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:

Using fractions as operators

In $p=\frac{2}{3}\times q$ and $q=\frac{3}{2}\times p$, 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 $p=\frac{2}{3}\times q$ can be read ‘p equals two thirds of q’. The operator ‘$\frac{2}{3}\times$’ 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 $q=\frac{3}{2}\times p$ can be read ‘q equals 3 halves of p’. The operator ‘$\frac{3}{2}\times$’ 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.