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Multiplication 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 multiplication of fractions. By the end of the module students should be able to understand the concept of multiplication of fractions, multiply one fraction by another, multiply mixed numbers and find powers and roots of various fractions.

Section Overview

  • Fractions of Fractions
  • Multiplication of Fractions
  • Multiplication of Fractions by Dividing Out Common Factors
  • Multiplication of Mixed Numbers
  • Powers and Roots of Fractions

Fractions of Fractions

We know that a fraction represents a part of a whole quantity. For example, two fifths of one unit can be represented by

A rectangle equally divided into five parts. Each part is labeled one-fifth. Two of the parts are shaded.2525 size 12{ { {2} over {5} } } {} of the whole is shaded.

A natural question is, what is a fractional part of a fractional quantity, or, what is a fraction of a fraction? For example, what 2323 size 12{ { {2} over {3} } } {} of 1212 size 12{ { {1} over {2} } } {}?

We can suggest an answer to this question by using a picture to examine 2323 size 12{ { {2} over {3} } } {} of 1212 size 12{ { {1} over {2} } } {}.

First, let’s represent 1212 size 12{ { {1} over {2} } } {}.

A rectangle equally divided into two parts. Both parts are labeled one-half. One of the parts is shaded.1212 size 12{ { {1} over {2} } } {} of the whole is shaded.

Then divide each of the 1212 size 12{ { {1} over {2} } } {} parts into 3 equal parts.

A rectangle divided into six equal parts in a gridlike fashion, with three rows and two columns. Each part is labeled one-sixth. Below the rectangles are brackets showing that each column of sixths is equal to one-half. Each part is 1616 size 12{ { {1} over {6} } } {} of the whole.

Now we’ll take 2323 size 12{ { {2} over {3} } } {} of the 1212 size 12{ { {1} over {2} } } {} unit.

A rectangle divided into six equal parts in a gridlike fashion, with three rows and two columns. Each part is labeled one-sixth. Below the rectangles are brackets showing that each column of sixths is equal to one-half. The first and second boxes in the left column are shaded. 2323 size 12{ { {2} over {3} } } {} of 1212 size 12{ { {1} over {2} } } {} is 2626 size 12{ { {2} over {6} } } {}, which reduces to 1313 size 12{ { {1} over {3} } } {}.

Multiplication of Fractions

Now we ask, what arithmetic operation (+, –, ×, ÷) will produce 2626 size 12{ { {2} over {6} } } {} from 2323 size 12{ { {2} over {3} } } {} of 1212 size 12{ { {1} over {2} } } {}?

Notice that, if in the fractions 2323 size 12{ { {2} over {3} } } {} and 1212 size 12{ { {1} over {2} } } {}, we multiply the numerators together and the denominators together, we get precisely 2626 size 12{ { {2} over {6} } } {}.

2 ⋅ 1 3 ⋅ 2 = 2 6 2 ⋅ 1 3 ⋅ 2 = 2 6 size 12{ { {2 cdot 1} over {3 cdot 2} } = { {2} over {6} } } {}

This reduces to 1313 size 12{ { {1} over {3} } } {} as before.

Using this observation, we can suggest the following:

  1. The Word "OF" Indicates Multiplication: The word "of" translates to the arithmetic operation "times."
  2. The Method of Multiplying Fractions: To multiply two or more fractions, multiply the numerators together and then multiply the denominators together. Reduce if necessary.

numerator 1 denominator 1 ⋅ numerator 2 denominator 2 = numerator 1 denominator 1 ⋅ numerator 2 denominator 2 numerator 1 denominator 1 ⋅ numerator 2 denominator 2 = numerator 1 denominator 1 ⋅ numerator 2 denominator 2

Sample Set A

Perform the following multiplications.

Example 1

34⋅16=3⋅14⋅6 =324 Now, reduce. 34⋅16=3⋅14⋅6 =324 Now, reduce.

= 3 1 24 8 = 1 8 = 3 1 24 8 = 1 8 size 12{ {}= { { { { {3}}} cSup { size 8{1} } } over { { { {2}} { {4}}} cSub { size 8{8} } } } = { {1} over {8} } } {}

Thus

3 4 ⋅ 1 6 = 1 8 3 4 ⋅ 1 6 = 1 8 size 12{ { {3} over {4} } cdot { {1} over {6} } = { {1} over {8} } } {}

This means that 3434 size 12{ { {3} over {4} } } {} of 1616 size 12{ { {1} over {6} } } {} is 1818 size 12{ { {1} over {8} } } {}, that is, 3434 size 12{ { {3} over {4} } } {} of 1616 size 12{ { {1} over {6} } } {} of a unit is 1818 size 12{ { {1} over {8} } } {} of the original unit.

Example 2

38⋅438⋅4 size 12{ { {3} over {8} } cdot 4} {}. Write 4 as a fraction by writing 4141 size 12{ { {4} over {1} } } {}

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

3 8 ⋅ 4 = 3 2 3 8 ⋅ 4 = 3 2 size 12{ { {3} over {8} } cdot 4= { {3} over {2} } } {}

This means that 3838 size 12{ { {3} over {8} } } {} of 4 whole units is 3232 size 12{ { {3} over {2} } } {} of one whole unit.

Example 3

25⋅58⋅14=2⋅5⋅15⋅8⋅4=10116016=11625⋅58⋅14=2⋅5⋅15⋅8⋅4=10116016=116 size 12{ { {2} over {5} } cdot { {5} over {8} } cdot { {1} over {4} } = { {2 cdot 5 cdot 1} over {5 cdot 8 cdot 4} } = { { { { {1}} { {0}}} cSup { size 8{1} } } over { { { {1}} { {6}} { {0}}} cSub { size 8{"16"} } } } = { { {1} cSup {} } over { {"16"} cSub {} } } } {}

This means that 2525 size 12{ { {2} over {5} } } {} of 5858 size 12{ { {5} over {8} } } {} of 1414 size 12{ { {1} over {4} } } {} of a whole unit is 116116 size 12{ { {1} over {"16"} } } {} of the original unit.

Practice Set A

Perform the following multiplications.

Exercise 1

25⋅1625⋅16 size 12{ { {2} over {5} } cdot { {1} over {6} } } {}

Solution

115115 size 12{ { {1} over {"15"} } } {}

Exercise 2

14⋅8914⋅89 size 12{ { {1} over {4} } cdot { {8} over {9} } } {}

Solution

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

Exercise 3

49⋅151649⋅1516 size 12{ { {4} over {9} } cdot { {15} over {16} } } {}

Solution

512512 size 12{ { {5} over {"12"} } } {}

Exercise 4

23232323 size 12{ left ( { {2} over {3} } right ) left ( { {2} over {3} } right )} {}

Solution

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

Exercise 5

74857485 size 12{ left ( { {7} over {4} } right ) left ( { {8} over {5} } right )} {}

Solution

145145 size 12{ { {"14"} over {5} } } {}

Exercise 6

56⋅7856⋅78 size 12{ { {5} over {6} } cdot { {7} over {8} } } {}

Solution

35483548 size 12{ { {"35"} over {"48"} } } {}

Exercise 7

23⋅523⋅5 size 12{ { {2} over {3} } cdot 5} {}

Solution

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

Exercise 8

34103410 size 12{ left ( { {3} over {4} } right ) left ("10" right )} {}

Solution

152152 size 12{ { {"15"} over {2} } } {}

Exercise 9

34⋅89⋅51234⋅89⋅512 size 12{ { {3} over {4} } cdot { {8} over {9} } cdot { {5} over {"12"} } } {}

Solution

518518 size 12{ { {5} over {"18"} } } {}

Multiplying Fractions by Dividing Out Common Factors

We have seen that to multiply two fractions together, we multiply numerators together, then denominators together, then reduce to lowest terms, if necessary. The reduction can be tedious if the numbers in the fractions are large. For example,

9 16 ⋅ 10 21 = 9 ⋅ 10 16 ⋅ 21 = 90 336 = 45 168 = 15 28 9 16 ⋅ 10 21 = 9 ⋅ 10 16 ⋅ 21 = 90 336 = 45 168 = 15 28 size 12{ { {9} over {"16"} } cdot { {"10"} over {"21"} } = { {9 cdot "10"} over {"16" cdot "21"} } = { {"90"} over {"336"} } = { {"45"} over {"168"} } = { {"15"} over {"28"} } } {}

We avoid the process of reducing if we divide out common factors before we multi­ply.

9 16 ⋅ 10 21 = 9 3 16 8 ⋅ 10 5 21 7 = 3 ⋅ 5 8 ⋅ 7 = 15 56 9 16 ⋅ 10 21 = 9 3 16 8 ⋅ 10 5 21 7 = 3 ⋅ 5 8 ⋅ 7 = 15 56 size 12{ { {9} over {"16"} } cdot { {"10"} over {"21"} } = { { { { {9}}} cSup { size 8{3} } } over { { { {1}} { {6}}} cSub { size 8{8} } } } cdot { { { { {1}} { {0}}} cSup { size 8{5} } } over { { { {2}} { {1}}} cSub { size 8{7} } } } = { {3 cdot 5} over {8 cdot 7} } = { {"15"} over {"56"} } } {}

Divide 3 into 9 and 21, and divide 2 into 10 and 16. The product is a fraction that is reduced to lowest terms.

The Process of Multiplication by Dividing Out Common Factors

To multiply fractions by dividing out common factors, divide out factors that are common to both a numerator and a denominator. The factor being divided out can appear in any numerator and any denominator.

Sample Set B

Perform the following multiplications.

Example 4

45⋅5645⋅56 size 12{ { {4} over {5} } cdot { {5} over {6} } } {}

4251⋅5163=2⋅11⋅3=234251⋅5163=2⋅11⋅3=23 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

Example 5

812⋅810812⋅810 size 12{ { {8} over {"12"} } cdot { {8} over {"10"} } } {}

84123⋅82105=4⋅23⋅5=81584123⋅82105=4⋅23⋅5=815 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.

Example 6

8⋅512=821⋅5123=2⋅51⋅3=1038⋅512=821⋅5123=2⋅51⋅3=103 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} } } {}

Example 7

3518⋅631053518⋅63105 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 35 7 1 18 2 63 7 105 21 3 = 1 ⋅ 7 2 ⋅ 3 = 7 6

Example 8

139⋅639⋅112139⋅639⋅112 size 12{ { {"13"} over {9} } cdot { {6} over {"39"} } cdot { {1} over {"12"} } } {}

13 1 9 ⋅ 6 2 1 39 3 1 ⋅ 1 126 = 1 ⋅ 1 ⋅ 1 9 ⋅ 1 ⋅ 6 = 1 54 13 1 9 ⋅ 6 2 1 39 3 1 ⋅ 1 126 = 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"} } }} {}

Practice Set B

Perform the following multiplications.

Exercise 10

23⋅7823⋅78 size 12{ { {2} over {3} } cdot { {7} over {8} } } {}

Solution

712712 size 12{ { {7} over {"12"} } } {}

Exercise 11

2512⋅10452512⋅1045 size 12{ { {"25"} over {"12"} } cdot { {"10"} over {"45"} } } {}

Solution

25542554 size 12{ { {"25"} over {"54"} } } {}

Exercise 12

4048⋅72904048⋅7290 size 12{ { {"40"} over {"48"} } cdot { {"72"} over {"90"} } } {}

Solution

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

Exercise 13

7⋅2497⋅249 size 12{7 cdot { {2} over {"49"} } } {}

Solution

2727 size 12{ { {2} over {7} } } {}

Exercise 14

12⋅3812⋅38 size 12{"12" cdot { {3} over {8} } } {}

Solution

9292 size 12{ { {9} over {2} } } {}

Exercise 15

13714261371426 size 12{ left ( { {"13"} over {7} } right ) left ( { {"14"} over {"26"} } right )} {}

Solution

1

Exercise 16

1610⋅226⋅21441610⋅226⋅2144 size 12{ { {"16"} over {"10"} } cdot { {"22"} over {6} } cdot { {"21"} over {"44"} } } {}

Solution

145145 size 12{ { {"14"} over {5} } } {}

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.

Example 9

118⋅423118⋅423 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 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 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 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} } } {}

Example 10

16⋅81516⋅815 size 12{"16" cdot 8 { {1} over {5} } } {}

Convert 815815 size 12{8 { {1} over {5} } } {} to an improper fraction.

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

161⋅415 161⋅415 .

There are no common factors to divide out.

16 1 ⋅ 41 5 = 16 ⋅ 41 1 ⋅ 5 = 656 5 = 131 1 5 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} } } {}

Example 11

916⋅1235916⋅1235 size 12{9 { {1} over {6} } cdot "12" { {3} over {5} } } {}

Convert to improper fractions.

9 1 6 = 6 ⋅ 9 + 1 6 = 55 6 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 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 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} } } {}

Example 12

118 ⋅ 4 12 ⋅3 18 = 118 ⋅ 93 21 ⋅ 105 3 1 = 11⋅3⋅5 8⋅1⋅1 = 1658 =20 58 118 ⋅ 4 12 ⋅3 18 = 118 ⋅ 93 21 ⋅ 105 3 1 = 11⋅3⋅5 8⋅1⋅1 = 1658 =20 58

Practice Set C

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

Exercise 17

223⋅214223⋅214 size 12{2 { {2} over {3} } cdot 2 { {1} over {4} } } {}

Solution

6

Exercise 18

623⋅3310623⋅3310 size 12{6 { {2} over {3} } cdot 3 { {3} over {"10"} } } {}

Solution

22

Exercise 19

718⋅12718⋅12 size 12{7 { {1} over {8} } cdot "12"} {}

Solution

85128512 size 12{"85" { {1} over {2} } } {}

Exercise 20

225⋅334⋅313225⋅334⋅313 size 12{2 { {2} over {5} } cdot 3 { {3} over {4} } cdot 3 { {1} over {3} } } {}

Solution

30

Powers and Roots of Fractions

Sample Set D

Find the value of each of the following.

Example 13

162=16⋅16=1⋅16⋅6=136162=16⋅16=1⋅16⋅6=136 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"} } } {}

Example 14

91009100 size 12{ sqrt { { {9} over {"100"} } } } {} . We’re looking for a number, call it ?, such that when it is squared, 91009100 size 12{ { {9} over {"100"} } } {} is produced.

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

We know that

32=932=9 size 12{3 rSup { size 8{2} } =9} {} and 102=100102=100 size 12{"10" rSup { size 8{2} } ="100"} {}

We’ll try 310310 size 12{ { {3} over {"10"} } } {}. Since

3 10 2 = 3 10 ⋅ 3 10 = 3 ⋅ 3 10 ⋅ 10 = 9 100 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 9 100 = 3 10 size 12{ sqrt { { {9} over {"100"} } ={}} { {3} over {"10"} } } {}

Example 15

425⋅100121425⋅100121 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 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 4 2 5 ⋅ 100 121 = 4 size 12{4 { {2} over {5} } cdot sqrt { { {"100"} over {"121"} } =4} } {}

Practice Set D

Find the value of each of the following.

Exercise 21

182182 size 12{ left ( { {1} over {8} } right ) rSup { size 8{2} } } {}

Solution

164164

Exercise 22

31023102 size 12{ left ( { {3} over {"10"} } right ) rSup { size 8{2} } } {}

Solution

91009100

Exercise 23

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

Solution

2323

Exercise 24

1414 size 12{ sqrt { { {1} over {4} } } } {}

Solution

1212

Exercise 25

38⋅1938⋅19 size 12{ { {3} over {8} } cdot sqrt { { {1} over {9} } } } {}

Solution

1818

Exercise 26

913⋅81100913⋅81100 size 12{9 { {1} over {3} } cdot sqrt { { {"81"} over {"100"} } } } {}

Solution

825825

Exercise 27

2813⋅169162813⋅16916 size 12{2 { {8} over {"13"} } cdot sqrt { { {"169"} over {"16"} } } } {}

Solution

812812

Exercises

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

Exercise 28

3434 size 12{ { {3} over {4} } } {} of 1313 size 12{ { {1} over {3} } } {}

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

Solution

1414 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.

Exercise 29

2323 size 12{ { {2} over {3} } } {} of 3535 size 12{ { {3} over {5} } } {}

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

Exercise 30

2727 size 12{ { {2} over {7} } } {} of 7878 size 12{ { {7} over {8} } } {}

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

Solution

1414 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.

Exercise 31

5656 size 12{ { {5} over {6} } } {} of 3434 size 12{ { {3} over {4} } } {}

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

Exercise 32

1818 size 12{ { {1} over {8} } } {} of 1818 size 12{ { {1} over {8} } } {}

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

Solution

164164 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.

Exercise 33

712712 size 12{ { {7} over {"12"} } } {} of 6767 size 12{ { {6} over {7} } } {}

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

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

Exercise 34

1212 size 12{ { {1} over {2} } } {} of 4545 size 12{ { {4} over {5} } } {}

Solution

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

Exercise 35

3535 size 12{ { {3} over {5} } } {} of 512512 size 12{ { {5} over {"12"} } } {}

Exercise 36

1414 size 12{ { {1} over {4} } } {} of 8989 size 12{ { {8} over {9} } } {}

Solution

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

Exercise 37

316316 size 12{ { {3} over {"16"} } } {} of 12151215 size 12{ { {"12"} over {"15"} } } {}

Exercise 38

29 of 6529 of 65 size 12{ { {2} over {9} } "of" { {6} over {5} } } {}

Solution

415415 size 12{ { {4} over {"15"} } } {}

Exercise 39

18 of 3818 of 38 size 12{ { {1} over {8} } ital "of" { {3} over {8} } } {}

Exercise 40

23  of  91023  of  910 size 12{ { {2} over {3} } ital "of" { {9} over {"10"} } } {}

Solution

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

Exercise 41

1819 of 38541819 of 3854 size 12{ { {"18"} over {"19"} } ital "of" { {"38"} over {"54"} } } {}

Exercise 42

56 of 22556 of 225 size 12{ { {5} over {6} } ital "of"2 { {2} over {5} } } {}

Solution

22 size 12{2} {}

Exercise 43

34 of 33534 of 335 size 12{ { {3} over {4} } ital "of"3 { {3} over {5} } } {}

Exercise 44

32 of 22932 of 229 size 12{ { {3} over {2} } ital "of"2 { {2} over {9} } } {}

Solution

103 or 313103 or 313 size 12{ { {"10"} over {3} } " or "3 { {1} over {3} } } {}

Exercise 45

154 of 445154 of 445 size 12{ { {"15"} over {4} } ital "of"4 { {4} over {5} } } {}

Exercise 46

513 of 934513 of 934 size 12{5 { {1} over {3} } ital "of"9 { {3} over {4} } } {}

Solution

52

Exercise 47

11315 of 83411315 of 834 size 12{1 { {"13"} over {"15"} } ital "of"8 { {3} over {4} } } {}

Exercise 48

89 of 34 of 2389 of 34 of 23 size 12{ { {8} over {9} } ital "of" { {3} over {4} } ital "of" { {2} over {3} } } {}

Solution

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

Exercise 49

16 of 1213 of 263616 of 1213 of 2636 size 12{ { {1} over {6} } " of " { {"12"} over {"13"} } " of " { {"26"} over {"36"} } } {}

Exercise 50

12 of 13 of 1412 of 13 of 14 size 12{ { {1} over {2} } " of " { {1} over {3} } " of " { {1} over {4} } } {}

Solution

124124 size 12{ { {1} over {"24"} } } {}

Exercise 51

137 of 515 of 813137 of 515 of 813 size 12{1 { {3} over {7} } " of 5" { {1} over {5} } " of 8" { {1} over {3} } } {}

Exercise 52

245 of 556 of 757245 of 556 of 757 size 12{2 { {4} over {5} } " of 5" { {5} over {6} } " of 7" { {5} over {7} } } {}

Solution

126

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

Exercise 53

13⋅2313⋅23 size 12{ { {1} over {3} } cdot { {2} over {3} } } {}

Exercise 54

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

Solution

1414 size 12{ { {1} over {4} } } {}

Exercise 55

34⋅3834⋅38 size 12{ { {3} over {4} } cdot { {3} over {8} } } {}

Exercise 56

25⋅5625⋅56 size 12{ { {2} over {5} } cdot { {5} over {6} } } {}

Solution

1313 size 12{ { {1} over {3} } } {}

Exercise 57

38⋅8938⋅89 size 12{ { {3} over {8} } cdot { {8} over {9} } } {}

Exercise 58

56⋅141556⋅1415 size 12{ { {5} over {6} } cdot { {"14"} over {"15"} } } {}

Solution

7979 size 12{ { {7} over {9} } } {}

Exercise 59

47⋅7447⋅74 size 12{ { {4} over {7} } cdot { {7} over {4} } } {}

Exercise 60

311⋅113311⋅113 size 12{ { {3} over {"11"} } cdot { {"11"} over {3} } } {}

Solution

1

Exercise 61

916⋅2027916⋅2027 size 12{ { {9} over {"16"} } cdot { {"20"} over {"27"} } } {}

Exercise 62

3536⋅48553536⋅4855 size 12{ { {"35"} over {"36"} } cdot { {"48"} over {"55"} } } {}

Solution

28332833 size 12{ { {"28"} over {"33"} } } {}

Exercise 63

2125⋅15142125⋅1514 size 12{ { {"21"} over {"25"} } cdot { {"15"} over {"14"} } } {}

Exercise 64

7699⋅66387699⋅6638 size 12{ { {"76"} over {"99"} } cdot { {"66"} over {"38"} } } {}

Solution

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

Exercise 65

37⋅1418⋅6237⋅1418⋅62 size 12{ { {3} over {7} } cdot { {"14"} over {"18"} } cdot { {6} over {2} } } {}

Exercise 66

415⋅103⋅272415⋅103⋅272 size 12{ { {4} over {"15"} } cdot { {"10"} over {3} } cdot { {"27"} over {2} } } {}

Solution

12

Exercise 67

1415⋅2128⋅4571415⋅2128⋅457 size 12{ { {"14"} over {"15"} } cdot { {"21"} over {"28"} } cdot { {"45"} over {7} } } {}

Exercise 68

83⋅154⋅162183⋅154⋅1621 size 12{ { {8} over {3} } cdot { {"15"} over {4} } cdot { {"16"} over {"21"} } } {}

Solution

71321 or 1602171321 or 16021 size 12{7 { {"13"} over {"21"} } " or " { {"160"} over {"21"} } } {}

Exercise 69

1814⋅2135⋅3671814⋅2135⋅367 size 12{ { {"18"} over {"14"} } cdot { {"21"} over {"35"} } cdot { {"36"} over {7} } } {}

Exercise 70

35⋅2035⋅20 size 12{ { {3} over {5} } cdot "20"} {}

Solution

12

Exercise 71

89⋅1889⋅18 size 12{ { {8} over {9} } cdot "18"} {}

Exercise 72

611⋅33611⋅33 size 12{ { {6} over {"11"} } cdot "33"} {}

Solution

18

Exercise 73

1819⋅381819⋅38 size 12{ { {"18"} over {"19"} } cdot "38"} {}

Exercise 74

56⋅1056⋅10 size 12{ { {5} over {6} } cdot "10"} {}

Solution

253 or 813253 or 813 size 12{ { {"25"} over {3} } " or 8" { {1} over {3} } } {}

Exercise 75

19⋅319⋅3 size 12{ { {1} over {9} } cdot 3} {}

Exercise 76

5⋅385⋅38 size 12{5 cdot { {3} over {8} } } {}

Solution

158=178158=178 size 12{ { {"15"} over {8} } "=1" { {7} over {8} } } {}

Exercise 77

16⋅1416⋅14 size 12{"16" cdot { {1} over {4} } } {}

Exercise 78

23⋅12⋅3423⋅12⋅34 size 12{ { {2} over {3} } cdot "12" cdot { {3} over {4} } } {}

Solution

6

Exercise 79

38⋅24⋅2338⋅24⋅23 size 12{ { {3} over {8} } cdot "24" cdot { {2} over {3} } } {}

Exercise 80

518⋅10⋅25518⋅10⋅25 size 12{ { {5} over {"18"} } cdot "10" cdot { {2} over {5} } } {}

Solution

109=119109=119 size 12{ { {"10"} over {9} } "=1" { {1} over {9} } } {}

Exercise 81

1615⋅50⋅3101615⋅50⋅310 size 12{ { {"16"} over {"15"} } cdot "50" cdot { {3} over {"10"} } } {}

Exercise 82

513⋅2732513⋅2732 size 12{5 { {1} over {3} } cdot { {"27"} over {"32"} } } {}

Solution

92=41292=412 size 12{ { {9} over {2} } "=4" { {1} over {2} } } {}

Exercise 83

267⋅535267⋅535 size 12{2 { {6} over {7} } cdot 5 { {3} over {5} } } {}

Exercise 84

614⋅2415614⋅2415 size 12{6 { {1} over {4} } cdot 2 { {4} over {"15"} } } {}

Solution

856=1416856=1416 size 12{ { {"85"} over {6} } "=14" { {1} over {6} } } {}

Exercise 85

913⋅916⋅113913⋅916⋅113 size 12{9 { {1} over {3} } cdot { {9} over {"16"} } cdot 1 { {1} over {3} } } {}

Exercise 86

359⋅11314⋅1012359⋅11314⋅1012 size 12{3 { {5} over {9} } cdot 1 { {"13"} over {"14"} } cdot "10" { {1} over {2} } } {}

Solution

72

Exercise 87

2014⋅823⋅16452014⋅823⋅1645 size 12{"20" { {1} over {4} } cdot 8 { {2} over {3} } cdot "16" { {4} over {5} } } {}

Exercise 88

232232 size 12{ left ( { {2} over {3} } right ) rSup { size 8{2} } } {}

Solution

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

Exercise 89

382382 size 12{ left ( { {3} over {8} } right ) rSup { size 8{2} } } {}

Exercise 90

21122112 size 12{ left ( { {2} over {"11"} } right ) rSup { size 8{2} } } {}

Solution

41214121 size 12{ { {4} over {"121"} } } {}

Exercise 91

892892 size 12{ left ( { {8} over {9} } right ) rSup { size 8{2} } } {}

Exercise 92

122122 size 12{ left ( { {1} over {2} } right ) rSup { size 8{2} } } {}

Solution

1414 size 12{ { {1} over {4} } } {}

Exercise 93

352⋅203352⋅203 size 12{ left ( { {3} over {5} } right ) rSup { size 8{2} } cdot { {"20"} over {3} } } {}

Exercise 94

142⋅1615142⋅1615 size 12{ left ( { {1} over {4} } right ) rSup { size 8{2} } cdot { {"16"} over {"15"} } } {}

Solution

115115 size 12{ { {1} over {"15"} } } {}

Exercise 95

122⋅89122⋅89 size 12{ left ( { {1} over {2} } right ) rSup { size 8{2} } cdot { {8} over {9} } } {}

Exercise 96

122⋅252122⋅252 size 12{ left ( { {1} over {2} } right ) rSup { size 8{2} } cdot left ( { {2} over {5} } right ) rSup { size 8{2} } } {}

Solution

125125 size 12{ { {1} over {"25"} } } {}

Exercise 97

372⋅192372⋅192 size 12{ left ( { {3} over {7} } right ) rSup { size 8{2} } cdot left ( { {1} over {9} } right ) rSup { size 8{2} } } {}

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

Exercise 98

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

Solution

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

Exercise 99

16251625 size 12{ sqrt { { {"16"} over {"25"} } } } {}

Exercise 100

8112181121 size 12{ sqrt { { {"81"} over {"121"} } } } {}

Solution

911911 size 12{ { {9} over {"11"} } } {}

Exercise 101

36493649 size 12{ sqrt { { {"36"} over {"49"} } } } {}

Exercise 102

1442514425 size 12{ sqrt { { {"144"} over {"25"} } } } {}

Solution

125=225125=225 size 12{ { {"12"} over {5} } =2 { {2} over {5} } } {}

Exercise 103

23⋅91623⋅916 size 12{ { {2} over {3} } cdot sqrt { { {9} over {"16"} } } } {}

Exercise 104

35⋅258135⋅2581 size 12{ { {3} over {5} } cdot sqrt { { {"25"} over {"81"} } } } {}

Solution

1313 size 12{ { {1} over {3} } } {}

Exercise 105

852⋅2564852⋅2564 size 12{ left ( { {8} over {5} } right ) rSup { size 8{2} } cdot sqrt { { {"25"} over {"64"} } } } {}

Exercise 106

1342⋅4491342⋅449 size 12{ left (1 { {3} over {4} } right ) rSup { size 8{2} } cdot sqrt { { {4} over {"49"} } } } {}

Solution

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

Exercise 107

2232⋅3649⋅64812232⋅3649⋅6481 size 12{ left (2 { {2} over {3} } right ) rSup { size 8{2} } cdot sqrt { { {"36"} over {"49"} } } cdot sqrt { { {"64"} over {"81"} } } } {}

Exercises for Review

Exercise 108

((Reference)) How many thousands in 342,810?

Solution

2

Exercise 109

((Reference)) Find the sum of 22, 42, and 101.

Exercise 110

((Reference)) Is 634,281 divisible by 3?

Solution

yes

Exercise 111

((Reference)) Is the whole number 51 prime or composite?

Exercise 112

((Reference)) Reduce 3615036150 size 12{ { {"36"} over {"150"} } } {} to lowest terms.

Solution

625625 size 12{ { {6} over {"25"} } } {}

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