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187

Dividing Fractions2.7

2.7 OBJECTIVES

1. Divide fractions2. Divide mixed numbers3. Solve applications involving division of fractions

We are now ready to look at the operation of division on fractions. First we will need a newconcept, the reciprocal of a fraction.

We invert, or turn over, a fraction to write its reciprocal.

Rules and Properties: The Reciprocal of a Fraction

The product of any number and its reciprocal is 1. (Every number except zerohas a reciprocal.)

Rules and Properties: Reciprocal Products

NOTE In general, the

reciprocal of the fraction is .

Neither a nor b can be 0.

ba

ab

Example 1

Finding the Reciprocal of a Fraction

Find the reciprocal of (a) , (b) 5, and (c) .

(a) The reciprocal of is .

(b) The reciprocal of 5, or , is .

(c) The reciprocal of , or , is .3

5

5

31

2

3

1

5

5

1

Just invert, or turn over, thefraction.

4

3

3

4

12

3

3

4

C H E C K Y O U R S E L F 1

Find the reciprocal of (a) and (b) .314

58

An important property relating a number and its reciprocal follows.

Write 5 as and then turn

over the fraction.

51

Write as , then invert.53

123

We are now ready to use the reciprocal to find a rule for dividing fractions. Recall thatwe can represent the operation of division in several ways. We used the symbol � earlier.Remember that a fraction also indicates division. For instance,

3 � 5 �In this statement, 5 is called thedivisor. It follows the division sign �and is written below the fraction bar.

3

5

NOTE 3 � 5 and both mean

“3 divided by 5.”

35

188 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS

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Writing a Quotient as a Complex Fraction

Write as a complex fraction.

The numerator is .

The denominator is .45

A complex fraction is written by placing thedividend in the numerator and the divisor inthe denominator.

2

3

4

5

23

2

3�

4

5

Example 2

C H E C K Y O U R S E L F 2

Write � as a complex fraction.34

25

Let’s continue with the same division problem.

Dividing Two Fractions

(1)

(2)

We see from lines (1) and (2) that

2

5�

3

4�

2

5�

4

3

Recall that a number divided by 1 is justthat number.

�2

5�

4

3

The denominator becomes 1.�

2

5�

4

3

1

�

2

5�

4

3

3

4�

4

3

Write the original quotient as acomplex fraction.

�

2

5

3

4

2

5�

3

4

Multiply the numerator and denominator

by , the reciprocal of the denominator.

This does not change the value of thefraction.

43

Example 3

Using this information, we can write a statement involving fractions and division as acomplex fraction, which has a fraction as both its numerator and denominator, as Example 2illustrates.

NOTE Do you see a rulesuggested?

DIVIDING FRACTIONS SECTION 2.7 189©

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We would certainly like to be able to divide fractions easily without all the work of thelast example. Look carefully at the example. The following rule is suggested.

To divide one fraction by another, invert the divisor (the fraction after thedivision sign) and multiply.

Rules and Properties: To Divide Fractions

C H E C K Y O U R S E L F 3

Write � as a multiplication problem.78

35

C H E C K Y O U R S E L F 4

Divide.

2

5�

3

4

Example 4

Dividing Two Fractions

Divide.

We invert the divisor, , then multiply.

�1 � 7

3 � 4�

7

12

47

�1

3�

7

4

1

3�

4

7

Let’s look at another similar example.

Example 4 applies the rule for dividing fractions.

Dividing Two Fractions

Divide.

Write the quotient as a mixednumber if necessary.

�5

8�

5

3�

5 � 5

8 � 3�

25

24� 1

1

24

5

8�

3

5

Example 5

NOTE Remember, the numberinverted is the divisor. It followsthe division sign.

190 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS

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Simplifying will also be useful in dividing fractions. Consider the next example.

C H E C K Y O U R S E L F 6

Divide.

4

9�

8

15

Dividing Two Fractions

Divide.

1

2

�3 � 7

5 � 6�

7

10

Invert the divisor first! Thenyou can divide by thecommon factor of 3.

�3

5�

7

6

3

5�

6

7

When mixed or whole numbers are involved, the process is similar. Simply change themixed or whole numbers to improper fractions as the first step. Then proceed with thedivision rule. Example 7 illustrates this approach.

Dividing Two Mixed Numbers

Divide.

1

2

�19

14� 1

5

14

Invert the divisor and multiplyas before.

�19

8�

4

7

Write the mixed numbers asimproper fractions.

�19

8�

7

42

3

8� 1

3

4

Example 6

Example 7

NOTE Be careful! We mustinvert the divisor before anysimplification.

C H E C K Y O U R S E L F 5

Divide.

5

6�

3

7

DIVIDING FRACTIONS SECTION 2.7 191©

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Example 8 illustrates the division process when a whole number is involved.

Dividing a Mixed Number and a Whole Number

Divide and simplify.

3

2

�3

10

Invert the divisor, then divideby the common factor of 3.

�9

5�

1

6

�9

5�

6

11

4

5� 6

Example 8

C H E C K Y O U R S E L F 7

Divide.

31

5� 2

2

5

C H E C K Y O U R S E L F 8

Divide.

8 � 44

5

NOTE Write the whole

number 6 as .61

Units AnalysisWhen dividing by denominate numbers that have fractional units, we multiplyby the reciprocal of the number and its units.

Examples

500 mi � � 500 mi � � 20 gal

$24,000 � � 24,000 dol � � 24,000 dol � � 60 yr

(As always, note that in each case, the arithmetic of the units produces the finalunits.)

1 yr400 dol

400 dol1 yr

$4001 yr

1 gal25 mi

25 mi1 gal

As was the case with multiplication, our work with the division of fractions will be usedin the solution of a variety of applications. The steps of the problem-solving process remainthe same.

192 CHAPTER 2 MULTIPLYING AND DIVIDING FRACTIONS

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Some applications require both multiplication and division. Example 11 is such anapplication.

Example 9

An Application Involving the Division of Mixed Numbers

Jack traveled 140 kilometers (km) in hours (h). What was his average speed?

Distance Time

Speed � 140 km � h

� km � h

20

1

� 60 km/h

�140

1�

3

7

km

h

7

3

140

1

We know the distance traveled andthe time for that travel. To find theaverage speed, we must usedivision. Do you remember why?

21

3

21

3NOTE A kilometer,abbreviated km, is a metric unit

of distance. It is about mi.610

NOTE The important formulais Speed � distance � time.

NOTE We must divide thelength of the longer piece bythe desired length of theshorter piece.

C H E C K Y O U R S E L F 9

A light plane flew 280 mi in h. What was its average speed?134

Example 10

An Application Involving the Division of Mixed Numbers

An electrician needs pieces of wire inches (in.) long. If she has a -in. piece of wire,

how many of the shorter pieces can she cut?

8 1

1 1

� 8 pieces

�104

5�

5

13

204

5� 2

3

5�

104

5�

13

5

204

52

3

5

C H E C K Y O U R S E L F 1 0

A piece of plastic water pipe 63 in. long is to be cut into lengths of in. How many

of the shorter pieces can be cut?

312

is read “kilometers per hour.”

This is a unit of speed.

kmh

DIVIDING FRACTIONS SECTION 2.7 193©

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NOTE We are using thegardener/contractor definitionof a “yard” of mulch. It isactually 1 yd � 1 yd � 1 yd or1 yd3.

Example 11

An Application Involving the Division of Mixed Numbers

A parcel of land that is miles (mi) long and mi wide is to be divided into tracts that

are each square mile (mi2). How many of these tracts will the parcel make?

The area of the parcel is its length times its width:

We need to divide the total area of the parcel into mi2 tracts.

The land will provide 10 tracts, each with an area of mi2.1

3

10

3 mi2 �

1

3 mi2 �

10

3 mi2 �

3

1 mi2 � 10

1

3

�10

3 mi2

�5

2 mi �

4

3 mi

Area � 21

2 mi � 1

1

3 mi

1

3

11

32

1

2

C H E C K Y O U R S E L F 1 1

A parcel of land that is mi long and mi wide is to be divided into mi2 tracts.

How many of these tracts will the parcel make?

13

212

313

In our final example, we will look at a case in which the divisor has fractional units.

An Application Involving Mixed Numbers

Jackson has yard of mulch. His garden needs yard per row. How many rows can he

cover with the mulch?

We have yard and . Even if you don’t immediately see how to solve the prob-

lem, units analysis can help. The units of the answer will be “rows.” To get there, we need

to have the yard units cancel. That will happen if we divide the by the

He can cover all of nine rows and part (three-fourths) of the tenth row.

61

2 yard �

2 yard3 row

�132

yard �2 yard3 row

�132

yard �3 row2 yard

�394

row � 934

rows

2

3!6

1

2

2 yard

3 row6

1

2

2

36

1

2

Example 12

C H E C K Y O U R S E L F 1 2

Tangela has $4100 to invest in a certain stock. If the stock is selling at $ per

share, how many shares can she buy?

2558

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C H E C K Y O U R S E L F A N S W E R S

1. (a) (b) 2. 3. 4.

5. 6.

7. 8. 9. 160 mi/h

10. 18 pieces 11. 25 tracts 12. 160 shares

12

33

1

5� 2

2

5�

16

5�

12

5�

16

5�

5

12�

4

3� 1

1

3

4

9�

8

15�

4

9�

15

8�

4 � 15

9 � 8�

5

61

17

18

8

15

3

5�

8

7

2

5

3

4

31

4 is

13

4, so the reciprocal is

4

13

8

5;

1 5

3 24 1

1 3

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Exercises

Find the reciprocal of each of the following.

1. 2. 3. 4. 6

5. 6. 7. 8.

Divide. Write each result in simplest form.

9. 10. 11.

12. 13. 14.

15. 16. 17.

18. 19. 20.

21. 22. 23.

24. 25. 26.

27. 28. 29.

30. 31. 32.

33. 34. 35.

36. 37. 38. 51

3� 2

2

53

3

4� 1

3

83

1

2� 2

4

5

11

3� 1

1

7

7

18� 5

5

65

3

5�

7

15

13

8�

5

12

7

12� 2

1

3

9

14� 2

4

7

13

5�

4

152

4

7� 1215 � 3

1

3

3

4� 9

12

17� 6

5

8� 5

12 �2

327 �

3

7

4

5� 4

9

28�

27

35

5

27�

25

36

5

27�

15

54

8

15�

2

5

8

9�

11

15

7

10�

5

9

5

9�

8

11

8

9�

4

3

5

8�

3

4

2

5�

3

4

2

5�

1

3

1

5�

3

4

1

89

3

44

3

52

1

3

1

29

5

7

8

2.7

Name

Section Date

ANSWERS

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

13. 14.

15. 16.

17. 18.

19. 20.

21. 22.

23. 24.

25. 26.

27. 28.

29. 30.

31. 32.

33. 34.

35. 36.

37. 38.

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Divide the following. Be sure to attach the proper units.

39. 900 mi � 15 40. 1500 joules � 75

41. 8750 watts � 350 42. $75,744 �

Solve the following applications.

43. Wire cutting. A wire feet (ft) long is to be cut into 7 pieces of the same length.

How long will each piece be?

44. Quantity. A potter uses pound (lb) of clay in making a bowl. How many bowls

can be made from 16 lb of clay?

45. Speed. Virginia made a trip of 95 miles (mi) in hours (h). What was her average

speed?

46. Unit pricing. A piece of land measures acres and is for sale at $60,000. What is

the price per acre?

47. Number of servings. A roast weighs lb. How many lb servings will the roast

provide?

48. Number of books. A bookshelf is 55 inches (in.) long. If the books have an average

thickness of how many books can be put on the shelf?11

4 in.,

1

43

1

4

33

4

11

4

2

3

51

4

$3156

monthwatts

s

jouless

migal

ANSWERS

39.

40.

41.

42.

43.

44.

45.

46.

47.

48.

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49. Quantity. A butcher wants to wrap lb packages of ground beef from a cut of meat

weighing lb. How many packages can be prepared?

50. Quantity. A manufacturer has yards (yd) of imported cotton fabric. A shirt

pattern uses yd. How many shirts can be made?

51. Number of pieces. A stack of in. thick plywood is 48 in. high. How many sheets

of plywood are in the stack?

52. Area. A landfill occupies land that measures mi by mi. If there are 144 cellsin the landfill, what is the area of each cell?

53. Manuel has yd of cloth. He wants to cut it into strips yd long. How many strips

will he have? How much cloth remains, if any?

54. Evette has ft of string. She wants to cut it into pieces ft long. How many

pieces of string will she have? How much string remains, if any?

55. In squeezing oranges for fresh juice, three oranges yield about of a cup.

(a) How much juice could you expect to obtain from a bag containing 24 oranges?

(b) If you needed 8 cups of orange juice, how many bags of oranges should you buy?

56. A farmer died and left 17 cows to be divided among three workers. The first worker

was to receive of the cows, the second worker was to receive of the cows, and the

third worker was to receive of the cows. The executor of the farmer’s estate realized

that 17 cows could not be divided into halves, thirds, or ninths and so added aneighbor’s cow to the farmer’s. With 18 cows, the executor gave 9 cows to the firstworker, 6 cows to the second worker, and 2 cows to the third worker. This accountedfor the 17 cows, so the executor returned the borrowed cow to the neighbor. Explainwhy this works.

1

9

1

3

1

2

1

3

33

441

1

2

13

47

1

2

63

410

2

3

3

4

13

4

451

2

191

8

3

8

ANSWERS

49.

50.

51.

52.

53.

54.

55.

56.

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57. Division of fractions is not commutative.

For example,

There could be an exception. Can you think of a situation in which division offractions would be commutative?

58. Josephine’s boss tells her that her salary is to be divided by . Should she quit?

59. Compare the English phrases: “divide in half ” and “divide by one-half.” Do they saythe same thing? Create examples to support your answer.

60. (a) Compute the following: ;

(b) As the divisor gets smaller (approaches 0), what happens to the quotient?

(c) What does this say about the answer to 5 � 0?

Answers

1. 3. 2 5. 7. 9. 11. 13. 15.

17. 19. 21. 23. 18 25. 27. 29. 6

31. 33. 12 35. 37. 39. 60 gal 41. 25 s

43. ft 45. 76 47. 13 servings 49. 51 packages

51. 64 sheets 53. 4; yd 55. cups; 3 bags 57.

59.

22

3

1

2

mih

3

4

28

111

1

6

1

4

41

2

2

17

1

5

4

151

1

3

113

50

2

3

8

15

4

15

4

39

3

7

8

7

5 �1

10,000.5 �

1

10; 5 �

1

100; 5 �

1

1000

1

3

3

4�

5

6�

5

6�

3

4.

ANSWERS

57.

58.

59.

60.

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199

Using Your Calculator to Divide Fractions

Dividing fractions on your calculator is almost exactly the same as multiplying them. Yousimply press the key instead of the key.

Scientific Calculator

Dividing fractions on a scientific calculator requires only that you enter the problemfollowed by the equal sign.

��

Example 1

Dividing Two Fractions

Find the quotient

The keystroke sequence is

23 24 13 16

Some scientific calculators display the results of dividing fractions as improper fractions,

and some display them as mixed numbers. Thus the answer to might be displayed

as or . Be sure you understand that the two answers are equivalent.1

7

39

46

39

23

24�

13

16

�a b/c�a b/c

23

24�

13

16

C H E C K Y O U R S E L F 1

Find the quotient

12

17�

3

16

Graphing Calculator

When using a graphing calculator, you must choose the fraction option fromthe menu before pressing .

The keystroke sequence for the fraction problem in Example 1, is

23 24 13 16 Enter1:� Frac)�(��

23

24�

13

16,

EnterMATH1:� Frac

C H E C K Y O U R S E L F A N S W E R

1.64

17 or 3

13

17

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Name

Section Date

ANSWERS

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Calculator Exercises

Find the following quotients using your calculator.

1. 2.

3. 4.

5. 6.

7. 8.

9. 10.

11. 12.

Answers

1. 3. 5. 7. 9. 11.3

4

1

2

16

21

10

3 or 3

1

3

4

3 or 1

1

3

3

2

86

24�

258

96

25

45�

100

135

19

63�

38

9

15

18�

45

27

2

3�

4

9

20

27�

35

36

13

15�

39

5

20

9�

10

15

3

7�

9

28

5

7�

15

28

13

17�

39

34

1

5�

2

15

200