Direct and inverse proportion

​​​​In a nutshell

Proportion is a way of describing relationships between two sets of quantities. It can be described as being direct or indirect.

Direct proportion

​​Definition

If two quantities, $$x$$ and $$y$$ are directly proportional, as one increases the other does too. Both increase by the same factor - this means that as one value doubles, so does the other. If one value halves, the other does too. 

Proportion can be written in two ways - both mean exactly the same thing: 

• Using the following symbol: $$\propto$$ which means 'is proportional to'.
• As an equation where k is some constant known as the constant of proportionality.

When $$x$$​ is directly proportional to $$y$$​, this creates a straight line on a graph. See below for some graphed examples.​

	Written using $$\propto$$ symbol​	Written as an equation	Graph representation
​'$$y$$​ is directly proportional to $$x$$​'​​​​	​$$y \propto x$$​​​ y \propto xy \propto x​	​​$$y=kx$$​
​'$$y$$​​ is directly proportional to the square of $$x$$​​'​​​	​​$$y\propto x^2$$​	​​$$y=kx^2$$​​​
​​'$$y$$​​ is directly proportional to the square root of $$x$$​​'​​​	​​$$y \propto \sqrt{x}$$​	​​$$y=k\sqrt{x}$$​

Note: Sometimes, the word 'directly' will be omitted and a question might just say 'is proportional to'. This means the same as 'is directly proportional to'.​

Example 1

Grace buys $$1000g$$​ of flour which is enough to bake $$40$$ cakes. Flour costs $$30p$$ per $$250g$$. How much would Grace need to spend on flour to make $$55$$ cakes?

Work out how much flour is in one cake.

$$1000\div 40 =25g$$​

Work out how much flour is in $$55$$ cakes.

$$55\times25=1375g$$

Work out how many lots of $$250g$$ are in $$1375g$$.

$$1375\div250=5.5$$​​

Multiply $$5.5$$​ by the cost of $$250g$$.

$$30p \times 5.5 = \underline{£1.65}$$

​​

Inverse proportion

​​Definition

​If two quantities, $$x$$​ and $$y$$​ are inversely proportional, as one increases the other decreases. As one increases by a factor, the other decreases by the same factor  - this means that as one value doubles, the other halves. If one value is multiplied by four, the other is divided by four (or multiplied by $$\frac{1}{4}$$​ ).

Just like with direct proportion, inverse proportion can be written in two ways - however this time $$x$$ is replaced by $$\frac{1}{x}$$. Again see the below graphed examples.

​	​Written using $$\propto$$​ symbol​	​Written as an equation​	​Graph representation​
​​'$$y$$​​ is inversely proportional to $$x$$​​'​​​​	​​​$$y \propto \frac{1}{x}$$​ ​	​​​$$y=\frac{k}{x}$$​
​​​'$$y$$​​​ is inversely proportional to the cube of $$x$$​​​'​​​	​​​$$y \propto \frac{1}{x^3}$$​	​​​$$y=\frac{k}{x^3}$$​

Example 2

It takes six decorators twelve weeks to decorate a mansion. How long would it take for $$18$$ decorators to decorate the same mansion?  Find a formula for the number of decorators, d, in terms of the number of weeks, t.

More decorators = less time so we are using inverse proportion to answer this question. 

$$6$$ decorators $$= 12$$ weeks

$$18$$ decorators $$= x$$ weeks

Work out what six has been multiplied by.

$$18\div6=3$$

Divide the weeks by the same number.

$$12\div3=x$$ 

​$$x=\underline4$$ weeks

To find a formula use the equation for inverse proportionality, using the letters from the question.

$$d=\frac{k}{t}$$​​

Substitute in the numbers you know and find the constant, k.

$$\begin{aligned}&&6&=\frac{k}{12}\\\times 12&&&&\times12\\&&72&=k\end{aligned}$$

Substitute k back into the original formula.

$$\underline{d=\frac{72}{t}}$$​​​