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1.3 Fractions: equivalent fractions  (Page 3/3)

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The fraction 3 4 size 12{ { {3} over {4} } } {} can be raised to 24 32 size 12{ { {"24"} over {"32"} } } {} by multiplying both the numerator and denominator by 8.

Three fourths equals three times eight, over four time eight, which is equal to twenty-four over thirty-two. Notice that eight over eight is equal to 1.

Most often, we will want to convert a given fraction to an equivalent fraction with a higher specified denominator. For example, we may wish to convert 5 8 size 12{ { {5} over {8} } } {} to an equivalent fraction that has denominator 32, that is,

5 8 = ? 32 size 12{ { {5} over {8} } = { {?} over {"32"} } } {}

This is possible to do because we know the process. We must multiply both the numerator and denominator of 5 8 size 12{ { {5} over {8} } } {} by the same nonzero whole number in order to 8 obtain an equivalent fraction.

We have some information. The denominator 8 was raised to 32 by multiplying it by some nonzero whole number. Division will give us the proper factor. Divide the original denominator into the new denominator.

32 ÷ 8 = 4 size 12{"32 " div " 8 "=" 4"} {}

Now, multiply the numerator 5 by 4.

5 ⋅ 4 = 20 size 12{"5 " cdot "4 "=" 20"} {}

Thus,

5 8 = 5 ⋅ 4 8 ⋅ 4 = 20 32 size 12{ { {5} over {8} } = { {5 cdot 4} over {8 cdot 4} } = { {"20"} over {"32"} } } {}

So,

5 8 = 20 32 size 12{ { {5} over {8} } = { {"20"} over {"32"} } } {}

Sample set d

Determine the missing numerator or denominator.

3 7 = ? 35 size 12{ { {3} over {7} } = { {?} over {"35"} } } {} . Divide the original denominator into the new denominator.

35 ÷ 7 = 5 size 12{"35"¸7=5} {} The quotient is 5. Multiply the original numerator by 5.

3 7 = 3 ⋅ 5 7 ⋅ 5 = 15 35 size 12{ { {3} over {7} } = { {3 cdot 5} over {7 cdot 5} } = { {"15"} over {"35"} } } {} The missing numerator is 15.

5 6 = 45 ? size 12{ { {5} over {6} } = { {"45"} over {?} } } {} . Divide the original numerator into the new numerator.

45 ÷ 5 = 9 size 12{"45"¸5=9} {} The quotient is 9. Multiply the original denominator by 9.

5 6 = 5 ⋅ 9 6 ⋅ 9 = 45 54 size 12{ { {5} over {6} } = { {5 cdot 9} over {6 cdot 9} } = { {"45"} over {"54"} } } {} The missing denominator is 45.

Practice set d

Determine the missing numerator or denominator.

4 5 = ? 40 size 12{ { {4} over {5} } = { {?} over {"40"} } } {}

32

3 7 = ? 28 size 12{ { {3} over {7} } = { {?} over {"28"} } } {}

12

1 6 = ? 24 size 12{ { {1} over {6} } = { {?} over {"24"} } } {}

4

3 10 = 45 ? size 12{ { {3} over {"10"} } = { {"45"} over {?} } } {}

150

8 15 = ? 165 size 12{ { {8} over {"15"} } = { {?} over {"165"} } } {}

88

Exercises

For the following problems, determine if the pairs of fractions are equivalent.

1 2 , 5 10 size 12{ { {1} over {2} } , { {5} over {"10"} } } {}

equivalent

2 3 , 8 12 size 12{ { {2} over {3} } , { {8} over {"12"} } } {}

5 12 , 10 24 size 12{ { {5} over {"12"} } , { {"10"} over {"24"} } } {}

equivalent

1 2 , 3 6 size 12{ { {1} over {2} } , { {3} over {6} } } {}

3 5 , 12 15 size 12{ { {3} over {5} } , { {"12"} over {"15"} } } {}

not equivalent

1 6 , 7 42 size 12{ { {1} over {6} } , { {7} over {"42"} } } {}

16 25 , 49 75 size 12{ { {"16"} over {"25"} } , { {"49"} over {"75"} } } {}

not equivalent

5 28 , 20 112 size 12{ { {5} over {"28"} } , { {"20"} over {"112"} } } {}

3 10 , 36 110 size 12{ { {3} over {"10"} } , { {"36"} over {"110"} } } {}

not equivalent

6 10 , 18 32 size 12{ { {6} over {"10"} } , { {"18"} over {"32"} } } {}

5 8 , 15 24 size 12{ { {5} over {8} } , { {"15"} over {"24"} } } {}

equivalent

10 16 , 15 24 size 12{ { {"10"} over {"16"} } , { {"15"} over {"24"} } } {}

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

not equivalent

5 7 , 15 21 size 12{ { {5} over {7} } , { {"15"} over {"21"} } } {}

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

not equivalent

For the following problems, determine the missing numerator or denominator.

1 3 = ? 12 size 12{ { {1} over {3} } = { {?} over {"12"} } } {}

1 5 = ? 30 size 12{ { {1} over {5} } = { {?} over {"30"} } } {}

6

2 3 = ? 9 size 12{ { {2} over {3} } = { {?} over {9} } } {}

1 5 = ? 30 size 12{ { {1} over {5} } = { {?} over {"30"} } } {}

12

2 3 = ? 9

3 4 = ? 16

12

5 6 = ? 18 size 12{ { {5} over {6} } = { {?} over {"18"} } } {}

4 5 = ? 25 size 12{ { {4} over {5} } = { {?} over {"25"} } } {}

20

1 2 = 4 ? size 12{ { {1} over {2} } = { {4} over {?} } } {}

9 25 = 27 ? size 12{ { {9} over {"25"} } = { {"27"} over {?} } } {}

75

3 2 = 18 ? size 12{ { {3} over {2} } = { {"18"} over {?} } } {}

5 3 = 80 ? size 12{ { {5} over {3} } = { {"80"} over {?} } } {}

48

1 8 = 3 ? size 12{ { {1} over {8} } = { {3} over {?} } } {}

4 5 = ? 100 size 12{ { {4} over {5} } = { {?} over {"100"} } } {}

80

1 2 = 25 ? size 12{ { {1} over {2} } = { {"25"} over {?} } } {}

3 16 = ? 96 size 12{ { {3} over {"16"} } = { {?} over {"96"} } } {}

18

15 16 = 225 ? size 12{ { {"15"} over {"16"} } = { {"225"} over {?} } } {}

11 12 = ? 168 size 12{ { {"11"} over {"12"} } = { {?} over {"168"} } } {}

154

9 13 = ? 286 size 12{ { {9} over {"13"} } = { {?} over {"286"} } } {}

32 33 = ? 1518 size 12{ { {"32"} over {"33"} } = { {?} over {"1518"} } } {}

1,472

19 20 = 1045 ? size 12{ { {"19"} over {"20"} } = { {"1045"} over {?} } } {}

37 50 = 1369 ? size 12{ { {"37"} over {"50"} } = { {"1369"} over {?} } } {}

1,850

For the following problems, reduce, if possible, each of the fractions to lowest terms.

6 8 size 12{ { {6} over {8} } } {}

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

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

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

6 14 size 12{ { {6} over {"14"} } } {}

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

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

4 14 size 12{ { {4} over {"14"} } } {}

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

1 6 size 12{ { {1} over {6} } } {}

4 6 size 12{ { {4} over {6} } } {}

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

18 14 size 12{ { {"18"} over {"14"} } } {}

20 8 size 12{ { {"20"} over {8} } } {}

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

4 6 size 12{ { {4} over {6} } } {}

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

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

6 14 size 12{ { {6} over {"14"} } } {}

14 6 size 12{ { {"14"} over {6} } } {}

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

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

16 70 size 12{ { {"16"} over {"70"} } } {}

8 35 size 12{ { {8} over {"35"} } } {}

40 60 size 12{ { {"40"} over {"60"} } } {}

20 12 size 12{ { {"20"} over {"12"} } } {}

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

32 28 size 12{ { {"32"} over {"28"} } } {}

36 10 size 12{ { {"36"} over {"10"} } } {}

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

36 60 size 12{ { {"36"} over {"60"} } } {}

12 18 size 12{ { {"12"} over {"18"} } } {}

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

18 27 size 12{ { {"18"} over {"27"} } } {}

18 24 size 12{ { {"18"} over {"24"} } } {}

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

32 40 size 12{ { {"32"} over {"40"} } } {}

11 22 size 12{ { {"11"} over {"22"} } } {}

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

27 81 size 12{ { {"27"} over {"81"} } } {}

17 51 size 12{ { {"17"} over {"51"} } } {}

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

16 42 size 12{ { {"16"} over {"42"} } } {}

39 13 size 12{ { {"39"} over {"13"} } } {}

3

44 11 size 12{ { {"44"} over {"11"} } } {}

66 33 size 12{ { {"66"} over {"33"} } } {}

2

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

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

already reduced

15 40 size 12{ { {"15"} over {"40"} } } {}

36 100 size 12{ { {"36"} over {"100"} } } {}

9 25 size 12{ { {9} over {"25"} } } {}

45 32 size 12{ { {"45"} over {"32"} } } {}

30 75 size 12{ { {"30"} over {"75"} } } {}

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

121 132 size 12{ { {"121"} over {"132"} } } {}

72 64 size 12{ { {"72"} over {"64"} } } {}

9 8 size 12{ { {9} over {8} } } {}

30 105 size 12{ { {"30"} over {"105"} } } {}

46 60 size 12{ { {"46"} over {"60"} } } {}

23 30 size 12{ { {"23"} over {"30"} } } {}

75 45 size 12{ { {"75"} over {"45"} } } {}

40 18 size 12{ { {"40"} over {"18"} } } {}

20 9 size 12{ { {"20"} over {9} } } {}

108 76 size 12{ { {"108"} over {"76"} } } {}

7 21 size 12{ { {7} over {"21"} } } {}

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

6 51 size 12{ { {6} over {"51"} } } {}

51 12 size 12{ { {"51"} over {"12"} } } {}

17 4 size 12{ { {"17"} over {4} } } {}

8 100 size 12{ { {8} over {"100"} } } {}

51 54 size 12{ { {"51"} over {"54"} } } {}

17 18 size 12{ { {"17"} over {"18"} } } {}

A ream of paper contains 500 sheets. What frac­tion of a ream of paper is 200 sheets? Be sure to reduce.

There are 24 hours in a day. What fraction of a day is 14 hours?

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

A full box contains 80 calculators. How many calculators are in 1 4 size 12{ { {1} over {4} } } {} of a box?

There are 48 plants per flat. How many plants are there in 1 3 size 12{ { {1} over {3} } } {} of a flat?

16

A person making $18,000 per year must pay $3,960 in income tax. What fraction of this per­son's yearly salary goes to the IRS?

For the following problems, find the mistake.

3 24 = 3 3 ⋅ 8 = 0 8 = 0 size 12{ { {3} over {"24"} } = { { { {3}}} over { { {3}} cdot 8} } = { {0} over {8} } =0} {}

Should be 1 8 size 12{ { {1} over {8} } } {} ; the cancellation is division, so the numerator should be 1.

8 10 = 2 + 6 2 + 8 = 6 8 = 3 4 size 12{ { {8} over {"10"} } = { { { {2}}+6} over { { {2}}+8} } = { {6} over {8} } = { {3} over {4} } } {}

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

Cancel factors only, not addends; 7 15 size 12{ { {7} over {"15"} } } {} is already reduced.

6 7 = 5 + 1 5 + 2 = 1 2 size 12{ { {6} over {7} } = { { { {5}}+1} over { { {5}}+2} } = { {1} over {2} } } {}

9 9 = 0 0 = 0 size 12{ { { { {9}}} over { { {9}}} } = { {0} over {0} } =0} {}

Same as [link] ; answer is 1 1 size 12{ { {1} over {1} } } {} or 1.

Exercises for review

( [link] ) Round 816 to the nearest thousand.

( [link] ) Perform the division: 0 ÷ 6 size 12{0 div 6} {} .

0

( [link] ) Find all the factors of 24.

( [link] ) Find the greatest common factor of 12 and 18.

6

( [link] ) Convert 15 8 size 12{ { {"15"} over {8} } } {} to a mixed number.

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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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