Ratios

Definition

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.

Simplifying Ratios

Like fractions, ratios can often be simplified. To simplify a ratio, divide all parts of the ratio by their highest common factor. For example, the highest common factor of both parts of the ratio $4:2$ is $2$, so $4:2 = 2:1$.

Example 1

Simplify the ratio $16:4$.

Solution

$16$ and $4$ have a common factor of $4$; dividing through by this gives $4:1$.

Example 2

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

Solution

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

The ratio can be simplified again. 3 is a factor of every component; dividing through gives the ratio $5:3:2:4$.

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.

Example 3

Write the ratio $9:14$ in the form $1:a$ where $a$ is a fraction.

Solution

This time we have to cancel the left part of the ratio down to be $1$. This is done by dividing by $9$. We must also divide the right part of the ratio by $9$ so that the parts are in the same proportion. This gives $1:\frac{14}{9}$.

Using Ratios

Ratios can be used to share or divide quantities.

Worked Examples
Example 1

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

Solution

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, $40+24 = 64$.

Example 2

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?

Solution

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:

\[3\times 2 = 6\text{m³}.\]

Example 3

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?

Solution

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 ratios 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

Converting Between Fractions and Ratios

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.

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 $\frac{40}{40+24}=\frac{40}{60}$. This fraction can then be simplified to $\frac{2}{3}$.

External Resources