Solutions to Questions on How to Reduce Fractions in Math

Detailed solutions and explanations to the questions on how to reduce fractions are presented.

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Detailed Solutions to questions below
  1. Reduce the fractions

    a) 24 / 36

    b) 52 / 120

    c) 156 / 208

    d) 122 / 6100


    Solution

    a) We start by the prime factorization of the numerator 24 and denominator 36 as follows:

    24 = 2 2 2 3

    36 = 2 2 3 3

    Rewrite the given fraction using the prime factorization of the numerator and denominator found above:

    \( \dfrac{24}{36} = \dfrac{2 2 2 3}{2 2 3 3} \)

    Simplify

    \( \dfrac{24}{36} = \dfrac{\cancel{2 2} 2 \cancel{3}}{\cancel{2 2} \cancel{3} 3} = \dfrac{2}{3} \)


    b) The prime factorization of the numerator 52 and denominator 120 is as follows:

    52 = 2 2 13

    120 = 2 2 2 3 5

    Rewrite the given fraction using the prime factorization of 52 and 120:

    \( \dfrac{52}{120} = \dfrac{2 2 13}{2 2 2 3 5} \)

    Simplify

    \( \dfrac{52}{120} = \dfrac{\cancel{2 2} 13}{\cancel{2 2} 2 3 5} = \dfrac{13}{30} \)


    c) We start with the prime factorization of the numerator 156 and denominator 208:

    156 = 2 2 3 13

    208 = 2 2 2 2 13

    Use the prime factorization of 156 and 208:

    \( \dfrac{156}{208} = \dfrac{2 2 3 13 }{2 2 2 2 13} \)

    Simplify

    \( \dfrac{156}{208} = \dfrac{\cancel{2 2} 3 \cancel{13}}{\cancel{2 2} 2 2 \cancel{13}} = \dfrac{3}{4} \)


    d) We start with the prime factorization of the numerator 122 and denominator 6100 as follows:

    122 = 2 61

    6100 = 2 2 5 5 61

    Rewrite the given fraction using the prime factorization of 122 and 6100:

    \( \dfrac{122}{6100} = \dfrac{2 61}{2 2 5 5 61 } \)

    Simplify

    \( \dfrac{122}{6100} = \dfrac{\cancel{2} \cancel{61}}{\cancel{2} 2 5 5 \cancel{61}} = \dfrac{1}{50} \)


  2. Reduce and compare each pair of fractions.

    a) 26 / 39 and 14 / 42

    b) 45 / 75 and 52 / 65


    Solution

    a) We start by the prime factorization and simplification of the pair of fractions:

    \( \dfrac{26}{39} = \dfrac{2 13}{3 13} = \dfrac{2}{3} \)

    \( \dfrac{14}{42} = \dfrac{2 7}{2 3 7} = \dfrac{1}{3} \)

    Comparing the reduced fractions 2/3 is greater than 1/3 and therefore the fraction 26 / 39 is greater than 14 / 42.


    b) The prime factorization and simplification of the pair of fractions gives:

    \( \dfrac{45}{75} = \dfrac{3 3 5}{3 5 5} = \dfrac{3}{5} \)

    \( \dfrac{52}{65} = \dfrac{2 2 13}{5 13} = \dfrac{4}{5} \)

    Comparing the reduced fractions 4/5 is greater than 3/5 and therefore the fraction 52 / 65 is greater than 45 / 75.

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Updated: 20 January 2017 (A Dendane)
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