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.
$\underline{d=\frac{72}{t}}$