Direct Proportion

Exam code: 4024

Written by: Ashika|Reviewed by: Caroline Carroll|Updated 2 July 2026

Direct Proportion

Direct proportion

Proportion is a way of talking about how two variables are related to each other
Direct proportion means that as one variable goes up the other goes up by the same factor
- The ratio between the two amounts will always stay the same
The symbol $\propto$ means "proportional to"
- E.g. y is directly proportional to x, y $\propto$ x
If x and y are directly proportional, then
- x : y will always be the same
- there will be some value, k, such that y = kx
- the graph relating x and y is a linear graph, with gradient k
k is called the constant of proportionality

Problems may involve a variable being directly proportional to a power or root of another variable
For example
- y is directly proportional to the square of x
  - $y \propto x^{2}$
  - means that $y = k x^{2}$
- y is directly proportional to the square root of x
  - $y \propto \sqrt{x}$
  - means that $y = k \sqrt{x}$
- y is directly proportional to the cube of x
  - $y \propto x^{3}$
  - means that equation $y = k x^{3}$
- y is directly proportional to the cube root of x
  - $y \propto \sqrt[3]{x}$
  - means that equation $y = k \sqrt[3]{x}$
- Each of these would have a different type of graph, depending on the power or root

Worked example

It is known that $y$ is directly proportional to the square of $x$ .

When $x = 3$ , $y = 18$ .

Find the value of $y$ when $x = 4$ .

Answer:

Identify the two variables

$y, x^{2}$

We are told this is direct proportion
Write down the formula involving k

$y = k x^{2}$

Find k by substituting in y = 18 when x = 3
Then solve the equation for k

$\begin{array}{rcl} 18 & = & k {(3)}^{2} \\ 18 & = & 9 k \\ \frac{18}{9} & = & k \\ 2 & = & k \end{array}$

Substitute this value of k back into the formula to get the full equation

$y = 2 x^{2}$

Use this formula to find y when x = 4

$\begin{array}{rcl} y & = & 2 \times 4^{2} \\ y & = & 2 \times 16 \\ y & = & 32 \end{array}$

$y = 32$