# Equivalent Fractions

Given two fractions, $$\frac{a}{b}$$ and $$\frac{c}{d}$$, we say they are equivalent if they are:

equal
and
written over different denominators
For instance the two fractions $$\frac{1}{2}$$ and $$\frac{4}{8}$$ are equivalent as they are both equal to $$0.5$$ and are written different denominators.

## Tutorial 1: defining equivalent fractions

In the following tutorial we explain what is meant by equivalent fractions as well as illustrate them with some examples.

This should help intriduce the topic, make sure to watch it now.

## Writing Fractions Over Different Denominators

Given a fraction, for example $$\frac{1}{3}$$, we'll somtimes need to express it over a different denominator. In other words we'll sometimes need to express it as equivalent fraction over a different denominator.
For example we may need to write $$\frac{1}{3}$$ as an equivalent fraction over $$12$$.

This leads to having to find the numerator, which we've called $$x$$ here: $\frac{1}{3} = \frac{x}{12}$ The method for doing this is explained in tutorial $$3$$.

## Tutorial 3: Finding equivalent fractions

In the following tutorial we learn how to write a fraction as an equivalent fraction over a different denominator.

In particular we see how to write $$\frac{1}{2}$$ as an equivalent fraction over $$6$$, by solving $$\frac{1}{2} = \frac{x}{6}$$.
Similarly we show how to write $$\frac{2}{3}$$ as an equivalent fraction over $$12$$ by solving $$\frac{2}{3} = \frac{x}{12}$$ as well as how to write $$\frac{9}{15}$$ as an equivalent fraction over $$5$$ by solving$$\frac{9}{15} = \frac{x}{5}$$.

## Method: how to find an equivalent fraction over a given denominator

Given a fraction, like $$\frac{2}{3}$$, we can write it as an equivalent fraction over a given denominator, for instance$$\frac{x}{12}$$, in two steps:

• Step 1: look at the fraction we know and focus on its denominator. Find how much you would have to multiply, or divide, it by to get the other denominator.
• Step 2: Multiply the numerator (of the fraction we know) by the number found in step 1.

## Exercise

1. $$\frac{2}{3} = \frac{x}{9}$$
2. $$\frac{3}{5} = \frac{x}{20}$$
3. $$\frac{9}{24} = \frac{x}{8}$$
4. $$\frac{2}{3} = \frac{x}{9}$$
5. $$\frac{20}{25} = \frac{x}{5}$$

## Finding equivalent fractions with imposed numerators

At times we'll need to write a fraction as an equivalent fraction for which the numerator has to equal a certain value.

For instance we may need to write $$\frac{4}{5}$$ as an equivalent fraction with $$28$$ as the numerator.

We do this by finding the value of $$x$$ in: $\frac{4}{5} = \frac{28}{x}$

## Method: finding equivalent fractions when the numerator is given

Given a fraction, such as $$\frac{1}{3}$$, to write it as an equivalent fraction, whose numertor is imposed we can use the following two steps:

• Step 1: starting from the fraction whose numerator and denominator are known, find what number we have to multiply (or divide) the numerator by to obtain the imposed numerator.
• Step 2: Multiply the denominator that we know by that number. The value obtained is the value of the unknown denominator.

## Exercise

1. $$\frac{1}{2} = \frac{3}{x}$$
2. $$\frac{4}{5} = \frac{16}{x}$$
3. $$\frac{4}{5} = \frac{16}{x}$$
4. $$\frac{20}{15} = \frac{4}{x}$$