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Sample set b

Perform the following multiplications.

4 5 5 6 size 12{ { {4} over {5} } cdot { {5} over {6} } } {}

4 2 5 1 5 1 6 3 = 2 1 1 3 = 2 3 size 12{ { { { { {4}}} cSup { size 8{2} } } over { { { {5}}} cSub { size 8{1} } } } cdot { { { { {5}}} cSup { size 8{1} } } over { { { {6}}} cSub { size 8{3} } } } = { {2 cdot 1} over {1 cdot 3} } = { {2} over {3} } } {}

Divide 4 and 6 by 2
Divide 5 and 5 by 5

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8 12 8 10 size 12{ { {8} over {"12"} } cdot { {8} over {"10"} } } {}

8 4 12 3 8 2 10 5 = 4 2 3 5 = 8 15 size 12{ { { { { {8}}} cSup { size 8{4} } } over { { { {1}} { {2}}} cSub { size 8{3} } } } cdot { { { { {8}}} cSup { size 8{2} } } over { { { {1}} { {0}}} cSub { size 8{5} } } } = { {4 cdot 2} over {3 cdot 5} } = { {8} over {"15"} } } {}

Divide 8 and 10 by 2.
Divide 8 and 12 by 4.

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

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35 18 63 105 size 12{ { {"35"} over {"18"} } cdot { {"63"} over {"105"} } } {}

35 7 1 18 2 63 7 105 21 3 = 1 7 2 3 = 7 6

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13 9 6 39 1 12 size 12{ { {"13"} over {9} } cdot { {6} over {"39"} } cdot { {1} over {"12"} } } {}

13 1 9 6 2 1 39 3 1 1 12 6 = 1 1 1 9 1 6 = 1 54 size 12{ { { { { {1}} { {3}}} cSup { size 8{1} } } over {9} } cdot { { { { {6}}} cSup { size 8{ { { {2}}} cSup { size 6{1} } } } } over { { { {3}} { {9}}} cSub { { { {3}}} cSub { size 6{1} } } } } size 12{ cdot { {1} over {"12"} } = { {1 cdot 1 cdot 1} over {9 cdot 1 cdot 6} } = { {1} over {"54"} } }} {}

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Practice set b

Perform the following multiplications.

2 3 7 8 size 12{ { {2} over {3} } cdot { {7} over {8} } } {}

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

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25 12 10 45 size 12{ { {"25"} over {"12"} } cdot { {"10"} over {"45"} } } {}

25 54 size 12{ { {"25"} over {"54"} } } {}

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40 48 72 90 size 12{ { {"40"} over {"48"} } cdot { {"72"} over {"90"} } } {}

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

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7 2 49 size 12{7 cdot { {2} over {"49"} } } {}

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

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12 3 8 size 12{"12" cdot { {3} over {8} } } {}

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

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13 7 14 26 size 12{ left ( { {"13"} over {7} } right ) left ( { {"14"} over {"26"} } right )} {}

1

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16 10 22 6 21 44 size 12{ { {"16"} over {"10"} } cdot { {"22"} over {6} } cdot { {"21"} over {"44"} } } {}

14 5 size 12{ { {"14"} over {5} } } {}

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Multiplication of mixed numbers

Multiplying mixed numbers

To perform a multiplication in which there are mixed numbers, it is convenient to first convert each mixed number to an improper fraction, then multiply.

Sample set c

Perform the following multiplications. Convert improper fractions to mixed numbers.

1 1 8 4 2 3 size 12{1 { {1} over {8} } cdot 4 { {2} over {3} } } {}

Convert each mixed number to an improper fraction.

1 1 8 = 8 1 + 1 8 = 9 8 size 12{1 { {1} over {8} } = { {8 cdot 1+1} over {8} } = { {9} over {8} } } {}

4 2 3 = 4 3 + 2 3 = 14 3 size 12{4 { {2} over {3} } = { {4 cdot 3+2} over {3} } = { {"14"} over {3} } } {}

9 3 8 4 14 7 3 1 = 3 7 4 1 = 21 4 = 5 1 4 size 12{ { { { { {9}}} cSup { size 8{3} } } over { { { {8}}} cSub { size 8{4} } } } cdot { { { { {1}} { {4}}} cSup { size 8{7} } } over { {3} cSub { size 8{1} } } } = { {3 cdot 7} over {4 cdot 1} } = { {"21"} over {4} } =5 { {1} over {4} } } {}

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16 8 1 5 size 12{"16" cdot 8 { {1} over {5} } } {}

Convert 8 1 5 size 12{8 { {1} over {5} } } {} to an improper fraction.

8 1 5 = 5 8 + 1 5 = 41 5 size 12{8 { {1} over {5} } = { {5 cdot 8+1} over {5} } = { {"41"} over {5} } } {}

16 1 41 5 .

There are no common factors to divide out.

16 1 41 5 = 16 41 1 5 = 656 5 = 131 1 5 size 12{ { {"16"} over {1} } cdot { {"41"} over {5} } = { {"16" cdot "41"} over {1 cdot 5} } = { {"656"} over {5} } ="131" { {1} over {5} } } {}

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9 1 6 12 3 5 size 12{9 { {1} over {6} } cdot "12" { {3} over {5} } } {}

Convert to improper fractions.

9 1 6 = 6 9 + 1 6 = 55 6 size 12{9 { {1} over {6} } = { {6 cdot 9+1} over {6} } = { {"55"} over {6} } } {}

12 3 5 = 5 12 + 3 5 = 63 5 size 12{"12" { {3} over {5} } = { {5 cdot "12"+3} over {5} } = { {"63"} over {5} } } {}

55 11 6 2 63 21 5 1 = 11 21 2 1 = 231 2 = 115 1 2 size 12{ { { { { {5}} { {5}}} cSup { size 8{"11"} } } over { { { {6}}} cSub { size 8{2} } } } cdot { { { { {6}} { {3}}} cSup { size 8{"21"} } } over { { { {5}}} cSub { size 8{1} } } } = { {"11" cdot "21"} over {2 cdot 1} } = { {"231"} over {2} } ="115" { {1} over {2} } } {}

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11 8 4 1 2 3 1 8 = 11 8 9 3 2 1 10 5 3 1 = 11 3 5 8 1 1 = 165 8 = 20 5 8

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Practice set c

Perform the following multiplications. Convert improper fractions to mixed numbers.

2 2 3 2 1 4 size 12{2 { {2} over {3} } cdot 2 { {1} over {4} } } {}

6

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6 2 3 3 3 10 size 12{6 { {2} over {3} } cdot 3 { {3} over {"10"} } } {}

22

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7 1 8 12 size 12{7 { {1} over {8} } cdot "12"} {}

85 1 2 size 12{"85" { {1} over {2} } } {}

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2 2 5 3 3 4 3 1 3 size 12{2 { {2} over {5} } cdot 3 { {3} over {4} } cdot 3 { {1} over {3} } } {}

30

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Powers and roots of fractions

Sample set d

Find the value of each of the following.

1 6 2 = 1 6 1 6 = 1 1 6 6 = 1 36 size 12{ left ( { {1} over {6} } right ) rSup { size 8{2} } = { {1} over {6} } cdot { {1} over {6} } = { {1 cdot 1} over {6 cdot 6} } = { {1} over {"36"} } } {}

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9 100 size 12{ sqrt { { {9} over {"100"} } } } {} . We’re looking for a number, call it ?, such that when it is squared, 9 100 size 12{ { {9} over {"100"} } } {} is produced.

? 2 = 9 100 size 12{ left (? right ) rSup { size 8{2} } = { {9} over {"100"} } } {}

We know that

3 2 = 9 size 12{3 rSup { size 8{2} } =9} {} and 10 2 = 100 size 12{"10" rSup { size 8{2} } ="100"} {}

We’ll try 3 10 size 12{ { {3} over {"10"} } } {} . Since

3 10 2 = 3 10 3 10 = 3 3 10 10 = 9 100 size 12{ left ( { {3} over {"10"} } right ) rSup { size 8{2} } = { {3} over {"10"} } cdot { {3} over {"10"} } = { {3 cdot 3} over {"10" cdot "10"} } = { {9} over {"100"} } } {}

9 100 = 3 10 size 12{ sqrt { { {9} over {"100"} } ={}} { {3} over {"10"} } } {}

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4 2 5 100 121 size 12{4 { {2} over {5} } cdot sqrt { { {"100"} over {"121"} } } } {}

22 2 5 1 10 2 11 1 = 2 2 1 1 = 4 1 = 4 size 12{ { { { { {2}} { {2}}} cSup { size 8{2} } } over { { { {5}}} cSub { size 8{1} } } } cdot { { { { {1}} { {0}}} cSup { size 8{2} } } over { { { {1}} { {1}}} cSub { size 8{1} } } } = { {2 cdot 2} over {1 cdot 1} } = { {4} over {1} } =4} {}

4 2 5 100 121 = 4 size 12{4 { {2} over {5} } cdot sqrt { { {"100"} over {"121"} } =4} } {}

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Practice set d

Find the value of each of the following.

1 8 2 size 12{ left ( { {1} over {8} } right ) rSup { size 8{2} } } {}

1 64

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3 10 2 size 12{ left ( { {3} over {"10"} } right ) rSup { size 8{2} } } {}

9 100

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

2 3

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1 4 size 12{ sqrt { { {1} over {4} } } } {}

1 2

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3 8 1 9 size 12{ { {3} over {8} } cdot sqrt { { {1} over {9} } } } {}

1 8

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9 1 3 81 100 size 12{9 { {1} over {3} } cdot sqrt { { {"81"} over {"100"} } } } {}

8 2 5

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2 8 13 169 16 size 12{2 { {8} over {"13"} } cdot sqrt { { {"169"} over {"16"} } } } {}

8 1 2

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Exercises

For the following six problems, use the diagrams to find each of the following parts. Use multiplication to verify your re­sult.

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

A rectangle divided into twelve parts in a pattern of four rows and three columns.

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

A rectangle divided into twelve parts in a pattern of four rows and three columns. Three of the parts are shaded.

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2 3 size 12{ { {2} over {3} } } {} of 3 5 size 12{ { {3} over {5} } } {}

A rectangle divided into twelve parts in a pattern of three rows and four columns.

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2 7 size 12{ { {2} over {7} } } {} of 7 8 size 12{ { {7} over {8} } } {}

A rectangle divided into fifty-six parts in a pattern of seven rows and eight columns.

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

A rectangle divided into fifty-six parts in a pattern of seven rows and eight columns. Fourteen of the parts are shaded.

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5 6 size 12{ { {5} over {6} } } {} of 3 4 size 12{ { {3} over {4} } } {}

A rectangle divided into twenty-four parts in a pattern of six rows and four columns.

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1 8 size 12{ { {1} over {8} } } {} of 1 8 size 12{ { {1} over {8} } } {}

A rectangle divided into sixty-four parts in a pattern of eight rows and eight columns.

1 64 size 12{ { {1} over {"64"} } } {}

A rectangle divided into sixty-four parts in a pattern of eight rows and eight columns. One part is shaded.

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7 12 size 12{ { {7} over {"12"} } } {} of 6 7 size 12{ { {6} over {7} } } {}

A rectangle divided into eighty-four parts in a pattern of twelve rows and seven columns.

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For the following problems, find each part without using a diagram.

1 2 size 12{ { {1} over {2} } } {} of 4 5 size 12{ { {4} over {5} } } {}

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

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3 5 size 12{ { {3} over {5} } } {} of 5 12 size 12{ { {5} over {"12"} } } {}

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1 4 size 12{ { {1} over {4} } } {} of 8 9 size 12{ { {8} over {9} } } {}

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

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3 16 size 12{ { {3} over {"16"} } } {} of 12 15 size 12{ { {"12"} over {"15"} } } {}

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2 9  of  6 5 size 12{ { {2} over {9} } "of" { {6} over {5} } } {}

4 15 size 12{ { {4} over {"15"} } } {}

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1 8  of  3 8 size 12{ { {1} over {8} } ital "of" { {3} over {8} } } {}

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2 3  of  9 10 size 12{ { {2} over {3} } ital "of" { {9} over {"10"} } } {}

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

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18 19  of  38 54 size 12{ { {"18"} over {"19"} } ital "of" { {"38"} over {"54"} } } {}

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5 6  of  2 2 5 size 12{ { {5} over {6} } ital "of"2 { {2} over {5} } } {}

2 size 12{2} {}

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3 4  of  3 3 5 size 12{ { {3} over {4} } ital "of"3 { {3} over {5} } } {}

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3 2  of  2 2 9 size 12{ { {3} over {2} } ital "of"2 { {2} over {9} } } {}

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

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15 4  of  4 4 5 size 12{ { {"15"} over {4} } ital "of"4 { {4} over {5} } } {}

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5 1 3  of  9 3 4 size 12{5 { {1} over {3} } ital "of"9 { {3} over {4} } } {}

52

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1 13 15  of  8 3 4 size 12{1 { {"13"} over {"15"} } ital "of"8 { {3} over {4} } } {}

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8 9  of  3 4  of  2 3 size 12{ { {8} over {9} } ital "of" { {3} over {4} } ital "of" { {2} over {3} } } {}

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

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1 6 of 12 13 of 26 36 size 12{ { {1} over {6} } " of " { {"12"} over {"13"} } " of " { {"26"} over {"36"} } } {}

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1 2 of 1 3 of 1 4 size 12{ { {1} over {2} } " of " { {1} over {3} } " of " { {1} over {4} } } {}

1 24 size 12{ { {1} over {"24"} } } {}

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1 3 7 of 5 1 5 of 8 1 3 size 12{1 { {3} over {7} } " of 5" { {1} over {5} } " of 8" { {1} over {3} } } {}

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2 4 5 of 5 5 6 of 7 5 7 size 12{2 { {4} over {5} } " of 5" { {5} over {6} } " of 7" { {5} over {7} } } {}

126

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For the following problems, find the products. Be sure to reduce.

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

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

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

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3 4 3 8 size 12{ { {3} over {4} } cdot { {3} over {8} } } {}

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2 5 5 6 size 12{ { {2} over {5} } cdot { {5} over {6} } } {}

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

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3 8 8 9 size 12{ { {3} over {8} } cdot { {8} over {9} } } {}

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5 6 14 15 size 12{ { {5} over {6} } cdot { {"14"} over {"15"} } } {}

7 9 size 12{ { {7} over {9} } } {}

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4 7 7 4 size 12{ { {4} over {7} } cdot { {7} over {4} } } {}

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3 11 11 3 size 12{ { {3} over {"11"} } cdot { {"11"} over {3} } } {}

1

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9 16 20 27 size 12{ { {9} over {"16"} } cdot { {"20"} over {"27"} } } {}

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35 36 48 55 size 12{ { {"35"} over {"36"} } cdot { {"48"} over {"55"} } } {}

28 33 size 12{ { {"28"} over {"33"} } } {}

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21 25 15 14 size 12{ { {"21"} over {"25"} } cdot { {"15"} over {"14"} } } {}

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76 99 66 38 size 12{ { {"76"} over {"99"} } cdot { {"66"} over {"38"} } } {}

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

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3 7 14 18 6 2 size 12{ { {3} over {7} } cdot { {"14"} over {"18"} } cdot { {6} over {2} } } {}

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4 15 10 3 27 2 size 12{ { {4} over {"15"} } cdot { {"10"} over {3} } cdot { {"27"} over {2} } } {}

12

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14 15 21 28 45 7 size 12{ { {"14"} over {"15"} } cdot { {"21"} over {"28"} } cdot { {"45"} over {7} } } {}

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8 3 15 4 16 21 size 12{ { {8} over {3} } cdot { {"15"} over {4} } cdot { {"16"} over {"21"} } } {}

7 13 21 or 160 21 size 12{7 { {"13"} over {"21"} } " or " { {"160"} over {"21"} } } {}

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18 14 21 35 36 7 size 12{ { {"18"} over {"14"} } cdot { {"21"} over {"35"} } cdot { {"36"} over {7} } } {}

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3 5 20 size 12{ { {3} over {5} } cdot "20"} {}

12

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8 9 18 size 12{ { {8} over {9} } cdot "18"} {}

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6 11 33 size 12{ { {6} over {"11"} } cdot "33"} {}

18

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18 19 38 size 12{ { {"18"} over {"19"} } cdot "38"} {}

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5 6 10 size 12{ { {5} over {6} } cdot "10"} {}

25 3 or 8 1 3 size 12{ { {"25"} over {3} } " or 8" { {1} over {3} } } {}

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1 9 3 size 12{ { {1} over {9} } cdot 3} {}

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5 3 8 size 12{5 cdot { {3} over {8} } } {}

15 8 =1 7 8 size 12{ { {"15"} over {8} } "=1" { {7} over {8} } } {}

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16 1 4 size 12{"16" cdot { {1} over {4} } } {}

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2 3 12 3 4 size 12{ { {2} over {3} } cdot "12" cdot { {3} over {4} } } {}

6

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3 8 24 2 3 size 12{ { {3} over {8} } cdot "24" cdot { {2} over {3} } } {}

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5 18 10 2 5 size 12{ { {5} over {"18"} } cdot "10" cdot { {2} over {5} } } {}

10 9 =1 1 9 size 12{ { {"10"} over {9} } "=1" { {1} over {9} } } {}

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16 15 50 3 10 size 12{ { {"16"} over {"15"} } cdot "50" cdot { {3} over {"10"} } } {}

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5 1 3 27 32 size 12{5 { {1} over {3} } cdot { {"27"} over {"32"} } } {}

9 2 =4 1 2 size 12{ { {9} over {2} } "=4" { {1} over {2} } } {}

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2 6 7 5 3 5 size 12{2 { {6} over {7} } cdot 5 { {3} over {5} } } {}

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6 1 4 2 4 15 size 12{6 { {1} over {4} } cdot 2 { {4} over {"15"} } } {}

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

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9 1 3 9 16 1 1 3 size 12{9 { {1} over {3} } cdot { {9} over {"16"} } cdot 1 { {1} over {3} } } {}

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3 5 9 1 13 14 10 1 2 size 12{3 { {5} over {9} } cdot 1 { {"13"} over {"14"} } cdot "10" { {1} over {2} } } {}

72

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20 1 4 8 2 3 16 4 5 size 12{"20" { {1} over {4} } cdot 8 { {2} over {3} } cdot "16" { {4} over {5} } } {}

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2 3 2 size 12{ left ( { {2} over {3} } right ) rSup { size 8{2} } } {}

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

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3 8 2 size 12{ left ( { {3} over {8} } right ) rSup { size 8{2} } } {}

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2 11 2 size 12{ left ( { {2} over {"11"} } right ) rSup { size 8{2} } } {}

4 121 size 12{ { {4} over {"121"} } } {}

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8 9 2 size 12{ left ( { {8} over {9} } right ) rSup { size 8{2} } } {}

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1 2 2 size 12{ left ( { {1} over {2} } right ) rSup { size 8{2} } } {}

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

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3 5 2 20 3 size 12{ left ( { {3} over {5} } right ) rSup { size 8{2} } cdot { {"20"} over {3} } } {}

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1 4 2 16 15 size 12{ left ( { {1} over {4} } right ) rSup { size 8{2} } cdot { {"16"} over {"15"} } } {}

1 15 size 12{ { {1} over {"15"} } } {}

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1 2 2 8 9 size 12{ left ( { {1} over {2} } right ) rSup { size 8{2} } cdot { {8} over {9} } } {}

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1 2 2 2 5 2 size 12{ left ( { {1} over {2} } right ) rSup { size 8{2} } cdot left ( { {2} over {5} } right ) rSup { size 8{2} } } {}

1 25 size 12{ { {1} over {"25"} } } {}

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3 7 2 1 9 2 size 12{ left ( { {3} over {7} } right ) rSup { size 8{2} } cdot left ( { {1} over {9} } right ) rSup { size 8{2} } } {}

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For the following problems, find each value. Reduce answers to lowest terms or convert to mixed numbers.

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

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

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16 25 size 12{ sqrt { { {"16"} over {"25"} } } } {}

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81 121 size 12{ sqrt { { {"81"} over {"121"} } } } {}

9 11 size 12{ { {9} over {"11"} } } {}

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36 49 size 12{ sqrt { { {"36"} over {"49"} } } } {}

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144 25 size 12{ sqrt { { {"144"} over {"25"} } } } {}

12 5 = 2 2 5 size 12{ { {"12"} over {5} } =2 { {2} over {5} } } {}

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2 3 9 16 size 12{ { {2} over {3} } cdot sqrt { { {9} over {"16"} } } } {}

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3 5 25 81 size 12{ { {3} over {5} } cdot sqrt { { {"25"} over {"81"} } } } {}

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

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8 5 2 25 64 size 12{ left ( { {8} over {5} } right ) rSup { size 8{2} } cdot sqrt { { {"25"} over {"64"} } } } {}

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1 3 4 2 4 49 size 12{ left (1 { {3} over {4} } right ) rSup { size 8{2} } cdot sqrt { { {4} over {"49"} } } } {}

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

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2 2 3 2 36 49 64 81 size 12{ left (2 { {2} over {3} } right ) rSup { size 8{2} } cdot sqrt { { {"36"} over {"49"} } } cdot sqrt { { {"64"} over {"81"} } } } {}

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Exercises for review

( [link] ) How many thousands in 342,810?

2

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( [link] ) Find the sum of 22, 42, and 101.

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( [link] ) Is 634,281 divisible by 3?

yes

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( [link] ) Is the whole number 51 prime or composite?

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( [link] ) Reduce 36 150 size 12{ { {"36"} over {"150"} } } {} to lowest terms.

6 25 size 12{ { {6} over {"25"} } } {}

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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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