Solutions to Questions on How to Reduce Fractions in Maths

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
    1. We start with the prime factorization of numerator 24 and denominator 36: \[ 24 = 2 \times 2 \times 2 \times 3, \quad 36 = 2 \times 2 \times 3 \times 3 \] Rewrite the fraction using the factorization: \[ \dfrac{24}{36} = \dfrac{2 \times 2 \times 2 \times 3}{2 \times 2 \times 3 \times 3} \] Simplify by canceling common factors: \[ \dfrac{24}{36} = \dfrac{\cancel{2} \times \cancel{2} \times 2 \times \cancel{3}}{\cancel{2} \times \cancel{2} \times \cancel{3} \times 3} = \dfrac{2}{3} \]
    2. Prime factorization of numerator 52 and denominator 120: \[ 52 = 2 \times 2 \times 13, \quad 120 = 2 \times 2 \times 2 \times 3 \times 5 \] Rewrite the fraction: \[ \dfrac{52}{120} = \dfrac{2 \times 2 \times 13}{2 \times 2 \times 2 \times 3 \times 5} \] Simplify: \[ \dfrac{52}{120} = \dfrac{\cancel{2} \times \cancel{2} \times 13}{\cancel{2} \times \cancel{2} \times 2 \times 3 \times 5} = \dfrac{13}{30} \]
    3. Prime factorization of numerator 156 and denominator 208: \[ 156 = 2 \times 2 \times 3 \times 13, \quad 208 = 2 \times 2 \times 2 \times 2 \times 13 \] Rewrite the fraction: \[ \dfrac{156}{208} = \dfrac{2 \times 2 \times 3 \times 13}{2 \times 2 \times 2 \times 2 \times 13} \] Simplify: \[ \dfrac{156}{208} = \dfrac{\cancel{2} \times \cancel{2} \times 3 \times \cancel{13}}{\cancel{2} \times \cancel{2} \times 2 \times 2 \times \cancel{13}} = \dfrac{3}{4} \]
    4. Prime factorization of numerator 122 and denominator 6100: \[ 122 = 2 \times 61, \quad 6100 = 2 \times 2 \times 5 \times 5 \times 61 \] Rewrite the fraction: \[ \dfrac{122}{6100} = \dfrac{2 \times 61}{2 \times 2 \times 5 \times 5 \times 61} \] Simplify: \[ \dfrac{122}{6100} = \dfrac{\cancel{2} \times \cancel{61}}{\cancel{2} \times 2 \times 5 \times 5 \times \cancel{61}} = \dfrac{1}{50} \]
  2. Reduce and compare each pair of fractions.
    1. Prime factorization and simplification of the pair: \[ \dfrac{26}{39} = \dfrac{2 \times 13}{3 \times 13} = \dfrac{2}{3} \] \[ \dfrac{14}{42} = \dfrac{2 \times 7}{2 \times 3 \times 7} = \dfrac{1}{3} \] Comparing, \( \dfrac{2}{3} > \dfrac{1}{3} \), so \( \dfrac{26}{39} > \dfrac{14}{42} \).
    2. Prime factorization and simplification: \[ \dfrac{45}{75} = \dfrac{3 \times 3 \times 5}{3 \times 5 \times 5} = \dfrac{3}{5} \] \[ \dfrac{52}{65} = \dfrac{2 \times 2 \times 13}{5 \times 13} = \dfrac{4}{5} \] Comparing, \( \dfrac{4}{5} > \dfrac{3}{5} \), so \( \dfrac{52}{65} > \dfrac{45}{75} \).

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