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# Fractions as ratios

Focus on the different ways of expressing fractions as ratios, including using fractions as operators (multipliers).

## Fractions as ratios

You can interpret division as grouping, as well as sharing. A fraction, when viewed in terms of grouping, is called a ratio – the number of times one number can be divided by another.

So the fraction $\frac{18}{6}$ can be interpreted as:

• ‘How many times can 18 be divided by 6?’ or
• ‘What does 6 have to be multiplied by to make 18?’

In this case the answer is a whole number (3) but this is not always the case.

For example, $\frac{2}{3}$ can be thought of as the number of times 2 can be divided by 3 ($\frac{2}{3}$ of a time) and, inversely, $\frac{3}{2}$ can be thought of as the number of times 3 can be divided by 2 ($\frac{3}{2}$ or $1\frac{1}{2}$ times).

It is useful to consider the divisions $\frac{2}{3}$ and $\frac{3}{2}$ together. Moving away from the part/whole aspect, the comparison of one whole number with another can be illustrated by two line segments, of length p and q for example. (If you wanted to, you could make a concrete form for these representations with Cuisenaire rods, linking cubes, or strips of card.)

There are several equivalent ways in which these relationships can be expressed:

 ratio of p to q = 2 : 3 ratio of q to p = 3 : 2 $\frac{p}{q}=\frac{2}{3}$ $\frac{q}{p}=\frac{3}{2}$ $p=\frac{2}{3}×q$ $q=\frac{3}{2}×p$

## Using fractions as operators

In $p=\frac{2}{3}×q$ and $q=\frac{3}{2}×p$, fractions are used as operators (multipliers).

To understand this, first think about the meaning of $\frac{p}{q}$. ‘What is p divided by q?’ is equivalent to ‘What does q have to be multiplied by to give p?’. In this example, the answer is $\frac{2}{3}$. So the equation $p=\frac{2}{3}×q$ can be read ‘p equals two thirds of q’. The operator ‘$\frac{2}{3}×$’ means ‘$\frac{2}{3}$ of’ or ‘2 lots of $\frac{1}{3}$ of’. It is equivalent to dividing by 3 and multiplying by 2.

Similarly, ‘What is q divided by p?’ is equivalent to ‘What does p have to be multiplied by to give q?’. In this example, the answer is $\frac{3}{2}$. So the equation $q=\frac{3}{2}×p$ can be read ‘q equals 3 halves of p’. The operator ‘$\frac{3}{2}×$’ means ‘$\frac{3}{2}$ of’ or ‘3 lots of $\frac{1}{2}$ of’. It is equivalent to dividing by 2 and multiplying by 3.

## More terminology

The term proportion is often used in the everyday sense of a fractional part.

Inverted fractions are called reciprocals. For example, $\frac{3}{2}$ is the reciprocal of $\frac{2}{3}$, and $\frac{2}{3}$ is the reciprocal of $\frac{3}{2}$.

## Dividing a number into parts in a given ratio

To divide a number into parts in a given ratio, first find the total number of parts involved.

For example, to divide 30 in the ratio of 2 : 3, there will be 5 parts altogether (2 + 3). Each part will be $\frac{1}{5}$ of 30 = 6. So $\frac{2}{5}\text{of}30=12$ and $\frac{3}{5}\text{of}30=18$ so the required numbers are 12 and 18.

Generalising, if n is to be divided into two parts in the ratio p : q, the parts will be $\frac{p}{\left(p+q\right)}$ of n and $\frac{q}{\left(p+q\right)}$ of n.