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