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Edexcel IGCSE Maths (4MA1) · Unit 3: Equations & Sequences

Edexcel IGCSE 4MA1 Direct and Inverse Proportion: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

If $y$ is directly proportional to $x$ , then $y = k x$ ; if $y$ is inversely proportional to $x$ , then $y = \frac{k}{x}$ . The same idea works with powers and roots, such as $y = k x^{2}$ or $y = \frac{k}{\sqrt{x}}$ . Every question follows three steps: write the formula with $k$ , use the pair of values you are given to find $k$ , then use the formula.

Unit 3: Equations & Sequences
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Notes
Formulas
Worked examples
Mistakes
FAQ

Direct and Inverse Proportion notes

Direct proportion

$y \propto x$ means $y = k x$ : double $x$ and $y$ doubles, and the graph is a straight line through the origin. If $y = 15$ when $x = 3$ , then $k = 5$ and $y = 5 x$ , so $x = 8$ gives $y = 40$ .

Inverse proportion

$y \propto \frac{1}{x}$ means $y = \frac{k}{x}$ , or $x y = k$ : double $x$ and $y$ halves. If 6 painters take 10 days to paint a building, $k = 60$ , so 4 painters working at the same rate take $\frac{60}{4} = 15$ days.

Powers and roots

Match the wording exactly. " $d$ is proportional to the square of $t$ " is $d = k t^{2}$ ; " $F$ is inversely proportional to the square of $r$ " is $F = \frac{k}{r^{2}}$ ; " $y$ is proportional to the square root of $x$ " is $y = k \sqrt{x}$ . With $d = k t^{2}$ and $d = 20$ when $t = 2$ : $k = 5$ , so $t = 3$ gives $d = 45$ .

Graphs and tables

In a table of direct proportion, $\frac{y}{x}$ is the same every time; in a table of inverse proportion, $x y$ is the same every time. The graph of $y = \frac{k}{x}$ is a reciprocal curve that never meets the axes.

Key formulas

Proportion Learn it: $y \propto x^{n} \Rightarrow y = k x^{n}; y \propto \frac{1}{x^{n}} \Rightarrow y = \frac{k}{x^{n}}$

Learn it = not printed on the 4MA1 Higher formulae sheet, so learn it by heart. The whole 4MA1 formula sheet (free PDF) → · 4MA1 formulas to learn

Direct and Inverse Proportion worked examples

Worked example 1: Direct proportion with a square

$y$ is directly proportional to $x^{2}$ . When $x = 5$ , $y = 50$ . Find $y$ when $x = 8$ .

Solution

$y = k x^{2}$ : $50 = 25 k$ , so $k = 2$ .
$y = 2 \times 64 = 128$ .

3 steps

Tip: Find k first, then write the formula.

Worked example 2: Inverse proportion

$y$ is inversely proportional to $x$ . When $x = 4$ , $y = 6$ . Find $y$ when $x = 3$ .

Solution

$y = \frac{k}{x}$ : $k = 24$ .
$y = \frac{24}{3} = 8$ .

3 steps · 1 to fill in

Tip: As x decreases, y increases.

Worked example 3: Inverse proportion with a square root · 3 marks

$p$ is inversely proportional to $\sqrt{q}$ . When $q = 9$ , $p = 12$ . Find the value of $p$ when $q = 16$ .

Mark scheme

M1 p = k/√q
M1 k = 36
A1 9

Answer: 9

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Direct and Inverse Proportion exam-style questions

Try each question before you open the answer (one has its full worked solution open). The mark schemes use Edexcel-style marks: M for method, A for accuracy (after the method mark), B for an independent result.

Question 1 · easy · 3 marks

$y$ is inversely proportional to $x$ . When $x = 4$ , $y = 6$ . Find the value of $y$ when $x = 3$ .

Show the answer and mark scheme

M1 y = k/x
M1 k = 24
A1 8

Answer: 8

Question 2 · easy · 2 marks

$y$ is inversely proportional to $x$ . $y = 10$ when $x = 2$ . Find the value of $x$ when $y = 4$ .

Show the answer and mark scheme

M1 k = 20 oe
A1 5

Answer: 5

Question 3 · easy · 2 marks

6 workers take 15 days to build a wall. All the workers work at the same rate. How many days would 9 workers take to build the same wall?

Show the answer and mark scheme

M1 6 × 15 = 90 worker-days
A1 10 days

Answer: 10 days

Question 4 · medium · 4 marks

$y$ is inversely proportional to $x^{2}$ . When $x = 4$ , $y = 5$ . (a) Find a formula for $y$ in terms of $x$ . (3) (b) Find the value of $y$ when $x = 10$ . (1)

Full worked solution Free, no sign-in

M1 (a) y = k/x²
M1 5 = k/16
A1 y = 80/x²
B1 (b) 0.8

Answer: (a) $y = \frac{80}{x^{2}}$ (b) 0.8

Every question here has a full worked solution. See them all →

Question 5 · hard · 3 marks

The intensity, $I$ , of light from a lamp is inversely proportional to the square of the distance, $d$ metres, from the lamp. When $d = 2$ , $I = 45$ . Find the distance from the lamp when $I = 5$ .

Show the answer and mark scheme

M1 I = k/d²
M1 k = 180
A1 6 m

Answer: 6 m

Question 6 · very hard · 3 marks

$y$ is inversely proportional to $x^{3}$ . When $x = 3$ , $y = 2$ . Find the value of $x$ when $y = 16$ .

Show the answer and mark scheme

M1 y = k/x³ with k = 54
M1 x³ = 54/16 = 3.375
A1 1.5

Answer: 1.5

Question 7 · very hard · 4 marks

Some corresponding values of $x$ and $y$ are: when $x = 2$ , $y = 12.5$ ; when $x = 5$ , $y = 2$ ; when $x = a$ , $y = 0.5$ . (a) Show that the values are consistent with $y$ being inversely proportional to $x^{2}$ . (2) (b) Find the value of $a$ , given $a > 0$ . (2)

Show the answer and mark scheme

M1 (a) 12.5 × 2² = 50
A1 2 × 5² = 50, same constant so consistent (cso)
M1 (b) 50/a² = 0.5 ⇒ a² = 100
A1 a = 10

Answer: (b) $a = 10$

Common direct and inverse proportion mistakes

Using the wrong power. "Proportional to the square of $x$ " means $y = k x^{2}$ . Writing $y = k x$ gives a different $k$ and the wrong answer.
Treating inverse as take away. Inverse proportion is $y = \frac{k}{x}$ , not "when $x$ goes up by 1, $y$ goes down by 1".
Not writing the formula. Find $k$ , then write the formula again with $k$ as a number, such as $y = \frac{80}{x^{2}}$ , before you use it.
Forgetting the square root. $p \propto \frac{1}{\sqrt{q}}$ means $p = \frac{k}{\sqrt{q}}$ : square root $q$ before you divide, both when you find $k$ and when you use the formula.

Where students lose marks across 4MA1, from the examiner reports →

Direct and Inverse Proportion FAQ

How do I recognise inverse proportion?

As one quantity increases the other decreases, so that their product stays the same, such as the number of workers and the time a job takes. The formula is y = k/x.

Is the proportion formula on the 4MA1 formula sheet?

No. y = kx, y = k/x and the versions with powers are not printed on the 4MA1 Higher formulae sheet, so learn to write them from the words in the question.

Can I use a calculator for direct and inverse proportion in 4MA1?

Yes. A calculator may be used in all four 4MA1 papers, so both Higher papers (1H and 2H) allow one. Most marks are for the method you write down, so show your working.

How long are the 4MA1 Higher papers?

Papers 1H and 2H are 2 hours each and 100 marks each, worth 50% each. Each tier has its own formulae sheet in the exam.

Next steps

Direct and Inverse Proportion revision notes · 4MA1 formula sheet · Taking Cambridge 0580 instead? Cambridge 0580 direct and inverse proportion questions

Every question and worked example here is original, written for IGCSE Math Revision rather than copied from Pearson papers, and every result was re-solved independently before publishing.

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Edexcel IGCSE 4MA1 Direct and Inverse Proportion: exam questions and answers, notes and worked examples

Try it first

Direct and Inverse Proportion notes

Direct proportion

Inverse proportion

Powers and roots

Graphs and tables

Key formulas

Direct and Inverse Proportion worked examples

Worked example 1: Direct proportion with a square

Solution

Worked example 2: Inverse proportion

Solution

Worked example 3: Inverse proportion with a square root · 3 marks

Mark scheme

Direct and Inverse Proportion exam-style questions

Question 1 · easy · 3 marks

Question 2 · easy · 2 marks

Question 3 · easy · 2 marks

Question 4 · medium · 4 marks

Full worked solution Free, no sign-in

Question 5 · hard · 3 marks

Question 6 · very hard · 3 marks

Question 7 · very hard · 4 marks

Common direct and inverse proportion mistakes

Direct and Inverse Proportion FAQ

Next steps

Practise direct and inverse proportion with your progress saved

More Unit 3 topics: Equations & Sequences