Ratio

​​In a nutshell

A ratio shows how much of one thing there is compared to another by separating a total into parts.

Expressing ratios

Ratios are written as two or more values separated by a colon (:) which is read as 'to'. In different kinds of problems they can be simplified, scaled up or converted from fractions and decimals to whole numbers.

Simplifying whole number ratios

Ratios are simplified just like fractions - by dividing each value by the same thing. Two ratios that share the same simplest form are equal ratios.

procedure

1. ​Find a number that all values in the ratio can be divided by - this is known as a common factor.
   
2. Divide all values in the ratio by this common factor.
   
3. Keep doing so until the ratio cannot be simplified any further. This is called the simplest form of the ratio.
   

Example 1

Simplify the ratio $$12:3:6$$ to be in its simplest form.

All values have a common factor of three. Divide each value in the ratio by three.

$$(12\div3):(3\div3):(6\div3)\\4: 1:2$$

Four, one and two have no more common factors and so the ratio can be simplified no further.

​$$\underline{4:1:2}$$

Scaling up ratios

In some questions you may know the simplest form of a ratio and need to scale it up in order to work out the actual number of parts. To do work out how much you need to multiply the values in the ratio by and multiply each by that same amount.

Example 2

John is making a bench. He needs pieces of wood, screws and nails in a ratio of $$2:7:9$$. If John needs eight pieces of wood to make his bench, how many screws and nails does he need?

Work out how much you need to multiply by. To do this divide the actual number of pieces of wood by the number in the ratio.

$$8\div2=4$$​ 

Multiply the rest of the parts of the ratio by four.

$$7\times4=28\\9\times4=36$$​​

Write as a ratio of wood : screws : nails. 

$$8:28:36$$​​

​

John needs $$\underline{28}$$ screws and $$\underline{36}$$ nails.

Ratios containing decimals or fractions

Ratios should be made of whole numbers and you may be asked to convert from fraction and decimal ratios to whole number ones. 

PROCEDURE

1. For a decimal: multiply every value in the ratio (usually by ten) until you have whole numbers.
   For a fraction: multiply every value by each denominator until you have whole numbers.
   
2. Simplify the ratio until it is in its simplest form.
   

Example 3

Simplify $$1.3:2.6$$.

​

Multiply the ratio by ten.

$$13:26$$​​

Simplify by finding a common factor: thirteen and divide.

$$\underline{1:2}$$​​

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Finding the form n : 1

The final way of expressing ratios is in the form $$n:1$$. To achieve this, simply divide the ratio by one of the values.

Example 4

Express $$8:12$$ in the form $$1:n$$​

Divide both values by eight.

$$\begin {aligned} (8\div8)&:(12\div8)\\\underline{1\enspace}&\underline{ :}\underline{\enspace1.5}\end {aligned}$$​​

Note: In this case it's fine to use decimals in your answer!

Common ratio questions

Some types of ratio question are very common and you will need to know how to deal with them. Given below are some examples.

1. Sharing into parts

This type of question involves dividing and sharing a quantity into a given ratio.

Example 5

Grace and Angela share $$£320$$​ in the ratio of $$3:5$$. How much money did each of them get?

Add up the parts of the ratio.

$$3+5=8$$

Divide the total by the total number of parts. This tells you what one 'part' is worth.

$$£320\div8=£40$$

Multiply each part of the ratio by the value of one part.

$$3\times£40=£120\\5\times £40 = £200$$

$$£120:£200$$

Grace gets $$\underline{£120}$$ and Angela gets $$\underline{£200}$$.​

​​

2. Differences between parts

In these questions you will be given a ratio along with the difference between two of the parts, rather than the total quantity.

Example 6

A piece of string is cut into three pieces in a ratio of $$4m:1m:6m$$​. The first piece is fifteen metres longer than the second piece. What was the total length of the string?

Calculate the difference between the first and second values in the ratio.

​$$4-1=3$$

Divide the difference given by this difference to work out the value of one 'part'.

​$$15\div3=5m$$​​

Add up the values in the ratio and multiply by the value of one part.

$$4m+1m+6m=11m\\11m\times5=55m$$​​

The total length of the string was $$\underline{55m}$$.