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Introduction to Fractions and Multiplication and Division of Fractions: Division of Fractions

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses division of fractions. By the end of the module students should be able to determine the reciprocal of a number and divide one fraction by another.

Section Overview

  • Reciprocals
  • Dividing Fractions

Reciprocals

Reciprocals

Two numbers whose product is 1 are called reciprocals of each other.

Sample Set A

The following pairs of numbers are reciprocals.

Example 1

34and43︸34⋅43=134and43︸34⋅43=1

Example 2

716and167︸716⋅167=1716and167︸716⋅167=1

Example 3

16and61︸16⋅61=116and61︸16⋅61=1

Notice that we can find the reciprocal of a nonzero number in fractional form by inverting it (exchanging positions of the numerator and denominator).

Practice Set A

Find the reciprocal of each number.

Exercise 1

310310 size 12{ { {3} over {"10"} } } {}

Solution

103103 size 12{ { {"10"} over {3} } } {}

Exercise 2

2323 size 12{ { {2} over {3} } } {}

Solution

3232 size 12{ { {3} over {2} } } {}

Exercise 3

7878 size 12{ { {7} over {8} } } {}

Solution

8787 size 12{ { {8} over {7} } } {}

Exercise 4

1515 size 12{ { {1} over {5} } } {}

Solution

5

Exercise 5

227227 size 12{2 { {2} over {7} } } {}

Hint:
Write this number as an improper fraction first.

Solution

716716 size 12{ { {7} over {"16"} } } {}

Exercise 6

514514 size 12{5 { {1} over {4} } } {}

Solution

421421 size 12{ { {4} over {"21"} } } {}

Exercise 7

1031610316 size 12{"10" { {3} over {"16"} } } {}

Solution

1616316163 size 12{ { {"16"} over {"163"} } } {}

Dividing Fractions

Our concept of division is that it indicates how many times one quantity is con­tained in another quantity. For example, using the diagram we can see that there are 6 one-thirds in 2.

Two rectangles, each divided into three parts. The rectangles are connected to each other.There are 6 one-thirds in 2.

Since 2 contains six 1313 size 12{ { {1} over {3} } } {}'s we express this as

Two divided by one-third is equal to six. Note also that two times three is equal to six, because one-third and three are reciprocals.

Using these observations, we can suggest the following method for dividing a number by a fraction.

Dividing One Fraction by Another Fraction

To divide a first fraction by a second, nonzero fraction, multiply the first traction by the reciprocal of the second fraction.

Invert and Multiply

This method is commonly referred to as "invert the divisor and multiply."

Sample Set B

Perform the following divisions.

Example 4

13÷3413÷34 size 12{ { {1} over {3} } div { {3} over {4} } } {}. The divisor is 3434 size 12{ { {3} over {4} } } {}. Its reciprocal is 4343 size 12{ { {4} over {3} } } {}. Multiply 1313 size 12{ { {1} over {3} } } {} by 4343 size 12{ { {4} over {3} } } {}.

1 3 ⋅ 4 3 = 1 ⋅ 4 3 ⋅ 3 = 4 9 1 3 ⋅ 4 3 = 1 ⋅ 4 3 ⋅ 3 = 4 9 size 12{ { {1} over {3} } cdot { {4} over {3} } = { {1 cdot 4} over {3 cdot 3} } = { {4} over {9} } } {}

1 3 ÷ 3 4 = 4 9 1 3 ÷ 3 4 = 4 9 size 12{ { {1} over {3} } div { {3} over {4} } = { {4} over {9} } } {}

Example 5

38÷5438÷54 size 12{ { {3} over {8} } div { {5} over {4} } } {} The divisor is 5454 size 12{ { {5} over {4} } } {}. Its reciprocal is 4545 size 12{ { {4} over {5} } } {}. Multiply 3838 size 12{ { {3} over {8} } } {} by 4545 size 12{ { {4} over {5} } } {}.

3 3 2 ⋅ 4 1 5 = 3 ⋅ 1 2 ⋅ 5 = 3 10 3 3 2 ⋅ 4 1 5 = 3 ⋅ 1 2 ⋅ 5 = 3 10 size 12{ { {3} over { { { {3}}} cSub { size 8{2} } } } cdot { { { { {4}}} cSup { size 8{1} } } over {5} } = { {3 cdot 1} over {2 cdot 5} } = { {3} over {"10"} } } {}

3 8 ÷ 5 4 = 3 10 3 8 ÷ 5 4 = 3 10 size 12{ { {3} over {8} } div { {5} over {4} } = { {3} over {"10"} } } {}

Example 6

56÷51256÷512 size 12{ { {5} over {6} } div { {5} over {"12"} } } {}. The divisor is 512512 size 12{ { {5} over {"12"} } } {}. Its reciprocal is 125125 size 12{ { {"12"} over {5} } } {}. Multiply 5656 size 12{ { {5} over {6} } } {} by 125125 size 12{ { {"12"} over {5} } } {}.

5 1 6 1 ⋅ 12 2 5 1 = 1 ⋅ 2 1 ⋅ 1 = 2 1 = 2 5 1 6 1 ⋅ 12 2 5 1 = 1 ⋅ 2 1 ⋅ 1 = 2 1 = 2 size 12{ { { { { {5}}} cSup { size 8{1} } } over { { { {6}}} cSub { size 8{1} } } } cdot { { {"12"} cSup { size 8{2} } } over { {5} cSub { size 8{1} } } } = { {1 cdot 2} over {1 cdot 1} } = { {2} over {1} } =2} {}

56÷512=256÷512=2

Example 7

229÷313229÷313 size 12{2 { {2} over {9} } div 3 { {1} over {3} } } {}. Convert each mixed number to an improper fraction.

229=9⋅2+29=209229=9⋅2+29=209 size 12{2 { {2} over {9} } = { {9 cdot 2+2} over {9} } = { {"20"} over {9} } } {}.

313=3⋅3+13=103313=3⋅3+13=103 size 12{3 { {1} over {3} } = { {3 cdot 3+1} over {3} } = { {10} over {3} } } {}.

209÷103209÷103 size 12{ { {"20"} over {9} } div { {"10"} over {3} } } {} The divisor is 103103 size 12{ { {"10"} over {3} } } {}. Its reciprocal is 310310 size 12{ { {3} over {"10"} } } {}. Multiply 209209 size 12{ { {"20"} over {9} } } {} by 310310 size 12{ { {3} over {"10"} } } {}.

20 2 9 3 ⋅ 3 1 10 1 = 2 ⋅ 1 3 ⋅ 1 = 2 3 20 2 9 3 ⋅ 3 1 10 1 = 2 ⋅ 1 3 ⋅ 1 = 2 3 size 12{ { { { { {2}} { {0}}} cSup { size 8{2} } } over { { { {9}}} cSub { size 8{3} } } } cdot { { { { {3}}} cSup { size 8{1} } } over { { { {1}} { {0}}} cSub { size 8{1} } } } = { {2 cdot 1} over {3 cdot 1} } = { {2} over {3} } } {}

2 2 9 ÷ 3 1 3 = 2 3 2 2 9 ÷ 3 1 3 = 2 3 size 12{2 { {2} over {9} } div 3 { {1} over {3} } = { {2} over {3} } } {}

Example 8

1211÷81211÷8 size 12{ { {"12"} over {"11"} } div 8} {}. First conveniently write 8 as 8181 size 12{ { {8} over {1} } } {}.

1211÷811211÷81 size 12{ { {"12"} over {"11"} } div { {8} over {1} } } {} The divisor is 8181 size 12{ { {8} over {1} } } {}. Its reciprocal is 1818 size 12{ { {1} over {8} } } {}. Multiply 12111211 size 12{ { {"12"} over {"11"} } } {} by 1818 size 12{ { {1} over {8} } } {}.

12 3 11 ⋅ 1 8 2 = 3 ⋅ 1 11 ⋅ 2 = 3 22 12 3 11 ⋅ 1 8 2 = 3 ⋅ 1 11 ⋅ 2 = 3 22 size 12{ { { { { {1}} { {2}}} cSup { size 8{3} } } over {"11"} } cdot { {1} over { { { {8}}} cSub { size 8{2} } } } = { {3 cdot 1} over {"11" cdot 2} } = { {3} over {"22"} } } {}

12 11 ÷ 8 = 3 22 12 11 ÷ 8 = 3 22 size 12{ { {"12"} over {"11"} } div 8= { {3} over {"22"} } } {}

Example 9

78÷2120⋅33578÷2120⋅335 size 12{ { {7} over {8} } div { {"21"} over {"20"} } cdot { {3} over {"35"} } } {}. The divisor is 21202120 size 12{ { {"21"} over {"20"} } } {}. Its reciprocal is 20212021 size 12{ { {"20"} over {"21"} } } {}.

7 1 8 2 ⋅ 20 5 1 21 3 1 3 1 35 7 = 1 ⋅ 1 ⋅ 1 2 ⋅ 1 ⋅ 7 = 1 14 7 1 8 2 ⋅ 20 5 1 21 3 1 3 1 35 7 = 1 ⋅ 1 ⋅ 1 2 ⋅ 1 ⋅ 7 = 1 14 size 12{ { { {7} cSup { size 8{1} } } over { {8} cSub { size 8{2} } } } cdot { { {"20"} cSup { size 8{ {5} cSup { size 6{1} } } } } over { {"21"} cSub { {3} cSub { size 6{1} } } } } size 12{ cdot { { {3} cSup {1} } over { size 12{ {"35"} cSub {7} } } } } size 12{ {}= { {1 cdot 1 cdot 1} over {2 cdot 1 cdot 7} } = { {1} over {"14"} } }} {}

7 8 ÷ 21 20 ⋅ 3 25 = 1 14 7 8 ÷ 21 20 ⋅ 3 25 = 1 14 size 12{ { {7} over {8} } div { {"21"} over {"20"} } cdot { {3} over {"25"} } = { {1} over {"14"} } } {}

Example 10

How many 238238 size 12{2 { {3} over {8} } } {}-inch-wide packages can be placed in a box 19 inches wide?

The problem is to determine how many two and three eighths are contained in 19, that is, what is 19÷23819÷238 size 12{"19" div 2 { {3} over {8} } } {}?

238=198238=198 size 12{2 { {3} over {8} } = { {"19"} over {8} } } {} Convert the divisor 238238 size 12{2 { {3} over {8} } } {} to an improper fraction.

19=19119=191 size 12{"19"= { {"19"} over {1} } } {} Write the dividend 19 as 191191 size 12{ { {"19"} over {1} } } {}.

191÷198191÷198 size 12{ { {"19"} over {1} } div { {"19"} over {8} } } {} The divisor is 198198 size 12{ { {"19"} over {8} } } {}. Its reciprocal is 819819 size 12{ { {8} over {"19"} } } {}.

19 1 1 ⋅ 8 19 1 = 1 ⋅ 8 1 ⋅ 1 = 8 1 = 8 19 1 1 ⋅ 8 19 1 = 1 ⋅ 8 1 ⋅ 1 = 8 1 = 8 size 12{ { { {"19"} cSup { size 8{1} } } over {1} } cdot { {8} over { {"19"} cSub { size 8{1} } } } = { {1 cdot 8} over {1 cdot 1} } = { {8} over {1} } =8} {}

Thus, 8 packages will fit into the box.

Practice Set B

Perform the following divisions.

Exercise 8

12÷9812÷98 size 12{ { {1} over {2} } div { {9} over {8} } } {}

Solution

4949 size 12{ { {4} over {9} } } {}

Exercise 9

38÷92438÷924 size 12{ { {3} over {8} } div { {9} over {"24"} } } {}

Solution

1

Exercise 10

715÷1415715÷1415 size 12{ { {7} over {"15"} } div { {"14"} over {"15"} } } {}

Solution

1212 size 12{ { {1} over {2} } } {}

Exercise 11

8÷8158÷815 size 12{8 div { {8} over {"15"} } } {}

Solution

15

Exercise 12

614÷512614÷512 size 12{6 { {1} over {4} } div { {5} over {"12"} } } {}

Solution

15

Exercise 13

313÷123313÷123 size 12{3 { {1} over {3} } div 1 { {2} over {3} } } {}

Solution

2

Exercise 14

56÷23⋅82556÷23⋅825 size 12{ { {5} over {6} } div { {2} over {3} } cdot { {8} over {"25"} } } {}

Solution

2525 size 12{ { {2} over {5} } } {}

Exercise 15

A container will hold 106 ounces of grape juice. How many 658658 size 12{6 { {5} over {8} } } {}-ounce glasses of grape juice can be served from this container?

Solution

16 glasses

Determine each of the following quotients and then write a rule for this type of division.

Exercise 16

1÷231÷23 size 12{1 div { {2} over {3} } } {}

Solution

3232 size 12{ { {3} over {2} } } {}

Exercise 17

1÷381÷38 size 12{1 div { {3} over {8} } } {}

Solution

8383 size 12{ { {8} over {3} } } {}

Exercise 18

1÷341÷34 size 12{1 div { {3} over {4} } } {}

Solution

4343 size 12{ { {4} over {3} } } {}

Exercise 19

1÷521÷52 size 12{1 div { {5} over {2} } } {}

Solution

2525 size 12{ { {2} over {5} } } {}

Exercise 20

When dividing 1 by a fraction, the quotient is the
.

Solution

is the reciprocal of the fraction.

Exercises

For the following problems, find the reciprocal of each number.

Exercise 21

4545 size 12{ { {4} over {5} } } {}

Solution

5454 size 12{ { {5} over {4} } } {} or 114114 size 12{1 { {1} over {4} } } {}

Exercise 22

811811 size 12{ { {8} over {"11"} } } {}

Exercise 23

2929 size 12{ { {2} over {9} } } {}

Solution

9292 size 12{ { {9} over {2} } } {} or 412412 size 12{4 { {1} over {2} } } {}

Exercise 24

1515 size 12{ { {1} over {5} } } {}

Exercise 25

314314 size 12{3 { {1} over {4} } } {}

Solution

413413 size 12{ { {4} over {"13"} } } {}

Exercise 26

814814 size 12{8 { {1} over {4} } } {}

Exercise 27

327327 size 12{3 { {2} over {7} } } {}

Solution

723723 size 12{ { {7} over {"23"} } } {}

Exercise 28

534534 size 12{5 { {3} over {4} } } {}

Exercise 30

4

For the following problems, find each value.

Exercise 31

38÷3538÷35 size 12{ { {3} over {8} } div { {3} over {5} } } {}

Solution

5858 size 12{ { {5} over {8} } } {}

Exercise 32

59÷5659÷56 size 12{ { {5} over {9} } div { {5} over {6} } } {}

Exercise 33

916÷158916÷158 size 12{ { {9} over {"16"} } div { {"15"} over {8} } } {}

Solution

310310 size 12{ { {3} over {"10"} } } {}

Exercise 34

49÷61549÷615 size 12{ { {4} over {9} } div { {6} over {"15"} } } {}

Exercise 35

2549÷492549÷49 size 12{ { {"25"} over {"49"} } div { {4} over {9} } } {}

Solution

225196225196 size 12{ { {"225"} over {"196"} } } {} or 129196129196 size 12{1 { {"29"} over {"196"} } } {}

Exercise 36

154÷278154÷278 size 12{ { {"15"} over {4} } div { {"27"} over {8} } } {}

Exercise 37

2475÷8152475÷815 size 12{ { {"24"} over {"75"} } div { {8} over {"15"} } } {}

Solution

3535 size 12{ { {3} over {5} } } {}

Exercise 38

57÷057÷0 size 12{ { {5} over {7} } div 0} {}

Exercise 39

78÷7878÷78 size 12{ { {7} over {8} } div { {7} over {8} } } {}

Solution

1

Exercise 40

0÷350÷35 size 12{0 div { {3} over {5} } } {}

Exercise 41

411÷411411÷411 size 12{ { {4} over {"11"} } div { {4} over {"11"} } } {}

Solution

1

Exercise 42

23÷2323÷23 size 12{ { {2} over {3} } div { {2} over {3} } } {}

Exercise 43

710÷107710÷107 size 12{ { {7} over {"10"} } div { {"10"} over {7} } } {}

Solution

4910049100 size 12{ { {"49"} over {"100"} } } {}

Exercise 44

34÷634÷6 size 12{ { {3} over {4} } div 6} {}

Exercise 45

95÷395÷3 size 12{ { {9} over {5} } div 3} {}

Solution

3535 size 12{ { {3} over {5} } } {}

Exercise 46

416÷313416÷313 size 12{4 { {1} over {6} } div 3 { {1} over {3} } } {}

Exercise 47

717÷813717÷813 size 12{7 { {1} over {7} } div 8 { {1} over {3} } } {}

Solution

6767 size 12{ { {6} over {7} } } {}

Exercise 48

112÷115112÷115 size 12{1 { {1} over {2} } div 1 { {1} over {5} } } {}

Exercise 49

325÷625325÷625 size 12{3 { {2} over {5} } div { {6} over {"25"} } } {}

Solution

856856 size 12{ { {"85"} over {6} } } {} or 14161416 size 12{"14" { {1} over {6} } } {}

Exercise 50

516÷316516÷316 size 12{5 { {1} over {6} } div { {"31"} over {6} } } {}

Exercise 51

356÷334356÷334 size 12{ { {"35"} over {6} } div 3 { {3} over {4} } } {}

Solution

2818=1492818=149 size 12{ { {"28"} over {"18"} } = { {"14"} over {9} } } {} or 159159 size 12{1 { {5} over {9} } } {}

Exercise 52

519÷118519÷118 size 12{5 { {1} over {9} } div { {1} over {"18"} } } {}

Exercise 53

834÷78834÷78 size 12{8 { {3} over {4} } div { {7} over {8} } } {}

Solution

10

Exercise 54

128÷112128÷112 size 12{ { {"12"} over {8} } div 1 { {1} over {2} } } {}

Exercise 55

318÷1516318÷1516 size 12{3 { {1} over {8} } div { {"15"} over {"16"} } } {}

Solution

103103 size 12{ { {"10"} over {3} } } {} or 313313 size 12{3 { {1} over {3} } } {}

Exercise 56

111112÷958111112÷958 size 12{"11" { {"11"} over {"12"} } div 9 { {5} over {8} } } {}

Exercise 57

229÷1123229÷1123 size 12{2 { {2} over {9} } div "11" { {2} over {3} } } {}

Solution

421421 size 12{ { {4} over {"21"} } } {}

Exercise 58

163÷625163÷625 size 12{ { {"16"} over {3} } div 6 { {2} over {5} } } {}

Exercise 59

4325÷256754325÷25675 size 12{4 { {3} over {"25"} } div 2 { {"56"} over {"75"} } } {}

Solution

3232 size 12{ { {3} over {2} } } {} or 112112 size 12{1 { {1} over {2} } } {}

Exercise 60

11000÷110011000÷1100 size 12{ { {1} over {"1000"} } div { {1} over {"100"} } } {}

Exercise 61

38÷916⋅6538÷916⋅65 size 12{ { {3} over {8} } div { {9} over {"16"} } cdot { {6} over {5} } } {}

Solution

4545 size 12{ { {4} over {5} } } {}

Exercise 62

316⋅98⋅65316⋅98⋅65 size 12{ { {3} over {"16"} } cdot { {9} over {8} } cdot { {6} over {5} } } {}

Exercise 63

415÷225⋅910415÷225⋅910 size 12{ { {4} over {"15"} } div { {2} over {"25"} } cdot { {9} over {"10"} } } {}

Solution

3

Exercise 64

2130⋅114÷9102130⋅114÷910 size 12{ { {"21"} over {"30"} } cdot 1 { {1} over {4} } div { {9} over {"10"} } } {}

Exercise 65

813⋅3675÷4813⋅3675÷4 size 12{8 { {1} over {3} } cdot { {"36"} over {"75"} } div 4} {}

Solution

1

Exercises for Review

Exercise 66

((Reference)) What is the value of 5 in the number 504,216?

Exercise 67

((Reference)) Find the product of 2,010 and 160.

Solution

321,600

Exercise 68

((Reference)) Use the numbers 8 and 5 to illustrate the commutative property of multiplication.

Exercise 69

((Reference)) Find the least common multiple of 6, 16, and 72.

Solution

144

Exercise 70

((Reference)) Find 8989 size 12{ { {8} over {9} } } {} of 634634 size 12{6 { {3} over {4} } } {}.

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