3.6 equations and fractions - bobprior.combobprior.com/mathhelp/livingmath3-6.pdf · equations and...

Equations and Fractions page 3.6 – 1

3.6 Equations and Fractions EQUATIONS INVOLVING THE MULTIPLICATION OF FRACTIONS In Section 1.5 you learned how to solve equations involving multiplication. Let’s take a look at an example as a reminder.

Example 1: Divide both sides of each equation by the coefficient so that it isolates the variable.

a) N x 6 = 18. b) 8 x N = 40. Answer: a) N x 6 = 18 To isolate the variable we need to divide

both sides by the coefficient, 6. N x 6 ÷ 6 = 18 ÷ 6 Apply division to both sides. N x 1 = 3 Since 1 is the identity for multiplication, N x 1 becomes just N. N = 3 We now know the value of the variable; it is no longer an unknown value. b) 8 x N = 40 This time, we need to rewrite the left side so that N is written first; in other words, we need to commute the left side. N x 8 = 40 Now we’re in a position to divide both side by the coefficient, 8. N x 8 ÷ 8 = 40 ÷ 8 Apply division to both sides. N x 1 = 5 Since 1 is the identity for multiplication, N x 1 becomes just N. N = 5 So, N = 5 is the solution.


YTI #1 Divide both sides by the coefficient of the equation so that it isolates the variable.

(You might not need all of the lines below.) a) N x 5 = 30 b) 3 x N = 21

Notice that, in isolating the variable, we were able to—toward the very end of the process—

multiply N by just 1; after that step, N was isolated. Of course, the way we were able to get a factor of 1 was to divide by the coefficient of N.

If we are, instead, presented with an equation in which the coefficient is a fraction, the process for

solving could be exactly the same: divide both sides by the coefficient, as in Example 2.

Example 2: Isolate the variable by dividing both sides by the coefficient.

N x 23 = 10

Answer: N x 23 = 10 To isolate the variable we need to divide both

sides by the coefficient, 23 .

N x 23 ÷

23 = 10 ÷

23 Apply division to both sides by using “invert

and multiply.”

N x 23 x

32 = 10 x

32

N x 23 x

32 =

101 x

32 Multiply the fractions together.

N x 66 =

302 Simplify each fraction.

N x 1 = 15 Again we get 1, the identity for multiplication; N x 1 becomes just N. N = 15 So, N = 15 is the solution.


A quicker way to isolate the variable—instead of dividing by the fraction—would be to multiply

by the fraction’s reciprocal. Recall from Sec. 3.5 that the product of a fraction and its reciprocal is always 1.

Example 3: Isolate the variable by multiplying both sides by the reciprocal of the coefficient.

a) N x 23 = 10. b)

85 x N =

163

Procedure: Multiply both sides by the reciprocal of the coefficient of N.

Answer: a) N x 23 = 10

32 is the reciprocal of

23 , so multiply both

sides by 32 .

N x 23 x

32 = 10 x

32

Make 10 into a fraction: 101

N x 23 x

32 =

101 x

32

Multiply the fractions together.

N x 66 =

302

Simplify each fraction. N x 1 = 15 Again we get 1 is the identity for

multiplication; N x 1 becomes just N. N = 15 So, N = 15 is the solution.

b) 85 x N =

163 This time, we can commute the left side

so that N is written first.

N x 85 =

163 The reciprocal of

85 is

58 , so multiply

both

sides by 58 .


N x 85 x

58 =

163 x

58

N x 4040 =

163 x

58 At this point we can cross-cancel the

extremes by as factor of 8.

N x 1 = 23 x

51

N = 103 So, N =

103 is the solution.

YTI #2 Isolate the variable by multiplying both sides by the reciprocal of the coefficient.

Use Example 3 as a guide.

a) N x 52 = 20 b)

34 x N = 9

c) N x 38 =

125 d)

12 x N =

59

e) N x 47 =

821 f)

34 x N = 6


You have seen two different types of coefficients in these equations. When the coefficient is a

whole number, it is easier to divide; when the coefficient is a fraction, it is easier to multiply by its reciprocal. APPLICATIONS

In Section 2.7 you were introduced to some situations that require using multiplication. At that time, we explored the formula

(Number of Same Parts) x Part = Whole.

or

Repetitions x Part = Whole. Some of the situations you’ll see here will be dividing something—the whole—into equal parts. Also remember that the whole and the parts are always of equal units of measure. For example, a dozen cookies can be divided easily between 4 children. The question might be,

“How many cookies will each child get?” It’s like asking for the number of cookies per child. You probably know that each child will receive 3 cookies, but we’ll use this simple situation to illustrate the proper way to approach the problem.

Notice that the word dividing is used. We will still set up the equation using multiplication, but—

in order to isolate the variable—we will need to divide to find the answer. Also, remember that the part can be identified by the words each and per.

WHEN THE PART IS UNKNOWN

Example 4: If a dozen cookies is to be divided evenly among four children, how many will each child receive?

Procedure: In this case, the whole is the total number of cookies. The part, the unknown, is how many cookies each child will receive. (Notice

that both the whole and the part are “measured” in cookies.) Since the part is shared (divided evenly) among 4 children, the number of times

the part is repeated is 4.


Answer Heading: Let N = the number of cookies each child receives. 4 x N = 12 Commute and divide. N x 4 ÷ 4 = 12 ÷ 4 N x 1 = 3

N = 3 So, Each child will receive 3 cookies.

It may seem that the last example is a bit too easy, especially since it doesn’t involve dividing

fractions. Consider this next example, though; similar, but different.

Example 5: Andy has a strip of wood that is 94 inches wide and needs to be divided into 3

smaller strips of equal width. How wide will each smaller strip be?

Procedure: Picture yourself in the situation. You’ve got a strip of wood that is 94 inches

wide. You are going to cut it (evenly) into smaller pieces.

Here, we’re given a width

94 inches and we’re asked to find a width. This

must mean that the “widths” (in inches) are the part and the whole.

The larger (original) piece must be the whole (total width of 94 inches);

Each smaller strip is a part, the width is unknown, and the part is repeated 3

times (there are to be 3 smaller strips) Answer Heading: Let N = the width of each smaller strip.

3 x N = 94 Commute and divide by the

coefficient.


N x 3 ÷ 3 = 94 ÷ 3

Change this to “invert and multiply”

N x 31 x

13 =

94 x

13 by the reciprocal of the coefficient.

N x 1 = 9

12 Reduce by a factor of 3.

N = 34

So, Each strip will be 34 inches wide.

Example 6: A mother has a sick child and needs to divide up the liquid cough medicine into 6

equal doses. If she has 34 of a cup of cough syrup, how much cough syrup will

be in each dose?

Procedure: Here we have 34 cup of cough syrup, and we’re asked to find the amount of

cough syrup (number of cups) in each dose. Cups is the common measure

between the part and the whole. So, the whole is the 34 cup of cough medicine.

The part (the amount of cough syrup in each dose) is unknown, and it is repeated

6 times (there are to be 6 doses) Answer: Heading: Let N = the amount (cups) of cough syrup in each dose.

6 x N = 34 Commute and divide by the coefficient.

N x 6 ÷ 6 = 94 ÷ 6 Change this to “invert and multiply”

by the reciprocal of the coefficient.


N x 61 x

16 =

34 x

16

N x 1 = 3

24 Reduce by a factor of 3.

N = 18 So, Each dose will be

18 cup.

Solve each of the following situation problems. Follow the outline one step at a time. YTI #3 Gail has a stack of flyers that need to be folded, and she asks Barbara and Jan to

help her. They decide to measure the height (thickness) of the stack; it turns out

to be 94 inches high. How high (how thick) should each person’s stack be if they

divide it evenly 3 ways? a) Identify the whole: b) Identify the part: c) Identify the number of repetitions: d) Write the

heading e) Write and solve the equation:


f) Write a sentence answering the question.

In the next few exercises, use the same outline, as above, as needed.

YTI #4 Hank works for a landscaper. Their most recent job requires them to prepare,

plant and cultivate 25 of an acre of land. The owner of the company is assigning

Hank and 3 others (4 workers in all) to the job and wants them to divide the land up evenly amongst themselves. What part of an acre will each employee be responsible for?

Write a sentence answering the question.

YTI #5 A huge soccer field is 38 of a mile long. Sandra works for the city’s Parks and

Recreation Department, and it is her job to mark off the fields appropriately. She needs to mark off 6 fields (of equal length) before the start of the soccer season. How long will each field be?



WHEN THE NUMBER OF REPETITIONS IS UNKNOWN Every situation presented in this section thus far has involved the formula

(Number of Same Parts) x Part = Whole.

or

Repetitions x Part = Whole.

To this point, the part has been the unknown. We have known the whole and the number of repetitions and needed to find the part, so we say, Let N = the part.

In other situations, we might know the whole and the size of the part we wish to obtain, but we

might not know the number of repetitions of the part that will be needed. Consider the following


Example 7: Andy needs small strips of wood that are only 38 inches wide. How many of

these strips of wood can he get from a board that is 154 inches wide?

Procedure: Put yourself in Andy’s position. He has a board that is 154 inches wide and

needs to cut it up into smaller strips. The board is the whole

154 inches and

the width of each strip

38 inches is the part. Both of these are in inches and

are known values, so the unknown value is the number of repetitions, which is mentioned in the question.

Answer: Heading: Let N = the number of strips.

N x 38 =

154 Multiply both sides by the reciprocal

of the coefficient.

N x 38 x

83 =

154 x

83 Simplify the right side by cross-canceling;

reduce the extremes by a factor of 3 and the means by a factor of 4.

N x 1 = 51 x

21

N = 101 = 10

So, Andy can get 10 smaller strips out of the board.


YTI #6 At a bicycle making factory, one bicycle can be assembled in 23 of an hour. The

factory is in operation 24 hours a day. How many bicycles can the factory make in 24 hours?

a) Identify the whole: b) Identify the part: c) Identify the number of repetitions: d) Write the

heading e) Write and solve the equation: f) Write a sentence answering the question.

In the next two exercises, use the same outline, as above, as needed.

YTI #7 Marie works in a candy shop. The shop advertises home made candies, and one

of their specialties is a “brick” of fudge. Marie is given a 6-inch brick of fudge,

and she needs to cut it into smaller bricks each having a width of 34 inches.

How many smaller bricks of fudge can she cut from the 6-inch brick?



YTI #8 At a gift wrapping booth, each package gets a ribbon bow made of 34 foot of

ribbon. If a spool has 24 feet of ribbon on it, how many bows can be made form one spool?


WHEN THE WHOLE IS UNKNOWN

There are some situations where the whole is unknown. Of course, when that occurs, the variable,

N, represents the whole.

Example 8: At a bookstore, Devra needs to place 32 copies of the new best-selling paperback

book tightly together on a shelf. If each book is 54 inches thick, how many inches

of the shelf will the 32 books take up? Procedure: As Devra is placing the books on the shelf, the part, the width (thickness) of each

is repeated over and over, 32 times in all. Taken together, they will add up to the whole, which is unknown. (The common unit of measure is inches, so the part and the whole will be in inches.)


Answer: Heading: Let N = the total number of inches.

The part is 54 inches, and it is repeated 32 times; so the equation is

32 x 54 = N

321 x

54 = N

Reduce the extremes by a factor of 4.

81 x

51 = N

40 = N So, The 32 books will take up 40 inches on the shelf.

YTI #9 Katie uses 56 yards of fabric for every doll dress she makes. How many yards of

fabric is needed to make 12 doll dresses? a) Identify the whole: b) Identify the part: c) Identify the number of repetitions: d) Write the

heading e) Write and solve the equation:

f) Write a sentence answering the question.


In the next two exercises, use the same outline, as above, as needed.

YTI #10 Nate’s dog, Banjo, eats 23 pound of dog food each day. How many pounds of

dog food does Banjo eat each month (30 days)?


YTI #11 Tim is a janitor for an elementary school. It takes him 14 hour to clean each

classroom at the end of the school day. How long (how many hours) should it take Tim to clean all 28 classrooms?



Answers to each “You try it” Exercise

Section 3.6

YTI #1: a) N x 5 = 30 b) 3 x N = 21 N = 6 N = 7 So, N = 6 is the solution So, N = 7 is the solution

YTI #2: a) N x 52 = 20 b)

34 x N = 9

N = 8 N = 12 So, N = 8 is the solution So, N = 12 is the solution

c) N x 38 =

125 d)

12 x N =

59

N = 325 N =

109

So, N = 325 is the solution So, N =

109 is the solution

e) N x 47 =

821 f)

34 x N = 6

N = 23 N = 8

So, N = 23 is the solution So, N = 8 is the solution

YTI #3: a) Whole: 94 YTI #4: Whole:

25

b) Part: N Part: N

c) Repetitions 3 Repetitions 4


d) Heading: Let N = the thickness Heading: Let N = the part of

of each person’s stack an acre each is responsible for

e) 3 x N = 94 4 x N =

25

N = 9

12 = 34 N =

220 =

110

So, Each person’s stack should be 34 inches high. So, Each will be responsible for

110 of an acre.

YTI #5: Whole: 38 YTI #6: Whole: 24

Part: N Part: 23

Repetitions 6 Repetitions N

Heading: Let N = the part of an Heading: Let N = the number

acre each is responsible for of bicycles

6 x N = 38 N x

23 = 24

N = 1

16 N = 36

So, Each field will be 1

16 of a mile long. So, They can make 36 bicycles

in 24 hours. YTI #7: Whole: 6 YTI #8: Whole: 24

Part: 34 Part:

34


Repetitions N Repetitions N

Heading: Let N = the number of Heading: Let N = the number

smaller bricks of fudge of bows

N x 34 = 6 N x

34 = 24

N = 8 N = 32 So, She can cut 8 smaller bricks of fudge. So, 32 bows can be made from

one spool. YTI #9: Whole: N YTI #10: Whole: N

Part: 56 Part:

23

Repetitions 12 Repetitions 30

Heading: Let N = the number of Heading: Let N = the number

yards of fabric of yards of fabric

12 x 56 = N 30 x

23 = N

N = 10 N = 20 So, Katie needs 10 yards of fabric. So, Banjo eats 20 pounds of

dog food each month.

YTI #11: Whole: N Part: 14 Repetitions 28

Heading: Let N = the number of yards of fabric

28 x 14 = N

N = 7 So, It should take Tim 7 hours to clean the rooms.


Section 3.6 Focus Exercises

1. Isolate the variable by multiplying both sides by the reciprocal of the coefficient. (You might not need all of the lines below.) Use Example 3 as a guide.

a) N x 115 = 22 b) N x

415 =

169

c) 89 x N = 24 d)

107 x N =

2542

For each application, use the outline described in this section. Be sure to write out the heading, set

up and solve the equation, and write a sentence answering the question. 2. Ramon has written a book that contains colorful graphs and charts. His computer printer can print

out the entire book in 23 hour. He needs to print out 6 copies for his team of editors. How long

will it take to print out 6 copies of his book?


Sentence:

3. The Music Room gives guitar lessons 12 hours each day. Each lesson lasts 34 hour. How many

lessons are there each day? Sentence:

4. At an amusement park, employees rotate from location to location so that they have a variety of

duties to do each day. One location, a hot dog cart, is open for eight hours. If 8 different employees share the 6 hours that the cart is open, how much time does each employee spend there?


Sentence:

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