Proportion

In a nutshell

Proportion is a way of describing a relationship between two quantities. In particular, two quantities in direct proportion will always increase with each other. But, if two quantities are in inverse proportion to each other, one will only increase if the other decreases.

Direct proportion

Definition

If two quantities $$y$$​ and $$x$$​ are in direct proportion with one another, then $$y$$ is directly proportional to $$x$$. ​Other ways of writing this are:

• ​$$y\propto x$$
• ​$$y=kx$$​, where $$k$$​ is called the constant of proportionality.

This also means that if $$y\propto x$$, then the graph of $$x$$​ and $$y$$​ will be a straight line through the origin.

The key property of direct proportion

The key property of direct proportion is that two quantities in direct proportion will always increase and decrease by the same factor. So, if one is doubled, then the other is doubled. And if one is divided by $$4$$​, the other is also divided by $$4$$​.

Direct proportion questions

To solve direct proportion questions, it is easiest to approach the simpler ones just by using the key property. However, harder proportion questions may require a more algebraic approach.

Example 1

If it costs $$£30$$​ an hour to employ $$4$$​ people to perform a job, how much would it cost to employ $$10$$​ people?

This question is using direct proportion because employing more people requires more money - the two quantities are increasing with each other. Hence, the number of employees is directly proportional to the cost. To find the cost for 10 employees, use the key property of direct proportion.

It is given that $$4$$​ people = $$£30.$$​

Divide both sides by $$4$$​ to find the cost of $$1$$​ employee:

$$1$$​ person = $$£30\div4=£7.50$$

Multiply both sides by $$10$$​ to get the cost of employing $$10$$​ people:

$$10$$​ people = $$£7.50 \times 10$$

$$10$$​ people = $$£75$$​

It will cost $$\underline{£75}$$ to employ $$10$$​ people.

Inverse proportion

Definition

If two quantities $$y$$ and $$x$$​ are in inverse proportion with one another, then $$y$$ is inversely proportional to $$x$$. Other ways of writing this are:

• $$y\propto \frac{1}{x}$$
  
• $$y=\frac{k}{x}$$, where $$k$$ is the constant of proportionality.

If $$y$$ and $$x$$​ are inversely proportional to each other, then their graph is going to look like this:

This is called a reciprocal graph, it is the graph of $$y=\frac{k}{x}$$​.

The key property of inverse proportion

The key property of two quantities in inverse proportion is that if one quantity increases by a factor, then the other quantity decreases by the same factor. So, if one quantity is doubled, the other is halved. If one quantity is multiplied by 3, the other is divided by 3.

Inverse proportion questions

To know whether or not you need to use inverse proportion, think about whether or not it makes sense for the two quantities to be inversely proportional. Will doubling one quantity halve the other? If so, then it's likely that you have to use inverse proportion.

Example 2

It takes $$4$$​ builders $$15$$​ weeks to build a house.

i) How long would it take for $$12$$​ builders to build the same house?

ii) Find a formula for the number of builders, $$b$$, in terms of the time taken in weeks, $$t$$.

i)

The more builders there are, the less time it would take to build the house. Hence, it can be assumed that the number of builders is inversely proportional to the time taken to build the house. So, use the key property to find the time taken for $$12$$​ builders.

$$4$$​ builders = $$15$$​ weeks

Divide the number of builders by $$4$$​ to find our the time for $$1$$​ builder. Due to the inverse relation, you must multiply the time by $$4$$​:

$$1$$​ builder = $$15\times4=60$$ weeks

Multiply the number of builders by $$12$$​ to find the time for $$12$$​ builders. Again, this means divide the time by $$12$$​:

$$12$$​ builders = $$60\div12=5$$ weeks

​

It will take $$12$$​ builders $$\underline{5\ \text{weeks}}$$​ to build the same house.

ii)

The values of $$b$$ and $$t$$ are inversely proportional to one another.  This gives a formula that involves the two quantities:

​$$b=\frac{k}{t}$$ $$(\star)$$​​

To find the value of $$k$$​, substitute a known value of $$b$$ and $$t$$. When $$b = 4, t = 15$$. Hence:

$$\begin{aligned}4&=\frac{k}{15}\\k&=4\times15=60\end{aligned}$$

Put this value of $$k$$​ into the original equation ($$\star$$) to give a full formula for $$b$$:

​$$\underline{b=\frac{60}{t}}$$