answer is 4, because .

Lets now try to find the cube root of 27, written 3 27 . Notice the little 3, called the index, on the

upper left part of the radical. To calculate 3 27 , we ask the question, What number must I raise to the

third power to get 27? Since = 27, it follows that 3 27 3 .

Similarly, 5 32 2 because and 2 49 7 because . Note that when the index of theradical is 2, we usually dont write the little 2 on the upper left part of the radical;therefore, when there

is no index in a radical, we assume that we are taking the square root . For example, we would write

49 instead of 2 49 , and we would write 16 instead of 2 16 .

Now, you try some examples.

Example 4: 100 10 since 102 = 100

Example 5: 144 12 since 122 = 144

Example 6: 36 6 since 62 = 36

Example 7:3

64 4 since 43

= 64

Example 8: 4 625 5 since 54 = 625

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1.1 Exponents and Roots Exercises

1.2.3.4.5.6.7.8.9.10.11. 4 12. 25 13. 64 14. 121 15. 3 8 16. 3 64 17. 3 27 18. 3 125 19. 4 81 20. 4 16

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1.1 Exponents and Roots Exercise Answers

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20. 2

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1.2 Order of Operations

Now that you have reviewed the basic operations and how they are related to each other, we can

examine expressions that contain more than one operation.

Back in the 1600s, there was a lack of uniformity with regard to how expressions were simplified.

This caused problems because two people asking the same question of two different mathematicians

would get two different answers, both of which seemed logical and correct. This presented a problem

that needed to be solved. So, mathematicians came together and agreed that computations should be

performed in the same way by everyone. That way, results would be consistent. Since multiplication is

the same as repeatedly adding the same value, and since division is the same as repeatedly subtracting

the same value, they determined that multiplication and division should be performed before singleadditions and subtractions. Because an exponent indicates the number of repeated multiplications of a

value, applying an exponent should be done before single multiplications.

So, the order in which the operations are performed was determined to be as follows:

1. Exponents2. Multiplication and Division3. Addition and Subtraction

Multiplication and division are inverse operations; addition and subtraction are inverse operations.

Therefore, multiplication and division were to be performed at the same time; addition and subtractionwere also determined to be considered together. Since the people who were making these decisions weremostly western Europeans (whose languages were and are written from left to written), the operationswere to be performed in the order that they appeared from left to right. That way, performingcalculations followed the same rules as reading text, yet subordinate to the determined order ofoperations.

There are always exceptions to every rule. So, mathematicians had to come up with a way to handlesituations where addition or subtraction would take place before multiplication, division or work withexponents. So, parentheses ( ) and other grouping symbols were incorporated into the prescribed

order. Mathematicians determined that if an operation had to be done out of order from the prescribedmethod above, they would use the grouping symbols to indicate that an exception was being made.Parentheses were given the honor of first choice of grouping symbol. If more than one exception neededto be made within a grouping, then brackets[ ] would surround the parenthetical group and theother components of the exception.

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This seemed to have solved all of the inconsistencies among mathematicians; they now had aprescribed order to follow when calculating expressions with more than one operation involved. Theycalled this the Order of Operations.

1. Parentheses2. Exponents3. Multiplication and Division4. Addition and Subtraction

In the schools, students learned this using a mnemonic: Please Excuse My Dear Aunt Sally, whichstands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction.

Example 1: Evaluate 7 + 4(3)

Since there are no operations inside the parentheses, you would perform the multiplication first,

then the addition.

7 + 4(3)

= 7 + 12

= 19

Example 2: Evaluate 8[24 (4 + 2)]

In this case, there are parentheses inside of brackets. Brackets are just like parentheses; in this

case, you work inside the parentheses first since they are inside of the brackets.

= 8[24 (4 + 2)]

= 8[24 6]

Next, the brackets (like parentheses):

= 8[4], which means 84

= 32

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Example 3: Evaluate 1042 (75)

Following the order of operations, you would simplify inside the parentheses first:

= 1042 (75)

= 1042 2

Next, the exponent:

= 1016 2

Although we do perform operations from left to right, dont forget that division comes before

subtraction. = 108

= 2

Example 4: Evaluate 4 33(35 7) + 24

Remember the process is always the same. Just pay attention to which operation you are

supposed to do next.

= 4 33(35 7) + 2

4

= 4 335 + 2

4

Exponents are next; which do you do first? Remember to move from left to right.

= 4 27 5 + 24

= 4 27 5 + 16

= 1085 + 16

= 103 + 16

= 119

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1.2 Order of Operations Exercises

1.2.

3.4.5.6.7.8.9. 6(2) + 9(8)10. 4(5)3(2)11.12.13.14.15.16.17.18.19.20.21.22.

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1.2 Order of Operations Exercise Answers

1. 13

2. 12

3. 64

4. 36

5. 43

6. 43

7. 1

8. 10

9. 84

10. 14

11. 9

12. 30

13. 61

14. 97

15. 29

16. 602

17. 21

18. 11

19. 124

20. 403

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1.3 Geometry

Geometry is the study of shapes and their mathematical properties. In almost every mathematics

course that you take, there is a basic geometric component. In this section, we will learn to calculate the

perimeter, area, and volume of a few basic geometric shapes.

Theperimeter of a geometric object is the measurement (meter) of the distance around

(peri) the boundary of the object. For example, to find the perimeter of a rectangle, two of whose

sides have length 4 and two of whose sides have length 5, we would simply find the sum 4 + 4 + 5 + 5 =

18 inches. The perimeter of this rectangle could also be calculated by adding 2 x 4 and 2 x 5, because of

the special property of rectangles that each pair of opposite sides have the same length.

Similarly, to find the perimeter of a square, each of whose sides has length 7 inches, we simply

compute 7 + 7 + 7 + 7 = 28 inches. Or, because of the special property of squares that all four sides

have the same length, we could multiply 4 x 7 = 28.

Thearea of a square or rectangle is found by multiplying the length and the width (also called

the base and the height). Therefore, if we want to find the area of the rectangle with base 5 centimeters

and height 4 centimeters, we simply multiply 4 by 5 to get 4 x 5 = 20 square centimeters, which can also

be written as 20 cm2.

Similarly, to find the area of a square with side length 5 cm, since the base and height are both 5, we

get 5 x 5 = 52

= 25 cm2.

Finally, the area of a triangle is found by multiplying the base by the height and then dividing by 2,

which we demonstrate later in a few examples.

Acube is a three-dimensional square, and arectangular solidis a three-dimensional

rectangle. Dice are examples of cubes, and most shoe boxes are examples of rectangular solids. To

find the volume of a cube or rectangular solid, we multiply the length by the width by the height. For

example, if you have a shoe box that is 12 inches long, 6 inches wide, and 5 inches high, to find the

volume we would compute 12 x 6 x 5 = 360 inches cubed, or 360 in3. Notice that the volume always has

cubic units, and area always has squared units.

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Example 1: Compute the perimeter of the given rectangle.

5 inches

3 inches

Step 1: Perimeter of a rectangle = 2 Length + 2 Width = 2L + 2W

Step 2: Perimeter = 2 5 + 2 3

= 10 + 6

= 16 Answer: Perimeter = 16 inches

Example 2: Compute the perimeter of the given triangle.

3 meters 2 meters

4 meters

Step 1: Perimeter = a + b + c

Step 2: Perimeter = 3 + 2 + 4

= 9 meters Answer: Perimeter = 9 meters

Example 3: Compute the perimeter of the given trapezoid.

9 m Perimeter = 9 + 15 + 11 + 17

= 52

Answer: Perimeter = 52 meters

15 m17 m

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11 m

Example 4: Compute the perimeter of the given square.

To calculate the perimeter, we add the lengths of the sides, each of which is the same.

18 + 18 + 18 + 18 = 4(18) = 72

Answer: Perimeter = 72 feet

Example 5: Find the length of the missing side of the triangle whose perimeter is 28 ft.

8 ft ?

14 ft

Step 1: Add the lengths of the sides that we know.

8 + 14 = 22

Step 2: Subtract that sum from the perimeter.

2822 = 6 Answer: the length of the missing side is 6 ft.

Example 6: Determine the area of the given rectangle.

22 cm

Area of a rectangle = Length Width = L W

14 cm = 22 14

= 308

Answer: Area = 308 cm2

18 ft

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Example 7: Determine the area of the given triangle.

Area = Base Height 2 = B H 2

4 in = 9 4 2

= 36 2

9 in = 18 Answer: Area = 18 in2

Example 8: Determine the volume of the given cube.

Since a cube is a rectangular solid, we multiply the length by the width by the height to get

the volume. However, the length and width and height of a cube are equal; we call this

common length S (for Side).

Volume of a cube = Side Side Side = S S S = S3

= 9 units 9 units 9 units = 93 units3

= 729 units3

Example 9: Determine the volume of the rectangular solid whose length is 12 units, width is

7 units, and depth is 4 units.

We multiply the length by the width by the height to get the volume of a rectangular solid.

Volume of a rectangular solid = Length Width Height = L W H

= 12 units 7 units 4 units

= 336 units3

9 units

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1.3 Geometry Exercises

1.2.

3.4.

5.6. 3

7.8.9.

2 units

2 units 2 units

2 units

10.

5 units 6 units

9 units

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11.

12.

13.

14.

3 units

7 units

4 units

5 units

6 units

10 units

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18.

19.

20.

3 units

1 unit

4 units

6 units

7 units

9 units

9 units

12 units

15 units

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1.3 Geometry Exercise Answers

1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.

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1.4 Statistics: Mean, Median, and Mode

You have probably heard people in the news talking about the mean and median. For example, the

news often mentions median incomes, mean house prices, and even the average price of gasoline per

gallon. Themean,median,andmodeare different numbers, each of which can be used to represent anentire list of data values.

You might already be familiar with the mean, because the mean is another word for average. We

find themean of a list of data values by adding all of the data values together and then dividing

this total by the number of data values in the list . Suppose that you want to find the mean of the

following quiz scores (each of which is out of ten points): 10, 9, 9, 8, and 4. We find the sum of the

numbers, 10 + 9 + 9 + 8 + 4 = 40, and then divide that sum by 5 (which is the number of quiz grades) to

get . Therefore, if someone asks you how you are doing on your quizzes, you might say that

the mean of your quiz scores is 8, which is a way of saying that, on average, you are getting 8 out of 10

on your quizzes.

Find the mean in the following problems.

Example 1: A car dealer recorded the miles-per-gallon ratings of six cars and the results were as

follows: 20, 30, 22, 32, 25, 21.

Step 1: Find the sum of the numbers.

20 + 30 + 22 + 32 + 25 + 21 = 150

Step 2: Divide the sum by the number of cars.

150 6 = 25 Answer: 25 miles per gallon

Example 2: Your scores on five math tests were 82, 79, 71, 73, and 90.

Step 1: 82 + 79 + 71 + 73 + 90 = 395

Step 2: 395 5 = 79 Answer: 79

Another (single) number that is used to represent a list of data values is the median. To find the

median of a set of data values, put the values in order and find the middle number. If there are two

numbers in the middle, find the mean of those two middle numbers. Lets go back to the quiz scoreexample. If your quiz scores are 10, 9, 9, 8, and 4, we first put the values in order: 4, 8, 9, 9, 10. Next,

we identify the value in the middle, which is 9, to get the median. Now you have another way to tell

someone how you are doing on your quizzes; you can say that the median of your quiz scores is 9.

Notice that it was easy to find the value in the middle because we were finding the median of a set of

five (an odd number) values.

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But, as we mentioned above, if there are two values in the middle (like if there are eightan even

numberof values in the list), we calculate the mean of the two middle values. In other words, we add

the two middle values and divide by 2.

Lets say that its later in the semester and you have had eight quizzes, and you are trying to find themedian of the numbers 10, 9, 9, 8, 4, 6, 3, and 0. Putting the quiz scores in order (0, 3, 4, 6, 8, 9, 9, 10)

we find that the two values in the middle are 6 and 8. Now we add 6 + 8 = 14 and then divide 14 by 2 to

get 7. The median of the eight quiz scores is 7.

Note that when we put the values in order, we usually list them from least to greatest. However, in order

to find the median of a list of values, the values can be listed either from least to greatest or from

greatest to least: the middle value will be the same middle value.

Example 3: Find the median number of passengers on seven US Airways flights.

309, 295, 311, 302, 400, 488, 344

Step 1: Put the values in order.

295, 302, 309, 311, 344, 400, 488

Step 2: Find the middle value.

295, 302, 309, 311, 344, 400, 488 Answer: The median is 311.

Example 4: Find the median number of laptops sold each month

18, 19, 20, 27, 19, 24, 20, 23, 30, 21, 25, 29

Step 1: Put the values in order.

18, 19, 19, 20, 20, 21, 23, 24, 25, 27, 29 30

Step 2: Find the middle value.

18, 19, 19, 20, 20, 21, 23, 24, 25, 27, 29 30

Step 3: Find the mean of the two middle values.

21 + 23 = 44

44 2 = 22 Answer: The median is 22.

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Another (single) number that is used to represent a list of data values is the mode. Themode of a list ofnumbers is the number that occurs most often. To return to our original example of five quiz scores,the mode of the numbers 10, 9, 9, 8, and 4 is 9 because 9 appears most oftentwicewhile each of therest of the numbers occurs just once. Note that the mode can be more than one value in a set of data,

or a set of data may have no mode.

Lets say that by the end of the semester, your quiz scores are 10, 9, 9, 8, 4, 6, 3, 0, 10, 5, 9, 10. Themode of these quiz scores is 10 and 9, because they occur equally often and more often than each of theother numbers: both 10 and 9 occur three times, while no other score appears more than once.

A set of data could be qualitative (color, make of car, etc.) instead of quantitative. If this is the case, you

find the color, make of car, etc. that occurs most often.

Example 5: Find the mode of the number of houses sold (per month).

15, 20, 25, 30, 33, 21, 25, 32, 19, 27, 35, 24

Looking at the data set: 15, 20, 25, 30, 33, 21, 25, 32, 19, 27, 35, 24, the number 25 appears twice.

Each of the rest of the numbers appears only once. The mode in this case would be 25.

Example 6: Find the mode of the number of students enrolled in seven sections of Math 081.

24, 22, 19, 23, 21, 11, 16

Notice that there are no repetitions in the data set, therefore there is no mode.

Example 7: Find the mode of the ages of ten recently hired employees at a company.

33, 54, 28, 33, 50, 25, 30, 28, 33, 28.

33 and 28 both appear three times, more often than any other number appears. Hence, the mode is

both 33 and 28.

Example 8: Find the mode of the makes of cars in a dealership.

Honda, Chevy, Ford, Ford, Toyota, Chevy, Honda, Ford

Since Ford appears more often than any other make of car, Ford is the mode of the data set.

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1.4 Statistics Exercises

1. Find the mean: 3, 4, 4, 5, 7, 8, 9, 9, 10, 112.

3.4.5.6.7.8.9.10. $18,000 $19,700 $18,600 $20,410 $21,78011. $18,000 $19,700 $18,600 $20,410 $21,78012.13.

$120,000 $90,000 $75,000 $10,000 $110,000 $80,000

14.$14,000 $12,500 $13,600 $11,900

15.16.17.18.19.20.

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1.4 Statistics Exercise Answers

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20. Blue

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1.5 Translation of Words into Expressions

Problem solving is a critical skill in all aspects of todays world. Everyday situations often call for

using one or more mathematical operations. Some key words and expressions suggest certain

operations. Use these translations as a guide when solving problems.

Operation Word or Expression Examples

Addition: + add, sum, plus, increased by,

more than

five more than a number

the temperature increased by 10 degrees

the sum of the sides of the triangle

Subtraction: subtract , less, less than,

minus, difference, decreasedby

two dollars less than the cost

the weight decreased by 5 pounds

the difference between revenue and cost

Multiplication: . multiply, multiplied by,product, times, of, twice

the product of length and width

three times the average cost

2

1of her salary

twice the length

Division: divide, divided by, quotient,per

the perimeter of a square divided by 4

the quotient of distance and time

miles per hour

There are various possibilities when the problem has is in it.

Greater Than: > is greater than, is more than 1,000 is greater than 80

Less Than: < is less than 40 < 5,000

Equal To: = is, are, was, were, equals, isequal to, results in, is thesame as

the total cost of the project is $5,000

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Some expressions require more than one operation or symbol in their translation.

four more than two times the width 4 + 2(w)

two less than the product of 4 and a number (4)(n)2

The total cost is the sum of fixed cost and variable cost

= +

Profit is the difference between revenue and cost

=

Here are some useful steps when solving problems.

Step 1: READ the problem carefully!

You may need to read the problem several times before you understand what is being stated orasked.

Step 2: If possible, VISUALIZE the solution.

Draw a diagram representing the situation.

Step 3: TRANSLATE the written problem into a math problem.

Determine what operations are used, what algebraic equation can be written, etc.

Step 4: SOLVE using mathematical techniques.

Perform the operation, solve the equation, etc.

Step 5: CHECK your answer.

Does the answer seem reasonable?

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Example 1: Translate the following expressions into mathematical statements.

a. 10 is more than 2

Answer: 10 > 2

b. 9 is less than 12

Answer: 9 < 12

c. $15 is more than $10

Answer: $15 > $10

Example 2: Translate and find the value.

a. the difference of 25 and 15

translation: the difference of 25 and 15 2515

value: 2515 = 10

b. the sum of 1553 and 1321

translation: the sum of 1553 and 1321 1553 + 1321

value: 1553 + 1321 = 2874

Example 3: The sum of three numbers is 121. One number is 25 and the other is 37. What is thethird number?

Taking the first two sentences together, we write: 25 + 37 + ? = 121. In order to find the missingnumber, we must subtract the sum of 25 and 37 from the total, 121.

This translates to: 121(25 + 37).

We make the calculation inside the parentheses: 25 + 37 = 62

Then we subtract: 12162 = 59.

Answer: 59

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Example 4: Translate and find the value.

a. the product of 6 and 11.

translation: the product of 6 and 11 6 11

value: 6 11 = 66

b. the product of 14 and $15

translation: the product of 14 and $15 14 $15

value: 14 $15 = $210

Example 5: Translate and find the value.

a. the square of 9

translation: the square of 9 92

value: 92 = 9 9 = 81

b. the cube of 8

translation: the cube of 8 83

value: 83

= 8 8 8 = 512

Example 6: Translate and find the value.

a. the quotient of 148 and 4

translation: the quotient of 148 and 4 148 4

value: 148 4 = 37

b. the quotient of 3675 and 3

translation: the quotient of 3675 and 3 3675 3

value: 3675 3 = 1225

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Example 7: The product of three numbers is 186. One number is 3 and the other is 2. What is thethird number?

Taking the first two sentences together, we write: 3 2 ? = 186. In order to find the missing

number, we must divide the product of 3 and 2 from 186.

This translates to: 186 (3 2).

We calculate inside the parentheses first: 186 6. Then we divide and obtain 31.

Answer: 31

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1.5 Translations Exercises

1.2.

3.4.5.6.7.8.9.10.11. Translate and find the value. Two of the

numbers are 10 and12. What is the third number?

12.13.

14.15.16.17.18.19.20. Translate and find the value. the

numbers are 2 and6, what is the third number?

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1.5 Exercises Answers

1.2.

3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.

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1.6 Applications

A good way to practice mathematics is to do applications in the context of everyday life. In many of the

mathematics courses you will take, applications will be important to the understanding of the underlying

theory.

Example 1: George can type 75 words a minute but Sally can only type 39 words per minute. How

many more words per minute can George type than Sally?

To find the difference between the number of words per minute that George can type and the number

of words per minute that Sally can type, we must subtract.

7539 = 36 Answer: 36 words per minute.

Example 2: A total of 188 employees attended four simultaneously occurring conference sessions.

What was the average number of employees per session?

To find the average, we must take the total number of employees and divide by the number of

sessions.

188 4 = 47 Answer: 47 employees.

Example 3: Jacque needs to buy additional material for her house remodeling project. She needs

to buy three extra gallons of paint, each of which costs $8. She must also buy four more paint brushes,

each of which costs $2. What is the total cost for all of these items?

We will calculate the cost of the extra paint, and then calculate the cost of the additional paint

brushes. Then we will add these two values.

Step 1: Find the cost of the paint.

3 $8 = $24

Step 2: Find the cost of the paint brushes.

4 $2 = $8

Step 3: Find the total cost of the items purchased.

$24 + $8 = $32 Answer: $32

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Example 4: A rectangular lot measures 69 meters by 110 meters. What is the perimeter of the

rectangular lot?

Step 1: Perimeter = 2 Length + 2 Width = 2 L + 2 W

Step 2: Length = 110 meters, Width = 69 meters

Substitute these values into the formula to get:

Perimeter = 2 110 + 2 69

= 220 + 138

= 358 Answer: 358 meters

Example 5: John wants to place tiles in a room that measures 4 feet by 8 feet. Each tile costs $3 per

square foot. How much will John spend on tiles?

Step 1: The room measures 4 by 8 feet and the cost of tile is measured in dollar per square foot. We

need to find the square footage of the room by calculating its area. Then we can determine the cost.

Step 2: Width = 4 and Length = 8 Area = Length Width

= 4 feet 8 feet

= 32 square feet

Step 3: Multiply the area of the room by the cost per square foot.

32 $3 = $96 Answer: $96

Example 6: The temperatures (in degrees Fahrenheit) for the last four days in Bangkok, Thailand

were 84, 86, 92, and 82. Find the mean temperature.

To find the mean, we need to calculate the sum of all the temperatures and then divide that sum by the

number of days.

Step 1: 84 + 86 + 92 + 82 = 344

Step 2: 344 4 = 86

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Answer: 86 degrees

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Example 7: Tony made several deposits into his account this week: $350, $45, and $120. If his balancewas $800 at the beginning of the week, and if he withdrew $400 at the end of the week, what is hisaccount balance at the end of the week?

Step 1: Find the amount deposited.

$350 + $45 + $120 = $515

Step 2: Find the total amount in his account after the deposit.

$800 + $515 = $1,315

Step 3: He withdrew money, which translates into subtracting the amount of the withdrawal from

the amount in the account.

$1315$400 = $915 Answer: $915

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1.6 Applications Exercises

1. of thegarden?

2.

3.

4.

5.have a

total of $500 in his account?

6.

7.

8.

9.

10.

11.

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12.

13.

14.

15.

16.

17.

18.

19.?

20.

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1.6 Exercises Answers

1.2.3.4.5.6.7.8.9. lots10.11.12.

13. yards14. patients15. cells16.17.18. friends19.20.

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1.7 Bar Graphs and Line Graphs

You have undoubtedly already worked with bar graphs and line graphs or have seen them as

graphics in newspapers, magazines, or television newscasts. Bar graphs and line graphs generally make

sets of data easier to compare to each other. They provide a useful picture of the numbers that describe

a given scenario. The best way to learn how to work with bar graphs and line graphs is to see a few

examples.

Example 1: Using the bar graph above, answer the following questions:

a. What is the mean salary for females in each of the three companies?

Answer: $42,000 for Company A; $50,000 for Company B; $36,000 for Company C

b. If you had the choice to be hired by one of these companies, which one would you work for and

why?

Answer: MaleCompany A because it has the highest male salary.

FemaleCompany B because it has the highest female salary.

c. What is the difference in female mean salary between Company B and Company C?

Answer: $50,000$36,000 = $ 14,000

0

10,000

20,000

30,000

40,000

50,000

60,000

Male Female

Dollars

Mean Salaries

Company A

Company B

Company C

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Example 2: The graph above describes the monthly snowfall in a city. Use the bar graph to answer the

following questions.

a. What was the snowfall in May? Answer: 1 inch

b. What was the snowfall in November? Answer: 6 inches

c. Which month had a snow fall that was greater than 5 inches but less than 10 inches?

Answer: November

Example 3: Use the line graph to answer the following questions.

a. Which month had the highest number of iPods sold? Answer: April

b. Which month had the lowest number of iPods sold? Answer: February

c. How many Ipods were sold in the month of March? Answer: 52

d. Over this four month period, did the number of iPods sold increase, decrease, or remain the same?

Answer: From January to February, the number sold decreased, but after February, there was an

increase in sales.

0

5

1015

20

25

Inchesofsnowfall

Month

Monthly Snowfall

0

10

20

30

40

50

6070

January February March April

iPod Sales

iPod Sales

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1.7 Bar Graphs and Line Graphs Exercises

1. What is the approximate number of people whose height is 63 inches?

6360575451

15

10

5

0

Heights (in inches)

Frequency

Frequency Distribution of Heights

2. Which age group has the greatest number of DUIs forSlate County?

65+46-6530-4521-2916-20

10

8

6

4

2

0

Age

#

ofDUI's(inhundreds)

2001 DUI figures for the County

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3. In what year was the greatest number of graduates?

200920082007200620052004200019901980

25

20

15

10

5

0

Year

Graduates

Chart of Graduates

4. In what year were there the fewest dropouts?

200920082007200620052004200019901980

35

30

25

20

15

10

5

0

Year

Dropouts

Chart of Dropouts

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5. How many tickets were sold in the third week?

654321

60

50

40

30

20

10

0

Week

#

ofTicketsSold

Tickets Sold Each Week

6. In what year was the number of graduates and the number of dropouts the same?

Year

2009

2008

2007

2006

2005

2004

2000

1990

1980

Drop

outs

Grad

uates

Drop

outs

Graduates

Drop

outs

Graduates

Drop

outs

Grad

uates

Drop

outs

Grad

uates

Drop

outs

Grad

uates

Drop

outs

Grad

uates

Drop

outs

Graduates

Drop

outs

Grad

uates

35

30

25

20

15

10

5

0

Data

Chart of Graduates, Dropouts

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7. In what year did the number of graduates and the number of dropouts differ most?

Year

2009

2008

2007

2006

2005

2004

D

ropouts

Gr

aduates

D

ropouts

Gr

aduates

D

ropouts

Gr

aduates

D

ropouts

Gr

aduates

D

ropouts

Gr

aduates

D

ropouts

Gr

aduates

35

30

25

20

15

10

5

0

Data

Chart of Graduates, Dropouts

Use this chart for problems 8 and 9.

157152147142137

15

10

5

0

Weights (in lbs)

Frequency

Distribution of Womens Weights

8. How many people weigh 152 pounds?

9. Which is greater: the number of people who weigh more than 152 pounds or the number of peoplewho weigh less than 152 pounds?

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10. Which search engine has the most users? Which search engine has the fewest users?

YahooBingGoogle

20

15

10

5

0

Search Engine

Count

Serach Engine Preference

11. In what year was the greatest number of graduates?

2010200520001995199019851980

26

24

22

20

18

16

14

12

10

Year

Graduates

Line graph of Graduates

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12. Are the sales of Widgets at Company X increasing or decreasing over time?

Use the graph for problems 13 and 14.

13. What was the value of Sarahs Car in 2004?

14. Is the value of Sarahs car increasing or decreasing over time?

0

5000

10000

15000

20000

25000

30000

35000

1 2 3 4 5 6 7 8 9 10

#ofWidgetsSold

Annual Sales

Year

Widgets Sold

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15. What was the temperature on Day 5 in New York City?

Use the graph for problems 16 and 17.

16. What was Sams weight in the month of March?

17. How much weight did Sam gain from February to April?

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18. How much did the value ofSarahs car depreciate from 2005 to 2006?

19. What is the stores busiest time?

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20. How many 18 year olds in Smalltown have a cell phone?

1.7 Bar Graphs and Line Graphs Exercises Answers

1. people2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.

#ofTeenswithCellPhones