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Observe a very important property of a fraction that has been reduced to lowest terms. The only whole number that divides both the numerator and denominator without a remainder is the number 1. When 1 is the only whole number that divides two whole numbers, the two whole numbers are said to be relatively prime .

Relatively prime

A fraction is reduced to lowest terms if its numerator and denominator are relatively prime .

Methods of reducing fractions to lowest terms

    Method 1: dividing out common primes

  1. Write the numerator and denominator as a product of primes.
  2. Divide the numerator and denominator by each of the common prime factors. We often indicate this division by drawing a slanted line through each divided out factor. This process is also called cancelling common factors .
  3. The product of the remaining factors in the numerator and the product of remaining factors of the denominator are relatively prime, and this fraction is reduced to lowest terms.

Sample set b

Reduce each fraction to lowest terms.

6 18 = 2 1 3 1 2 1 3 1 3 = 1 3 size 12{ { {6} over {"18"} } = { { { { {2}}} cSup { size 8{1} } cdot { { {3}}} cSup { size 8{1} } } over { { { {2}}} cSub { size 8{1} } cdot { { {3}}} cSub { size 8{1} } cdot 3} } = { {1} over {3} } } {} 1 and 3 are relatively prime.

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16 20 = 2 1 2 1 2 2 2 1 2 1 5 = 4 5 size 12{ { {"16"} over {"20"} } = { { { { {2}}} cSup { size 8{1} } cdot { { {2}}} cSup { size 8{1} } cdot 2 cdot 2} over { { { {2}}} cSub { size 8{1} } cdot { { {2}}} cSub { size 8{1} } cdot 5} } = { {4} over {5} } } {} 4 and 5 are relatively prime.

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56 104 = 2 1 2 1 2 1 7 2 1 2 1 2 1 13 = 7 13 size 12{ { {"56"} over {"104"} } = { { { { {2}}} cSup { size 8{1} } cdot { { {2}}} cSup { size 8{1} } cdot { { {2}}} cSup { size 8{1} } cdot 7} over { { { {2}}} cSub { size 8{1} } cdot { { {2}}} cSub { size 8{1} } cdot { { {2}}} cSub { size 8{1} } cdot "13"} } = { {7} over {"13"} } } {} 7 and 13 are relatively prime (and also truly prime)

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315 336 = 3 1 3 5 7 1 2 2 2 2 3 1 7 1 = 15 16 size 12{ { {"315"} over {"336"} } = { { { { {3}}} cSup { size 8{1} } cdot 3 cdot cdot 5 cdot { { {7}}} cSup { size 8{1} } } over {2 cdot 2 cdot 2 cdot 2 cdot { { {3}}} cSub { size 8{1} } cdot { { {7}}} cSub { size 8{1} } cdot } } = { {"15"} over {"16"} } } {} 15 and 16 are relatively prime.

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8 15 = 2 2 2 3 5 size 12{ { {8} over {"15"} } = { {2 cdot 2 cdot 2} over {3 cdot 5} } } {} No common prime factors, so 8 and 15 are relatively prime.

The fraction 8 15 size 12{ { {8} over {"15"} } } {} is reduced to lowest terms.

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Practice set b

Reduce each fraction to lowest terms.

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

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

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6 15 size 12{ { {6} over {"15"} } } {}

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

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6 48 size 12{ { {6} over {"48"} } } {}

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

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21 48 size 12{ { {"21"} over {"48"} } } {}

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

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72 42 size 12{ { {"72"} over {"42"} } } {}

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

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135 243 size 12{ { {"135"} over {"243"} } } {}

5 9 size 12{ { {5} over {9} } } {}

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    Method 2: dividing out common factors

  1. Mentally divide the numerator and the denominator by a factor that is com­mon to each. Write the quotient above the original number.
  2. Continue this process until the numerator and denominator are relatively prime.

Sample set c

Reduce each fraction to lowest terms.

25 30 size 12{ { {"25"} over {"30"} } } {} . 5 divides into both 25 and 30.

25 5 30 6 = 5 6 size 12{ { { { { {2}} { {5}}} cSup { size 8{5} } } over { { { {3}} { {0}}} cSub { size 8{6} } } } = { {5} over {6} } } {} 5 and 6 are relatively prime.

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18 24 size 12{ { {"18"} over {"24"} } } {} . Both numbers are even so we can divide by 2.

18 9 24 12 Now, both 9 and 12 are divisible by 3.

18 9 3 24 12 4 = 3 4 size 12{ { { { { {1}} { {8}}} cSup { size 8{ { { {9}}} cSup { size 6{3} } } } } over { { { {2}} { {4}}} cSub {"12"} } } size 12{ {}= { {3} over {4} } }} {} 3 and 4 are relatively prime.

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210 21 7 150 15 5 = 7 5 size 12{ { { { { {2}} { {1}} { {0}}} cSup { size 8{ { { {2}} { {1}}} cSup { size 6{7} } } } } over { { { {1}} { {5}} { {0}}} cSub { { { {1}} { {5}}} cSub { size 6{5} } } } } size 12{ {}= { {7} over {5} } }} {} . 7 and 5 are relatively prime.

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36 96 = 18 48 = 9 24 = 3 8 size 12{ { {"36"} over {"96"} } = { {"18"} over {"48"} } = { {9} over {"24"} } = { {3} over {8} } } {} . 3 and 8 are relatively prime.

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Practice set c

Reduce each fraction to lowest terms.

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

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

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9 24 size 12{ { {9} over {"24"} } } {}

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

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21 84 size 12{ { {"21"} over {"84"} } } {}

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

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48 64 size 12{ { {"48"} over {"64"} } } {}

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

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63 81 size 12{ { {"63"} over {"81"} } } {}

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

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150 240 size 12{ { {"150"} over {"240"} } } {}

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

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Raising fractions to higher terms

Equally as important as reducing fractions is raising fractions to higher terms. Raising a fraction to higher terms is the process of constructing an equivalent fraction that has higher values in the numerator and denominator than the original fraction.

The fractions 3 5 size 12{ { {3} over {5} } } {} and 9 15 size 12{ { {9} over {"15"} } } {} are equivalent, that is, 3 5 = 9 15 size 12{ { {3} over {5} } = { {9} over {"15"} } } {} . Notice also,

3 3 5 3 = 9 15 size 12{ { {3 cdot 3} over {5 cdot 3} } = { {9} over {"15"} } } {}

Notice that 3 3 = 1 size 12{ { {3} over {3} } =1} {} and that 3 5 1 = 3 5 size 12{ { {3} over {5} } cdot 1= { {3} over {5} } } {} . We are not changing the value of 3 5 size 12{ { {3} over {5} } } {} .

From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has higher values in the numerator and denominator. This method is called raising a fraction to higher terms .

Raising a fraction to higher terms

A fraction can be raised to an equivalent fraction that has higher terms in the numerator and denominator by multiplying both the numerator and denominator by the same nonzero whole number.

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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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