1MA1 · R10, R13, R16Ratio, Proportion & Rates of ChangeFoundation & HigherGrades 5–8

Direct & Inverse Proportion

Q: What does the symbol ∝ mean?

"Is proportional to". Replace it with "= k " to form an equation.

If 4 pens cost £3.20, then 10 pens cost £8 — doubling the pens doubles the cost. That is $\textbf{direct proportion}$ . If 4 workers take 12 days to build a wall, then 6 workers take 8 days — more workers, less time. That is $\textbf{inverse proportion}$ . At Higher tier, both extend to squares, cubes and roots, and are written as equations with a constant $k$ .

50 min Ratio & Proportion

The big picture

Direct proportion means one quantity is a constant multiple of the other: $y = kx$ . Inverse proportion means their product is constant: $y = \dfrac{k}{x}$ . Recognising which applies, then finding $k$ from one pair of values, turns any proportion question into substitution.

Edexcel Higher questions often start " $y$ is inversely proportional to the square of $x$ ", and the marks go to writing the correct equation, finding $k$ , and using it. Graphs also appear: direct proportion is a straight line through the origin; inverse proportion is a reciprocal curve.

What you'll learn

Solve direct and inverse proportion problems using unitary and multiplier methods
Recognise direct and inverse proportion from tables and graphs
Form equations such as $y = kx^{2}$ and $y = \dfrac{k}{\sqrt{x}}$ and use them (Higher)
Describe the effect of changing one variable on another

Direct proportion

Two quantities are in direct proportion if multiplying one by a number multiplies the other by the same number. Their ratio stays constant.

The unitary method finds the value for one unit: if 4 pens cost £3.20, one costs 80p, so 10 cost £8.

As an equation, $y \propto x$ means $y = kx$ . Its graph is a straight line through the origin with gradient $k$ .

y \propto x \iff y = kx

Direct proportion.

Your turnSame method, new numbers — try it before moving on.

5 kg of potatoes cost £12.50. How much do 8 kg cost, in pounds?

Your answer

Show worked solution

Solution

11 kg costs $12.50 \div 5 = £2.50$ .
28 kg cost $8 \times 2.50 = £20$ .

Answer: £20

Inverse proportion

Two quantities are inversely proportional if multiplying one by a number divides the other by the same number. Their product stays constant.

If 4 workers take 12 days, the job is $4 \times 12 = 48$ worker-days, so 6 workers take $48 \div 6 = 8$ days.

As an equation, $y \propto \dfrac{1}{x}$ means $y = \dfrac{k}{x}$ . Its graph is a reciprocal curve.

y \propto \tfrac{1}{x} \iff y = \tfrac{k}{x}

Inverse proportion.

Your turnSame method, new numbers — try it before moving on.

6 workers take 10 days to build a wall. How many days would 4 workers take?

Your answer

Show worked solution

Solution

1Total $= 6 \times 10 = 60$ worker-days.
2 $60 \div 4 = 15$ days.

Answer: $15$ days

Why this works

Direct proportion keeps the $\textit{ratio}$ constant; inverse proportion keeps the $\textit{product}$ constant. Checking which of $\tfrac{y}{x}$ or $xy$ stays the same in a table identifies the relationship immediately.

Proportion with powers and roots

" $y$ is directly proportional to the square of $x$ " means $y = kx^{2}$ . " $y$ is inversely proportional to the square root of $x$ " means $y = \dfrac{k}{\sqrt{x}}$ .

The method is always: write the equation with $k$ , substitute the given pair to find $k$ , write the equation with the value of $k$ , then use it.

Example 1

y

is directly proportional to

x^{2}

. When

x = 3

y = 18

. Find

y

when

x = 5

y = kx^{2}

, so

18 = 9k

and

k = 2

y = 2x^{2}

3When

x = 5

y = 2 \times 25

Answer

y = 50

Your turnSame method, new numbers — try it before moving on.

$y$ is directly proportional to $x^3$ . When $x = 2$ , $y = 24$ . Find $y$ when $x = 3$ .

Your answer

Show worked solution

Solution

1 $24 = k \times 8$ , so $k = 3$ .
2 $y = 3 \times 27 = 81$ .

Answer: $y = 81$

Example 2

y

is inversely proportional to

\sqrt{x}

. When

x = 4

y = 6

. Find

y

when

x = 9

y = \dfrac{k}{\sqrt{x}}

, so

6 = \dfrac{k}{2}

and

k = 12

2When

x = 9

y = \dfrac{12}{3}

Answer

y = 4

Tip — Write the equation with $k$ found as a separate line, e.g. " $y = 2x^{2}$ ". Mark schemes award a mark for it.

The effect of a change

If $y = kx^{2}$ and $x$ is doubled, $y$ is multiplied by $2^{2} = 4$ . If $y = \dfrac{k}{x^{2}}$ and $x$ is doubled, $y$ is divided by 4.

For a percentage change, use multipliers: if $x$ increases by 20% in $y = kx^{3}$ , $y$ is multiplied by $1.2^{3} = 1.728$ , a 72.8% increase.

Example 1The force

F

between two magnets is inversely proportional to the square of the distance

d

between them. The distance increases by 20%. By what percentage does the force change?

F = \dfrac{k}{d^{2}}

; new distance

1.2d

2New force

= \dfrac{k}{1.44d^{2}} = 0.694F

AnswerThe force decreases by about 30.6%.

Your turnSame method, new numbers — try it before moving on.

$y$ is directly proportional to $x^2$ . $x$ is increased by 20%. By what percentage does $y$ increase?

Your answer

Show worked solution

Solution

1The multiplier for $x$ is $1.2$ .
2 $y$ is multiplied by $1.2^2 = 1.44$ : a 44% increase.

Answer: $44\%$

Think like an examiner

Write the proportionality equation with $k$ before substituting.
Show the value of $k$ and the equation with $k$ replaced.
Use multipliers for percentage changes and explain the conclusion.

Proportion

y = kx

Direct proportion.

y = \tfrac{k}{x}

Inverse proportion.

y = kx^{n},\ y = \tfrac{k}{x^{n}}

With powers or roots.

Watch out for these

Writing "

y

inversely proportional to

x^{2}

" as

y = kx^{2}

y = \dfrac{k}{x^{2}}

Assuming

y

doubles when

x

doubles in

y = kx^{2}

It quadruples.

Using

\tfrac{y}{x}

to test inverse proportion.

For inverse proportion,

xy

is constant.

Forgetting to square

x

when finding

k

Substitute into the full equation:

18 = k \times 3^{2}

Stretch yourself

The time $t$ for a pendulum to swing is directly proportional to the square root of its length $l$ . A pendulum of length 64 cm takes 1.6 s. Find the time for a pendulum of length 1 m, and the length needed for a time of 2.4 s.

Hint — Use consistent units. Form $t = k\sqrt{l}$ with $l$ in cm.

Reveal solution

Solution

t = k\sqrt{l}

, so

1.6 = 8k

and

k = 0.2

, giving

t = 0.2\sqrt{l}

. For

l = 100

cm:

t = 0.2 \times 10 = 2

s. For

t = 2.4

\sqrt{l} = 12

, so

l = 144

cm. Check with scale factors: the time is multiplied by

\tfrac{2.4}{1.6} = 1.5

, so the length is multiplied by

1.5^{2} = 2.25

, and

64 \times 2.25 = 144

✓.

\blacksquare

Questions students ask

How do I tell direct from inverse proportion?

Direct: both increase together and the ratio is constant. Inverse: one increases as the other decreases and the product is constant.

What does the symbol ∝ mean?

"Is proportional to". Replace it with "

= k \times

" to form an equation.

Is every straight-line graph direct proportion?

No — only straight lines through the origin.

Key takeaways

Direct proportion: $y = kx$ , constant ratio, straight line through the origin.
Inverse proportion: $y = \tfrac{k}{x}$ , constant product, reciprocal curve.
With powers and roots, write the equation, find $k$ , then substitute.
Doubling $x$ in $y = kx^{n}$ multiplies $y$ by $2^{n}$ .
Use multipliers to find the effect of percentage changes.

Exam-style questions

Practise it the way it is examined

3 original questions · 8 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 53 marks

$d$ is directly proportional to $t^2$ . When $t = 2$ , $d = 20$ .

(a)Find a formula for $d$ in terms of $t$ .[2]
Show mark scheme (a)
- M1 $20 = 4k$
- A1 $d = 5t^2$
(b)Find $d$ when $t = 5$ .[1]
Show mark scheme (b)
- B1 $125$
Answer: $125$

Question 2Grade 72 marks

$P$ is inversely proportional to $V$ . When $V = 40$ , $P = 150$ . Find $V$ when $P = 250$ .[2]
Show mark scheme
- M1 $k = 6000$
- A1 $V = 24$
Answer: $V = 24$

Question 3Grade 83 marks

$y$ is inversely proportional to $x^2$ . $x$ is increased by 25%. Find the percentage decrease in $y$ .[3]
Show mark scheme
- M1 $y$ is multiplied by $\frac{1}{1.25^2}$
- M1 $= 0.64$
- A1 $36\%$ decrease
Answer: $36\%$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 5 lessons