Direct and Inverse Proportion

What you'll learn

How to turn proportion statements into equations using $k$ .
How to solve direct and inverse proportion problems.
How squares, cubes and square roots change the formula.
How to recognise proportion from graphs and tables.

The language of proportion

A variable is a letter that stands for a number that can change. A formula is an equation linking variables.

The symbol $∝\propto$ means “is proportional to”. In GCSE questions, you usually replace $∝\propto$ with an equals sign and a constant.

Definition

Constant of proportionality

The constant of proportionality is the fixed number, usually called $k$ , that connects the two variables in a proportion formula.

The constant k is the fixed multiplier or divider that links two changing variables.

Example

Changing words into equations

If $A$ is directly proportional to the square of $B$ , write:
$A = kB^2$
If $P$ is inversely proportional to the cube of $Q$ , write:
$\frac{k}{Q^3}$

Key Idea

The main method

For almost every proportion question: write the formula with $k$ , use the given pair of values to find $k$ , then substitute the new value and solve.

Direct proportion

Definition

Direct proportion

If $y$ is directly proportional to $x$ , then $\propto x$ , so $y = k x$ . As $x$ increases, $y$ increases by the same scale factor.

For example, if one variable doubles, the other doubles too. If one variable is multiplied by 5, the other is multiplied by 5.

Example

Direct proportion with a missing value

$p$ is directly proportional to $q$ . When $p = 8$ , $q = 20$ . Find $q$ when $p = 14$ .

In direct proportion, the ratio between the two variables stays the same as both values scale up or down.

Write the direct proportion formula:
$p = k q$
Substitute $p = 8$ and $q = 20$ :
$8 = 20 k$
Solve for $k$ :
$\frac{8}{20} = \frac{2}{5}$
Use $p = 14$ in the formula:
$\frac{2}{5}q$
Rearrange to find $q$ :
$\div \frac{2}{5} = 35$

Common Mistake

Swapping the variables

If the question says “ $p$ is directly proportional to $q$ ”, start with $p = k q$ . Keep the first variable on the left.

Inverse proportion

Definition

Inverse proportion

If $y$ is inversely proportional to $x$ , then $\propto \frac{1}{x}$ , so $\frac{k}{x}$ . As $x$ increases, $y$ decreases.

A useful fact is that for $\frac{k}{x}$ , the product $x y$ stays constant.

Example

Inverse proportion

$m$ is inversely proportional to $n$ . When $m = 18$ , $n = 5$ . Find $m$ when $n = 15$ .

In inverse proportion, one variable increases while the other decreases so the product stays constant.

Write the inverse proportion formula:
$\frac{k}{n}$
Substitute $m = 18$ and $n = 5$ :
$\frac{k}{5}$
Solve for $k$ :
$\times 5 = 90$
Substitute $n = 15$ :
$\frac{90}{15} = 6$

Tip

Sanity check

In inverse proportion, if the input gets bigger, the output should usually get smaller. If your answer goes the wrong way, check your formula.

Squares, cubes and square roots

A square means “to the power of 2”, so $x^2$ means $\times x$ .

A cube means “to the power of 3”, so $x^3$ means $\times x \times x$ .

A square root is the opposite of squaring. For example, $25=5\sqrt{25}=5$ .

Common formula patterns:

$y$ directly proportional to $x^2$ means $y = kx^2$ .
$y$ directly proportional to $x\sqrt{x}$ means $k\sqrt{x}$ .
$y$ inversely proportional to $x^3$ means $\frac{k}{x^3}$ .
$y$ inversely proportional to $x\sqrt{x}$ means $\frac{k}{\sqrt{x}}$ .

Example

Direct proportion to a square root

$r$ is directly proportional to the square root of $s$ . When $r = 10$ , $s = 25$ . Find $s$ when $r = 6$ .

$For square-root proportion, the formula links r to 1m\sqrt{s}0m, not directly to s.$

Write the formula:
$k\sqrt{s}$
Substitute $r = 10$ and $s = 25$ :
$k\sqrt{25}$
Since $25=5\sqrt{25}=5$ , find $k$ :
$\Rightarrow k = 2$
Substitute $r = 6$ :
$2\sqrt{s}$
Divide by 2, then square both sides:
$s=3⇒s=9\sqrt{s}=3 \Rightarrow s=9$

Example

Inverse proportion to a cube

$t$ is inversely proportional to the cube of $u$ . When $t = 48$ , $u = 0.5$ . Find $t$ when $u = 2$ .

For inverse proportion to a cube, the cube u^3 goes in the denominator.

Write the formula:
$\frac{k}{u^3}$
Substitute $t = 48$ and $u = 0.5$ :
$\frac{k}{0.5^3}$
Find $k$ :
$\times 0.5^3 = 6$
Substitute $u = 2$ :
$\frac{6}{2^3} = \frac{6}{8} = 0.75$

Common Mistake

Square roots

The symbol $s\sqrt{s}$ means the positive square root. If $s=3\sqrt{s}=3$ , then $s = 9$ , not $s=±3s=\pm 3$ .

Recognising graphs

The origin is the point (0, 0), where the x-axis and y-axis meet.

Graph shapes to remember:

The main proportion types have distinctive graph shapes.

$\propto x$ : a straight line through the origin.
$\propto x^2$ : a parabola, meaning a U-shaped curve, with its vertex at the origin.
$\propto \frac{1}{x}$ : two curved branches, usually in the top-right and bottom-left for positive $k$ .
$\propto \frac{1}{x^2}$ : two curved branches above the x-axis for positive $k$ , one on each side of the y-axis.

Example

Matching graph descriptions

Match each description to a proportion statement.

A straight line passing through the origin matches:
$\propto x$
A U-shaped curve with its lowest point at the origin matches:
$\propto x^2$
Two opposite curved branches in the top-right and bottom-left match:
$\propto \frac{1}{x}$
Two curved branches above the x-axis, symmetric about the y-axis, match:
$\propto \frac{1}{x^2}$

Common Mistake

Straight line trap

A direct proportion graph must be a straight line through the origin. A straight line that misses the origin is not direct proportion.

A straight line only represents direct proportion if it passes through the origin.

Using tables to find the formula

For table questions, test which expression gives the same value of $k$ each time.

If the options are $\propto x$ , $\propto x^2$ and $\propto x^3$ , check:

$\div x$
$\div x^2$
$\div x^3$

The correct one gives the same answer for every pair.

Example

Choosing from a table

Two pairs of values are: when $x = 2$ , $y = 24$ ; when $x = 3$ , $y = 81$ . Decide whether $\propto x$ , $\propto x^2$ or $\propto x^3$ , then write the formula.

$Testing y \div x^n in a table shows which power gives a constant value of k.$

Test $\propto x$ . The constants do not match:
$242=12,813=27\frac{24}{2}=12,\qquad \frac{81}{3}=27$
Test $\propto x^2$ . The constants do not match:
$2422=6,8132=9\frac{24}{2^2}=6,\qquad \frac{81}{3^2}=9$
Test $\propto x^3$ . The constants match:
$2423=3,8133=3\frac{24}{2^3}=3,\qquad \frac{81}{3^3}=3$
Therefore $k = 3$ , so the formula is:
$y = 3x^3$

Exam technique

In the exam

Write the proportion formula first, including $k$ .
Use the given pair of values to find $k$ before using the new value.
For table questions, test each possible power and look for the constant result.

Self review

Check yourself

Can you explain the difference between $y = k x$ and $\frac{k}{x}$ ?
What graph shape tells you that $\propto x^2$ ?
If $y$ is inversely proportional to $x^3$ , where does $x^3$ go in the formula?

Direct and Inverse Proportion

What you'll learn

How to turn proportion statements into equations using $k$ .
How to solve direct and inverse proportion problems.
How squares, cubes and square roots change the formula.
How to recognise proportion from graphs and tables.

The language of proportion

A variable is a letter that stands for a number that can change. A formula is an equation linking variables.

The symbol $∝\propto$ means “is proportional to”. In GCSE questions, you usually replace $∝\propto$ with an equals sign and a constant.

Definition

Constant of proportionality

The constant of proportionality is the fixed number, usually called $k$ , that connects the two variables in a proportion formula.

The constant k is the fixed multiplier or divider that links two changing variables.

Example

Changing words into equations

If $A$ is directly proportional to the square of $B$ , write:
$A = kB^2$
If $P$ is inversely proportional to the cube of $Q$ , write:
$\frac{k}{Q^3}$

Key Idea

The main method

For almost every proportion question: write the formula with $k$ , use the given pair of values to find $k$ , then substitute the new value and solve.

Direct proportion

Definition

Direct proportion

If $y$ is directly proportional to $x$ , then $\propto x$ , so $y = k x$ . As $x$ increases, $y$ increases by the same scale factor.

For example, if one variable doubles, the other doubles too. If one variable is multiplied by 5, the other is multiplied by 5.

Example

Direct proportion with a missing value

$p$ is directly proportional to $q$ . When $p = 8$ , $q = 20$ . Find $q$ when $p = 14$ .

In direct proportion, the ratio between the two variables stays the same as both values scale up or down.

Write the direct proportion formula:
$p = k q$
Substitute $p = 8$ and $q = 20$ :
$8 = 20 k$
Solve for $k$ :
$\frac{8}{20} = \frac{2}{5}$
Use $p = 14$ in the formula:
$\frac{2}{5}q$
Rearrange to find $q$ :
$\div \frac{2}{5} = 35$

Common Mistake

Swapping the variables

If the question says “ $p$ is directly proportional to $q$ ”, start with $p = k q$ . Keep the first variable on the left.

Inverse proportion

Definition

Inverse proportion

If $y$ is inversely proportional to $x$ , then $\propto \frac{1}{x}$ , so $\frac{k}{x}$ . As $x$ increases, $y$ decreases.

A useful fact is that for $\frac{k}{x}$ , the product $x y$ stays constant.

Example

Inverse proportion

$m$ is inversely proportional to $n$ . When $m = 18$ , $n = 5$ . Find $m$ when $n = 15$ .

In inverse proportion, one variable increases while the other decreases so the product stays constant.

Write the inverse proportion formula:
$\frac{k}{n}$
Substitute $m = 18$ and $n = 5$ :
$\frac{k}{5}$
Solve for $k$ :
$\times 5 = 90$
Substitute $n = 15$ :
$\frac{90}{15} = 6$

Tip

Sanity check

In inverse proportion, if the input gets bigger, the output should usually get smaller. If your answer goes the wrong way, check your formula.

Squares, cubes and square roots

A square means “to the power of 2”, so $x^2$ means $\times x$ .

A cube means “to the power of 3”, so $x^3$ means $\times x \times x$ .

A square root is the opposite of squaring. For example, $25=5\sqrt{25}=5$ .

Common formula patterns:

$y$ directly proportional to $x^2$ means $y = kx^2$ .
$y$ directly proportional to $x\sqrt{x}$ means $k\sqrt{x}$ .
$y$ inversely proportional to $x^3$ means $\frac{k}{x^3}$ .
$y$ inversely proportional to $x\sqrt{x}$ means $\frac{k}{\sqrt{x}}$ .

Example

Direct proportion to a square root

$r$ is directly proportional to the square root of $s$ . When $r = 10$ , $s = 25$ . Find $s$ when $r = 6$ .

$For square-root proportion, the formula links r to 1m\sqrt{s}0m, not directly to s.$

Write the formula:
$k\sqrt{s}$
Substitute $r = 10$ and $s = 25$ :
$k\sqrt{25}$
Since $25=5\sqrt{25}=5$ , find $k$ :
$\Rightarrow k = 2$
Substitute $r = 6$ :
$2\sqrt{s}$
Divide by 2, then square both sides:
$s=3⇒s=9\sqrt{s}=3 \Rightarrow s=9$

Example

Inverse proportion to a cube

$t$ is inversely proportional to the cube of $u$ . When $t = 48$ , $u = 0.5$ . Find $t$ when $u = 2$ .

For inverse proportion to a cube, the cube u^3 goes in the denominator.

Write the formula:
$\frac{k}{u^3}$
Substitute $t = 48$ and $u = 0.5$ :
$\frac{k}{0.5^3}$
Find $k$ :
$\times 0.5^3 = 6$
Substitute $u = 2$ :
$\frac{6}{2^3} = \frac{6}{8} = 0.75$

Common Mistake

Square roots

The symbol $s\sqrt{s}$ means the positive square root. If $s=3\sqrt{s}=3$ , then $s = 9$ , not $s=±3s=\pm 3$ .

Recognising graphs

The origin is the point (0, 0), where the x-axis and y-axis meet.

Graph shapes to remember:

The main proportion types have distinctive graph shapes.

$\propto x$ : a straight line through the origin.
$\propto x^2$ : a parabola, meaning a U-shaped curve, with its vertex at the origin.
$\propto \frac{1}{x}$ : two curved branches, usually in the top-right and bottom-left for positive $k$ .
$\propto \frac{1}{x^2}$ : two curved branches above the x-axis for positive $k$ , one on each side of the y-axis.

Example

Matching graph descriptions

Match each description to a proportion statement.

A straight line passing through the origin matches:
$\propto x$
A U-shaped curve with its lowest point at the origin matches:
$\propto x^2$
Two opposite curved branches in the top-right and bottom-left match:
$\propto \frac{1}{x}$
Two curved branches above the x-axis, symmetric about the y-axis, match:
$\propto \frac{1}{x^2}$

Common Mistake

Straight line trap

A direct proportion graph must be a straight line through the origin. A straight line that misses the origin is not direct proportion.

A straight line only represents direct proportion if it passes through the origin.

Using tables to find the formula

For table questions, test which expression gives the same value of $k$ each time.

If the options are $\propto x$ , $\propto x^2$ and $\propto x^3$ , check:

$\div x$
$\div x^2$
$\div x^3$

The correct one gives the same answer for every pair.

Example

Choosing from a table

Two pairs of values are: when $x = 2$ , $y = 24$ ; when $x = 3$ , $y = 81$ . Decide whether $\propto x$ , $\propto x^2$ or $\propto x^3$ , then write the formula.

$Testing y \div x^n in a table shows which power gives a constant value of k.$

Test $\propto x$ . The constants do not match:
$242=12,813=27\frac{24}{2}=12,\qquad \frac{81}{3}=27$
Test $\propto x^2$ . The constants do not match:
$2422=6,8132=9\frac{24}{2^2}=6,\qquad \frac{81}{3^2}=9$
Test $\propto x^3$ . The constants match:
$2423=3,8133=3\frac{24}{2^3}=3,\qquad \frac{81}{3^3}=3$
Therefore $k = 3$ , so the formula is:
$y = 3x^3$

Exam technique

In the exam

Write the proportion formula first, including $k$ .
Use the given pair of values to find $k$ before using the new value.
For table questions, test each possible power and look for the constant result.

Self review

Check yourself

Can you explain the difference between $y = k x$ and $\frac{k}{x}$ ?
What graph shape tells you that $\propto x^2$ ?
If $y$ is inversely proportional to $x^3$ , where does $x^3$ go in the formula?

Direct and Inverse Proportion

What you'll learn

The language of proportion

Direct proportion

Inverse proportion

Squares, cubes and square roots

Recognising graphs

Using tables to find the formula

Recap questions

You've reached the end

Direct and Inverse Proportion

What you'll learn

The language of proportion

Direct proportion

Inverse proportion

Squares, cubes and square roots

Recognising graphs

Using tables to find the formula

Recap questions

You've reached the end

Direct and Inverse Proportion

What you'll learn

The language of proportion

Direct proportion

Inverse proportion

Squares, cubes and square roots

Recognising graphs

Using tables to find the formula

Recap questions

You've reached the end

Direct and Inverse Proportion Revision Guide

Direct and Inverse Proportion

What you'll learn

The language of proportion

Direct proportion

Inverse proportion

Squares, cubes and square roots

Recognising graphs

Using tables to find the formula

Recap questions

You've reached the end

Direct and Inverse Proportion Revision Guide