Direct and Inverse Proportion

What you'll learn

How to turn proportionality statements into equations using a constant.
How to recognise direct and inverse proportion from formulas, graphs, and tables.
How to handle powers, cubes, and square roots in proportion questions.
How to avoid the common exam traps when solving for a new value.

The big idea: one quantity changes with another

In proportion questions, two variables are linked by a multiplier. A variable is a letter that can stand for different values, such as $x$ , $y$ , $a$ , or $b$ .

The key skill is to replace a sentence like “ $y$ is directly proportional to $x$ ” with an equation involving a constant.

Definition

Constant of proportionality

The constant of proportionality is the fixed number that connects two proportional variables. It is usually called $k$ .

For example, if $y$ is directly proportional to $x$ , then $y = k x$ .

Key Idea

The proportion method

Almost every proportion question follows the same pattern:

Write the proportionality as an equation using $k$ .
Use the given pair of values to find $k$ .
Substitute the new value and solve.

Direct proportion

If two quantities are in direct proportion, they increase or decrease together by the same scale factor.

For example, if the number of identical pens doubles, the total cost doubles. If the number of pens triples, the total cost triples.

Definition

Direct proportion

$y$ is directly proportional to $x$ means:

y = k x

where $k$ is a constant.

Worked example: direct proportion

Suppose $p$ is directly proportional to $q$ .

When $p = 9$ , $q = 36$ .

Find $q$ when $p = 4$ .

Example

Finding a missing value in direct proportion

Write the direct proportion as an equation.
$p = k q$
Substitute the given values to find $k$ .
$9 = 36 k$
Solve for $k$ .
$\frac{9}{36} = \frac{1}{4}$
Use the equation with $p = 4$ .
$\frac{1}{4}q$
Solve for $q$ .
$q = 16$

Common Mistake

Mixing up the variables

If the question says “ $p$ is directly proportional to $q$ ”, write $p = k q$ , not $q = k p$ . Sometimes both can still be rearranged, but using the wording correctly makes your working much safer.

Inverse proportion

If two quantities are in inverse proportion, one increases while the other decreases in a linked way.

For example, if more identical workers share a job, the time taken might decrease. Doubling one quantity halves the other.

Definition

Inverse proportion

$y$ is inversely proportional to $x$ means:

\frac{k}{x}

Equivalently, $x y = k$ .

Worked example: inverse proportion

Suppose $m$ is inversely proportional to $n$ .

When $m = 6$ , $n = 10$ .

Find $m$ when $n = 4$ .

Example

Finding a missing value in inverse proportion

Write the inverse proportion as an equation.
$\frac{k}{n}$
Substitute the given values to find $k$ .
$\frac{k}{10}$
Solve for $k$ .
$k = 60$
Substitute $n = 4$ .
$\frac{60}{4}$
Calculate $m$ .
$m = 15$

Tip

A quick check for inverse proportion

For inverse proportion, the product stays constant. In the example above, $mn = 60$ each time.

Proportion with powers and roots

IGCSE questions often go beyond plain $x$ . You may see phrases involving squares, cubes, or square roots.

Directly proportional to a power

If $y$ is directly proportional to the square of $x$ , then:

y = kx^2

If $y$ is directly proportional to the cube of $x$ , then:

y = kx^3

Inversely proportional to a power

If $y$ is inversely proportional to the square of $x$ , then:

\frac{k}{x^2}

If $y$ is inversely proportional to the cube of $x$ , then:

\frac{k}{x^3}

Proportional to a square root

If $y$ is directly proportional to the square root of $x$ , then:

k\sqrt{x}

If $y$ is inversely proportional to the square root of $x$ , then:

\frac{k}{\sqrt{x}}

Common Mistake

Forgetting the square, cube, or square root

The phrase “inversely proportional to the cube of $x$ ” means the cube goes in the denominator: $\frac{k}{x^3}$ . It does not mean $\frac{k^3}{x}$ .

Worked example: square root proportion

Suppose $r$ is directly proportional to the square root of $s$ .

When $r = 20$ , $s = 25$ .

Find $s$ when $r = 8$ .

Example

Direct proportion with a square root

Write the equation.
$k\sqrt{s}$
Substitute the given values.
$k\sqrt{25}$
Since $25=5\sqrt{25} = 5$ , find $k$ .
$\Rightarrow k = 4$
Substitute $r = 8$ .
$4\sqrt{s}$
Divide by 4, then square both sides.
$s=2⇒s=4\sqrt{s} = 2 \Rightarrow s = 4$

Worked example: inverse cube proportion

Suppose $y$ is inversely proportional to the cube of $x$ .

When $y = 160$ , $x = 0.5$ .

Find $y$ when $x = 2$ .

Example

Inverse proportion with a cube

Write the equation.
$\frac{k}{x^3}$
Substitute the given values.
$\frac{k}{0.5^3}$
Calculate $0.5^3$ , then find $k$ .
$\frac{k}{0.125} \Rightarrow k = 20$
Substitute $x = 2$ .
$\frac{20}{2^3}$
Calculate the final value.
$\frac{20}{8} = 2.5$

Recognising proportion graphs

The graph shape can tell you the type of proportion.

$\propto x$ gives a straight line through the origin.
$\propto x^2$ gives a U-shaped parabola with its vertex at the origin.
$\propto \frac{1}{x}$ gives a reciprocal curve in the first and third quadrants.
$\propto \frac{1}{x^2}$ gives two positive branches, symmetric about the $y$ -axis.

Four standard proportionality graphs: direct linear, direct square, inverse reciprocal, and inverse square.

Example

Matching a graph to a proportionality statement

A graph is a straight line passing through the origin with positive gradient. Decide which statement it represents.

A straight line suggests a linear relationship.
Passing through the origin is the key sign of direct proportion.
So the correct statement is:
$\propto x$

Common Mistake

Not every straight line is direct proportion

A straight line only represents $\propto x$ if it passes through the origin. A line such as $y = 2 x + 3$ is linear, but not directly proportional to $x$ .

Using tables to identify the relationship

Sometimes you are given two pairs of values and asked whether $y$ is proportional to $x$ , $x^2$ , or $x^3$ .

The method is to test which ratio stays constant:

For $\propto x$ , check whether $yx\frac{y}{x}$ is constant.
For $\propto x^2$ , check whether $yx2\frac{y}{x^2}$ is constant.
For $\propto x^3$ , check whether $yx3\frac{y}{x^3}$ is constant.

Worked example: choosing the right power

For two values of $x$ , the matching values of $y$ are:

When $x = 2$ , $y = 24$ .
When $x = 3$ , $y = 81$ .

Decide whether $\propto x$ , $\propto x^2$ , or $\propto x^3$ , then write a formula for $y$ .

Example

Testing ratios from a table

Test $\propto x$ by comparing $yx\frac{y}{x}$ .
$242=12,813=27\frac{24}{2} = 12,\qquad \frac{81}{3} = 27$
The ratios are not equal, so it is not $\propto x$ .
Test $\propto x^2$ by comparing $yx2\frac{y}{x^2}$ .
$2422=6,8132=9\frac{24}{2^2} = 6,\qquad \frac{81}{3^2} = 9$
The ratios are not equal, so it is not $\propto x^2$ .
Test $\propto x^3$ by comparing $yx3\frac{y}{x^3}$ .
$2423=3,8133=3\frac{24}{2^3} = 3,\qquad \frac{81}{3^3} = 3$
The constant is 3, so the formula is:
$y = 3x^3$

Tip

Table strategy

If the $x$ values are small, testing $x$ , then $x^2$ , then $x^3$ is usually faster than trying to guess the formula.

Rearranging when the missing value is inside a power

Sometimes the final value you need is the variable being squared, cubed, or square-rooted. Do not panic: solve the equation carefully.

Worked example: direct cube proportion

Suppose $a$ is directly proportional to the cube of $b$ .

When $a = 54$ , $b = 3$ .

Find $b$ when $a = 16$ .

Example

Solving for a variable inside a cube

Write the equation.
$a = kb^3$
Use the first pair of values to find $k$ .
$\times 3^3$
Since $3^3 = 27$ , calculate $k$ .
$\Rightarrow k = 2$
Substitute $a = 16$ .
$16 = 2b^3$
Divide by 2, then cube-root.
$b3=8⇒b=2b^3 = 8 \Rightarrow b = 2$

Exam technique

In the exam

Start by writing the correct formula with $k$ before substituting any numbers.
Use the first pair of values only to find $k$ , then use your formula again for the new value.
Check the direction: in direct proportion, bigger input usually gives bigger output; in inverse proportion, bigger input usually gives smaller output.

Self review

Check yourself

If $y$ is inversely proportional to $x^2$ , what equation should you write first?
How can you tell from a graph that $y$ is directly proportional to $x$ ?
In a table question, what ratio would you test for $\propto x^3$ ?

Direct and Inverse Proportion

What you'll learn

How to turn proportionality statements into equations using a constant.
How to recognise direct and inverse proportion from formulas, graphs, and tables.
How to handle powers, cubes, and square roots in proportion questions.
How to avoid the common exam traps when solving for a new value.

The big idea: one quantity changes with another

In proportion questions, two variables are linked by a multiplier. A variable is a letter that can stand for different values, such as $x$ , $y$ , $a$ , or $b$ .

The key skill is to replace a sentence like “ $y$ is directly proportional to $x$ ” with an equation involving a constant.

Definition

Constant of proportionality

The constant of proportionality is the fixed number that connects two proportional variables. It is usually called $k$ .

For example, if $y$ is directly proportional to $x$ , then $y = k x$ .

Key Idea

The proportion method

Almost every proportion question follows the same pattern:

Write the proportionality as an equation using $k$ .
Use the given pair of values to find $k$ .
Substitute the new value and solve.