A **ratio** is a way of comparing two or more quantites. Ratios can be used to compare costs, weights and sizes. For example, $2:3$ is a ratio, which means for every two parts of one thing, there are three parts of another. Note that ratios do not have units.

Like fractions, ratios can often be simplified. To simplify a ratio, divide all parts of the ratio by their highest common factor. A ratio which has been simplified is said to be written in its simplest form. For example, the highest common factor of both parts of the ratio $4:2$ is $2$, so $4:2 = 2:1$.

Simplify the ratio $16:4$.

$16$ and $4$ have a common factor of $4$. Dividing through by $4$ gives $4:1$.

Simplify the ratio $75:45:30:60$.

There is a common factor of $5$, since each component ends in a $0$ or $5$. Dividing through by $5$ gives $15:9:6:12$.

The ratio can be simplified again since $3$ is a factor of every component. Dividing through by $3$ gives the ratio $5:3:2:4$. This ratio is written in its simplest form.

**Note:** Alternatively, if $15$ was spotted as common factor at the start, then dividing through by $15$ straight away would give the same final answer - check this for yourself.

Sometimes it is useful to write a ratio in the form $1:n$ or $n:1$ where $n$ is any number, possibly a fraction or decimal.

For instance, suppose we are asked to write the ratio $9:14$ in the form $1:n$ where $n$ is a fraction. To do this, we need to make the left-hand side of the ratio equal to $1$. This is done by dividing both sides of the ratio by $9$. This gives $1:\frac{14}{9}$.

Ratios can be used to share or divide quantities.

£64 is to be shared between two people, **A** and **B**, in the ratio $5:3$. Work out how much they each get.

The ratio has a total of $8$ parts since $5+3=8$.

Dividing the full amount $64$ by $8$:

\[64 \div 8 = 8\]

So each part is equal to £8.

Person **A** has $5$ parts, so will receive \[5 \times £8 = £40.\]

Person **B** has $3$ parts, so will receive \[3 \times £8 = £24\] **Note**: The amounts received by **A** and **B** should sum to the original amount. That is, $40+24 = 64$.

Concrete is made by mixing gravel, sand and cement in the ratio $3:2:1$. How much gravel will be needed to make $12$m³ of concrete?

The ratio has a total of $3+2+1=6$ parts.

Divide the full amount $12$m³ by $6$:

\[12 \div 6 = 2\]

So we have six $2$m³ parts.

Gravel makes up three parts of the mixture:

\[3\times 2 = 6\text{m³}.\]

Here is a list of the ingredients to make a quantity of houmous sufficient for 6 people.

- 2 cloves garlic
- 4 oz chick peas
- 4 tbs olive oil
- 5 fl oz tahini paste

What amounts of each ingredient would be needed so that there will be enough for 9 people?

The ratio for 6 people is $2:4:4:5$. If we divide this by $6$, we would get the ratio for one person. Then multiply by $9$ to get the ratio for $9$ people.

\begin{array}{cccccccr} 2 & : & 4 & : & 4 & : & 5 & \text{(6 people)} \\ \frac{2}{6} & : & \frac{4}{6} & : & \frac{4}{6} & : & \frac{5}{6} & \text{(1 person)} \\ \frac{2}{6} \times 9 & : & \frac{4}{6} \times 9 & : & \frac{4}{6} \times 9 & : & \frac{5}{6} \times 9 & \text{(9 people)} \\ 3 & : & 6 & : & 6 & : & 7\tfrac{1}{2} & \text{(9 people)} \\ \end{array}

So the recipe for 9 people is:

- 3 cloves garlic
- 6 oz chick peas
- 6 tbs olive oil
- 7½ fl oz tahini paste

All ratios can be expressed in the form of a fraction. To write a ratio as a fraction in its *simplest form* we write the first part of the ratio as the numerator and the sum of the two parts as the denominator and then simplify the fraction: \[a:b \text{ is equivalent to } \frac{a}{a+b}\]

Using the example about the money shared between person $A$ and person $B$ above, the fraction or *proportion* of the total money ($£64$) received by person $A$ would be $\frac{40}{40+24}=\frac{40}{64}$. This fraction can then be simplified to $\frac{5}{8}$.

To convert a fraction into a ratio in its simplest form, write the numerator of the fraction as the first part of the ratio and the difference between the denominator and the numerator of the fraction as the second part of the ratio. Then simplify the ratio. \[\frac{c}{d}\text{ is equivalent to }c:(d-c)\]

For example, $\frac{5}{12}$ is equivalent to $5:7$.

- Various resources on ratios by mathcentre.
- Ratio and proportion videos, worksheets and quizzes at BBC Skillswise.