eolas/zk/Increasing_fractions_to_their_highest_terms.md

---
tags:
  - prealgebra
  - fractions
---

# 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](Equivalent%20fractions.md) 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:

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