THINK IT THROUGH Budgets

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3.4 Adding and Subtracting Fractions

251

Solve We must ﬁnd the sum of three fractions with different denominators. To ﬁnd the LCD, we prime factor the denominators and use each prime factor the greatest number of times it appears in any one factorization:

6 2 ~3

4 2 2 ¶LCD 2 2 3 5 60 15 3 ~5


The LCD for				1	,	1	, and			7	is 60.
The LCD for			6		,	4	, and		15		is 60.
1	1	7			1		10	1			15	7	4
6	4	15			6		10	4			15	15	4

2 appears twice in the factorization of 4.

3 appears once in the factorization of 6 and 15.

5 appears once in the factorization of 15.

Build each fraction so that its denominator is 60.

1060 1560 2860

10 15 28

5360

Multiply the numerators. Multiply the denominators. The denominators are now the same.

	Add the numerators and write the sum	1
		10
	over the common denominator 60.
		15

	This fraction is in simplest form.	28
		53

State The fraction of the student body that watches 0 to 2 hours of TV daily is 5360 .

Check We can check by estimation. The result, 5360 , is approximately 5060 , which simpliﬁes to 56 . The red, yellow, and blue shaded areas appear to shade

about 56 of the pie chart. The result seems reasonable.

ANSWERS TO SELF CHECKS
ANSWERS TO SELF CHECKS
1. a.	1	b.	7	2.	6	3.	2	4.	9	5.	9	6.	3	7.	45	8.	23	9.	3	10.	3	11.	7
1. a.	2	b.	9	2.	11	3.	3	4.	10	5.	35	6.	2	7.	8	8.	24	9.	20	10.	5	11.	12

Fill in the blanks.

5. To add (or subtract) fractions that have the same

denominator, add (or subtract) their

write the sum (or difference) over the

denominator.

the result, if possible.

“Putting together a budget is crucial if you don’t want to spend your way into serious problems.You’re also developing a habit that can serve you well throughout your life.”

Liz Pulliam Weston, MSN Money

The circle graph below shows a suggested budget for new college graduates as recommended by Springboard, a nonproﬁt consumer credit counseling service. What fraction of net take-home pay should be spent on housing?

	2
	Utilities: ––
	25
	3
	Transportation:––
	20
	1
	Food: ––	Housing: ?
	10	Housing: ?
	1
	Debt: ––
	10
	1	1
	Clothing: ––	Personal: ––
	25	20
	1	2
	1	Medical: ––
	Savings: ––	Medical: ––
	20	25
	20

252 Chapter 3 Fractions and Mixed Numbers

S E C T I O N 3.4 STUDY SET

VOCABULARY

Fill in the blanks.

1.Because the denominators of 38 and 78 are the same number, we say that they have a

denominator.

2.	The				common denominator for a set of
	fractions is the smallest number each denominator
	will divide exactly (no remainder).
3.	Consider the solution below. To									an equivalent
	fraction with a denominator of 18, we multiply 4 by a
	1 in the form of						.	9
	1 in the form of						.
	4		4	2

		9	9			2
			8
			18
4.	Consider the solution below. To										the fraction
	2715 , we factor 15 and 27, and then remove the common
	factor of 3 from the								and the
				.
	15			1
	15			3 5
		27		3 3 3
				1
				5

				9

CONCEPTS

6.To add (or subtract) fractions that have different denominators, we express each fraction as an

equivalent fraction that has the for its

denominator. Then we use the rule for adding (subtracting) fractions that have the denominator.

7.When adding (or subtracting) two fractions with different denominators, if the smaller denominator is a factor of the larger denominator, the

denominator is the LCD.

8.Write the subtraction as addition of the opposite:

18 a 58b

9.Consider 34 . By what form of 1 should we multiply the numerator and denominator to express it as an

equivalent fraction with a denominator of 36?

10.The denominators of two fractions are given. Find the least common denominator.

a.	2 and 3	b.	3 and 5
c.	4 and 8	d.	6 and 36

11. Consider the following prime factorizations:

24 2 2 2 3

90 2 3 3 5

For any one factorization, what is the greatest number of times

a.a 5 appears?

b.a 3 appears?

c.a 2 appears?

12.The denominators of two fractions have their primefactored forms shown below. Fill in the blanks to ﬁnd the LCD for the fractions.

20 2	2	5
30 2	3	5 fLCD
30 2	3	5 fLCD

13.The denominators of three fractions have their primefactored forms shown below. Fill in the blanks to ﬁnd the LCD for the fractions.

20 2 2 5

30 2 3 5 ¶LCD

90 2 3 3 5

14.Place a or symbol in the blank to make a true statement.

3231

a.35 35

b.1317 1117

NOTATION

Fill in the blanks to complete each solution.

15.25 17 25 17 55

35 5

16.78 23 78 33 23

21 16

GUIDED PRACTICE

Perform each operation and simplify, if possible. See Example 1.


17.	4	1	18.	3	1
17.	9	9	18.	7	7
19.	3	1	20.	7		1
19.	8	8	20.	12		12

21.	11	7	22.	10	5
21.	15	15	22.	21	21
23.	11	3	24.	7	5

	20	20		18	18

Subtract and simplify, if possible. See Example 2.


25.	11	a	8	b		26.	15	a	11	b
	5		5				9		9
27.	7	a	2		b	28.	21	a	9	b
	21		21				25		25

Perform the operations and simplify, if possible. See Example 3.


29.		19		3		1	30.	11	1	7
	40		40		40			24	24	24
31.	13		1		7		32.	21	1	13
	33		33		33			50	50	50

Add and simplify, if possible. See Example 4.

33.	1	1	34.	1	1
33.	3	7	34.	4	5
35.	2	1	36.	2	1

	5	2		7	2

		3.4	Adding and Subtracting Fractions			253
Subtract and simplify, if possible. See Example 5.
37.	4	3	38.	2	3
37.	5	4	38.	3	5
39.	3	2	40.	6	2
39.	4	7	40.	7	3

Subtract and simplify, if possible. See Example 6.

41.	11	2	42.	11	1
41.	12	3	42.	18	6
43.	9	1	44.	13	2

	14	7		15	3

Add and simplify, if possible. See Example 7.

45.	2	5	46.	3	5
		9			8

47.	3	9	48.	1	7
		4			10

Add and simplify, if possible. See Example 8.


49.	1	5	50.	7		3
	6	8		12		8
51.	4	5	52.	1	5
	9	12		9	6

Subtract and simplify, if possible. See Example 9.

53.	9	3	54.	11	11
	10	14		12	30
55.	11	7	56.	7	5
	12	15		15	12

Determine which fraction is larger. See Example 10.

57.	3	or	5	58.	5	or	7
	8		16		6		12
59.	4	or	2	60.	7	or	4
	5		3		9		5
61.	7	or	11	62.	3	or	5

	9		12		8		12
63.	23	or	7	64.	19	or	5

	20		6		15		4

Add and simplify, if possible. See Example 11.


65.	1		5		2	66.	1		1			1
	6	18			9		10		8		5
67.	4			2	1	68.	1	3			3
	15		3		6		2	5		20


254					Chapter 3							Fractions and Mixed Numbers
		TRY IT YOURSELF
		TRY IT YOURSELF
	Perform each operation.
69.					1		a					5	b	70.					1		a						15	b
69.				12			a					12	b	70.				16			a						16	b
71.			4				2							72.		1					2

			5				3									4					3
73.			12					1				1		74.		7					1					1

			25					25				25				9					9					9
75.					7					1				76.				5				1

				20						5									8				3
77.					7					1				78.				17				4

				16						4								20							5
79.			11					2						80.		2					1

			12					3								3					6
81.			2				4			5				82.		3					2					3

			3				5				6					4					5					10
83.			9					1						84.		5					3
83.			20					30						84.		6					10
85.			27					5						86.		49						15

			50					16								50						16
87.			13					1						88.		71						1

			20					5								100								10
89.			37							17				90.		54						52
89.			103						103					90.		53						53
91.			3			5								92.		2						7

				4																		8
93.			4					1						94.		8					7

			27					6								9					12
95.			7					19						96.		73						31

			30					75								75						30
97.			Find the difference of											11	and	2				.
97.			Find the difference of											60	and		45			.

98.Find the sum of 48 and 40 .

99.Subtract 12 from 15 .

100. What is the sum of	11	and	7	increased by	5	?
	24		36		48

APPLICATIONS

101.BOTANY To determine the effects of smog on tree development, a scientist cut down a pine tree and measured the width of the growth rings for the last two years.

a.What was the growth over this two-year period?

b.What is the difference in the widths of the two rings?

102.GARAGE DOOR OPENERS What is the difference in strength between a 13 -hp and a 12 -hp garage door opener?

103.MAGAZINE COVERS The page design for the magazine cover shown below includes a blank strip at the top, called a header, and a blank strip at the bottom of the page, called a footer. How much page length is lost because of the header and footer?

3 in. header

–

FRAUD & SAT EVALUATION | jon cheater 8

THE TRUTH BEHIND COLLEGE TESTING | issac icue WHAT REALLY HAPPENS IN DORMS | laura life lesson

colleg e life

Page TODAY

length The

TRUTH about college

A Real

Student

talking with with kids all over America and in depth intreviews with Colby students and teachers

all the news that’s fit to print and quite a bit that isn’t PLUS

articles and lots of pictures gossip and trash and misinformation

5 in. footer

––

104.DELIVERY TRUCKS A truck can safely carry a one-ton load. Should it be used to deliver one-half ton of sand, one-third ton of gravel, and one-ﬁfth ton of cement in one trip to a job site?

105.DINNERS A family bought two large pizzas for dinner. Some pieces of each pizza were not eaten, as shown.

a.What fraction of the ﬁrst pizza was not eaten?

b.What fraction of the second pizza was not eaten?

c.What fraction of a pizza was left?

d.Could the family have been fed with just one pizza?

106.GASOLINE BARRELS Three identical-sized barrels are shown below. If their contents of the two of the barrels are poured into the empty third barrel, what fraction of the third barrel will be ﬁlled?

107.WEIGHTS AND MEASURES A consumer protection agency determines the accuracy of butcher shop scales by placing a known three- quarter-pound weight on the scale and then comparing that to the scale’s readout. According to the illustration, by how much is this scale off? Does it result in undercharging or overcharging customers on their meat purchases?

– pound 4

weight

–

3.4 Adding and Subtracting Fractions

255

108.FIGURE DRAWING As an aid in drawing the human body, artists divide the body into three

parts. Each part is then expressed as a fraction of the total body height. For example, the torso is 154 of the body height. What fraction of body height

is the head?

Head

Torso:

––

Below the waist:

–

109. Suppose you work as a	from Campus to Careers
school guidance counselor	School Guidance Counselor	iStockphoto.com/Monkeybusinessimages
at a community college
and your department has
conducted a survey of the
full-time students to learn
more about their study
habits. As part of a Power
Point presentation of the
survey results to the
survey results to the
school board, you show the following circle
graph. At that time, you are asked, “What
fraction of the full-time students study 2 hours
or more daily?” What would you answer?
More than 2 hr	2 hr
3
––	2
10	2
10	–
	5

	1
	––	1
	Less than 1 hr 10	1
	Less than 1 hr 10	–
		5
		1 hr

1 pound

256Chapter 3 Fractions and Mixed Numbers

110.HEALTH STATISTICS The circle graph below

shows the leading causes of death in the United States for 2006. For example, 1350 of all of the deaths that year were caused by heart disease. What

fraction of all the deaths were caused by heart disease, cancer, or stroke, combined?

	Alzheimer’s
	disease		Diabetes
	3		Diabetes
	–––		3
	100		–––
			100
	Other		Heart
	13		Heart
	13		disease
	––		disease
	50		13
			––
Respiratory			50
diseases
1
––
20		Cancer	Flu
Accidents		6	Flu
		6	1
		––	1
1		25	––
––	Stroke		50
20	Stroke
	3
	––
	50

Source: National Center for Health Statistics

111.MUSICAL NOTES The notes used in music have fractional values. Their names and the symbols used to represent them are shown in illustration (a). In common time, the values of the notes in each measure must add to 1. Is the measure in illustration

(b) complete?

Half	Quarter	Eighth	Sixteenth
note	note	note	note

(a)

(b)

112.TOOLS A mechanic likes to hang his wrenches above his tool bench in order of narrowest to widest. What is the proper order of the wrenches in the illustration?

113.TIRE TREAD A mechanic measured the tire tread depth on each of the tires on a car and recorded them on the form shown below. (The letters LF stand for left front, RR stands for right rear, and so on.)

a.Which tire has the most tread?

b.Which tire has the least tread?

		Measure of tire tread depth
1/4 in.	LF	RF 5/16 in.
7/32 in.	LR	RR 21/64 in.

114.HIKING The illustration below shows the length of each part of a three-part hike. Rank the lengths of the parts from longest to shortest.

	4
	–	C
	5 mi	C
3	B	5
3	B	– mi
– mi		8
4		D
A		D
A

WRITING

115.Explain why we cannot add or subtract the fractions 29 and 25 as they are written.

116.To multiply fractions, must they have the same denominators? Explain why or why not. Give an example.

REVIEW

Perform each operation and simplify, if possible.


117. a.	1				1		b.	1	1
	4			8				4	8
c.	1		1				d.	1	1
	4	8						4	8
118. a.	5					3	b.	5		3

	21					14		21		14
c.	5			3			d.	5		3

	21			14				21		14

1	in.	3	in.	3	5
–	in.	–	in.	–– in.	–– in.
4		8		16	32

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57.	3	or	5	58.	5	or	7
	8		16		6		12
59.	4	or	2	60.	7	or	4
	5		3		9		5
61.	7	or	11	62.	3	or	5

	9		12		8		12
63.	23	or	7	64.	19	or	5

	20		6		15		4

57.	3	or	5	58.	5	or	7
	8		16		6		12
59.	4	or	2	60.	7	or	4
	5		3		9		5
61.	7	or	11	62.	3	or	5

	9		12		8		12
63.	23	or	7	64.	19	or	5

	20		6		15		4

57.	3	or	5	58.	5	or	7
	8		16		6		12
59.	4	or	2	60.	7	or	4
	5		3		9		5
61.	7	or	11	62.	3	or	5

	9		12		8		12
63.	23	or	7	64.	19	or	5

	20		6		15		4