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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.

3 4 and 4 3 3 4 4 3 = 1

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7 16 and 16 7 7 16 16 7 = 1

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1 6 and 6 1 1 6 6 1 = 1

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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.

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

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

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

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

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

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

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

5

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

Write this number as an improper fraction first.

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

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

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

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

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

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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 1 3 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.

1 3 ÷ 3 4 size 12{ { {1} over {3} } div { {3} over {4} } } {} . The divisor is 3 4 size 12{ { {3} over {4} } } {} . Its reciprocal is 4 3 size 12{ { {4} over {3} } } {} . Multiply 1 3 size 12{ { {1} over {3} } } {} by 4 3 size 12{ { {4} over {3} } } {} .

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

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3 8 ÷ 5 4 size 12{ { {3} over {8} } div { {5} over {4} } } {} The divisor is 5 4 size 12{ { {5} over {4} } } {} . Its reciprocal is 4 5 size 12{ { {4} over {5} } } {} . Multiply 3 8 size 12{ { {3} over {8} } } {} by 4 5 size 12{ { {4} over {5} } } {} .

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

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5 6 ÷ 5 12 size 12{ { {5} over {6} } div { {5} over {"12"} } } {} . The divisor is 5 12 size 12{ { {5} over {"12"} } } {} . Its reciprocal is 12 5 size 12{ { {"12"} over {5} } } {} . Multiply 5 6 size 12{ { {5} over {6} } } {} by 12 5 size 12{ { {"12"} over {5} } } {} .

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} {}

5 6 ÷ 5 12 = 2

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

2 2 9 = 9 2 + 2 9 = 20 9 size 12{2 { {2} over {9} } = { {9 cdot 2+2} over {9} } = { {"20"} over {9} } } {} .

3 1 3 = 3 3 + 1 3 = 10 3 size 12{3 { {1} over {3} } = { {3 cdot 3+1} over {3} } = { {10} over {3} } } {} .

20 9 ÷ 10 3 size 12{ { {"20"} over {9} } div { {"10"} over {3} } } {} The divisor is 10 3 size 12{ { {"10"} over {3} } } {} . Its reciprocal is 3 10 size 12{ { {3} over {"10"} } } {} . Multiply 20 9 size 12{ { {"20"} over {9} } } {} by 3 10 size 12{ { {3} over {"10"} } } {} .

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

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

12 11 ÷ 8 1 size 12{ { {"12"} over {"11"} } div { {8} over {1} } } {} The divisor is 8 1 size 12{ { {8} over {1} } } {} . Its reciprocal is 1 8 size 12{ { {1} over {8} } } {} . Multiply 12 11 size 12{ { {"12"} over {"11"} } } {} by 1 8 size 12{ { {1} over {8} } } {} .

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

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7 8 ÷ 21 20 3 35 size 12{ { {7} over {8} } div { {"21"} over {"20"} } cdot { {3} over {"35"} } } {} . The divisor is 21 20 size 12{ { {"21"} over {"20"} } } {} . Its reciprocal is 20 21 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 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 size 12{ { {7} over {8} } div { {"21"} over {"20"} } cdot { {3} over {"25"} } = { {1} over {"14"} } } {}

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How many 2 3 8 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 ÷ 2 3 8 size 12{"19" div 2 { {3} over {8} } } {} ?

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

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

19 1 ÷ 19 8 size 12{ { {"19"} over {1} } div { {"19"} over {8} } } {} The divisor is 19 8 size 12{ { {"19"} over {8} } } {} . Its reciprocal is 8 19 size 12{ { {8} over {"19"} } } {} .

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.

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Practice set b

Perform the following divisions.

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

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

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

1

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7 15 ÷ 14 15 size 12{ { {7} over {"15"} } div { {"14"} over {"15"} } } {}

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

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

15

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

15

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

2

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5 6 ÷ 2 3 8 25 size 12{ { {5} over {6} } div { {2} over {3} } cdot { {8} over {"25"} } } {}

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

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

16 glasses

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

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

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

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

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

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

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

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

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

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

is the reciprocal of the fraction.

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Exercises

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

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

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

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

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

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

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

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

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

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

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

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

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

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For the following problems, find each value.

3 8 ÷ 3 5 size 12{ { {3} over {8} } div { {3} over {5} } } {}

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

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5 9 ÷ 5 6 size 12{ { {5} over {9} } div { {5} over {6} } } {}

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

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

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

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

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

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

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

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

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

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

1

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

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

1

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

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7 10 ÷ 10 7 size 12{ { {7} over {"10"} } div { {"10"} over {7} } } {}

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

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

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

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

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4 1 6 ÷ 3 1 3 size 12{4 { {1} over {6} } div 3 { {1} over {3} } } {}

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7 1 7 ÷ 8 1 3 size 12{7 { {1} over {7} } div 8 { {1} over {3} } } {}

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

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

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

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

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

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

28 18 = 14 9 size 12{ { {"28"} over {"18"} } = { {"14"} over {9} } } {} or 1 5 9 size 12{1 { {5} over {9} } } {}

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

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8 3 4 ÷ 7 8 size 12{8 { {3} over {4} } div { {7} over {8} } } {}

10

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12 8 ÷ 1 1 2 size 12{ { {"12"} over {8} } div 1 { {1} over {2} } } {}

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

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

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

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

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

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

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

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

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1 1000 ÷ 1 100 size 12{ { {1} over {"1000"} } div { {1} over {"100"} } } {}

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3 8 ÷ 9 16 6 5 size 12{ { {3} over {8} } div { {9} over {"16"} } cdot { {6} over {5} } } {}

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

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3 16 9 8 6 5 size 12{ { {3} over {"16"} } cdot { {9} over {8} } cdot { {6} over {5} } } {}

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4 15 ÷ 2 25 9 10 size 12{ { {4} over {"15"} } div { {2} over {"25"} } cdot { {9} over {"10"} } } {}

3

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21 30 1 1 4 ÷ 9 10 size 12{ { {"21"} over {"30"} } cdot 1 { {1} over {4} } div { {9} over {"10"} } } {}

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8 1 3 36 75 ÷ 4 size 12{8 { {1} over {3} } cdot { {"36"} over {"75"} } div 4} {}

1

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Exercises for review

( [link] ) What is the value of 5 in the number 504,216?

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( [link] ) Find the product of 2,010 and 160.

321,600

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( [link] ) Use the numbers 8 and 5 to illustrate the commutative property of multiplication.

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( [link] ) Find the least common multiple of 6, 16, and 72.

144

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( [link] ) Find 8 9 size 12{ { {8} over {9} } } {} of 6 3 4 size 12{6 { {3} over {4} } } {} .

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Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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