Increasing fractions to their highest terms

Given the equivalence between factors and divisors we can increase fractions to higher terms in a very similar way to when we reduce fractions. In the latter case we are dividing by divisors to reduce. In the former, we are multiplying by factors to increase.

Whenever we increase a fraction, the resultant fraction will always be equivalent to the fraction we started with.

Demonstration

Express \(\frac{3}{4}\) as an equivalent fraction having the denominator 20

$$ \\frac{3 \cdot 4}{5 \cdot 4} = \frac{12}{20} $$

Express \(\frac{2}{3}\) as an equivalent fraction having the denominator 21

$$ \\frac{2 \cdot 7}{3 \cdot 7} = \frac{14}{21} $$

Increasing fractions with variables to higher terms

Express \(\frac{2}{9}\) as an equivalent fraction having the denominator 18a

In these cases, just append the variable to the factor:

$$ \\frac{2 \cdot 2a}{9 \cdot 2a} = \frac{4a}{18a} $$