IGCSE · 4MA1 · Topic

Direct & Inverse Proportion: past paper questions & worked solutions

Every Direct & Inverse Proportion question from the 4MA1 papers in our bank, 2021–2026, sorted from easiest to hardest. Try each one first, then open the worked solution.

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18questions

69marks in total

33%of papers (18 of 54) include it

0/14/4easy / medium / hard

Questions, easiest first

Level is estimated from marks: 1–2 easy, 3–4 medium, 5+ hard.

1. Jun 2026 4WM2HR · Q16

Medium3 marks

Jun 2026 4WM2HR Question 16 (3 marks): Direct & Inverse Proportion – 16 m is directly proportional to r2. m = 180 when r = 6. Find a formula for m in terms of r

Show worked solution

Write the proportion
\[m=kr^2\]
Find k
\[180=k(6)^2\Rightarrow k=5\]
Formula
\[m=5r^2\]

Final answer

$m=5r^2$

2. Nov 2025 4WM2H · Q17

Medium3 marks

Nov 2025 4WM2H Question 17 (3 marks): Direct & Inverse Proportion – 17 T is inversely proportional to m T = 15 when m = 36 Find a formula for T in terms of m (Total for Question 17 is 3 marks)

Show worked solution

Solve equation
$$T = \\frac{k}{\\sqrt{m}}$$
**Step 2: Substitute the given values to find constant *k*.** $$15 = \\frac{k}{\\sqrt{36}}$$ $$15 = \\frac{k}{6}$$ *k* = 15 × 6 = 90
Rearrange formula
$$T = \\frac{90}{\\sqrt{m}}$$

Final answer

$T = \\frac{90}{\\sqrt{m}}$

3. Nov 2025 P2H · Q17

Medium3 marks

Nov 2025 P2H Question 17 (3 marks): Direct & Inverse Proportion – 17 T is inversely proportional to m T = 15 when m = 36 Find a formula for T in terms of m (Total for Question 17 is 3 marks)

Show worked solution

Find inverse function
$T$ is inversely proportional to $m$, so
\[ T=\frac{k}{m} \]
Use when
Use $T=15$ when $m=36$:
\[ 15=\frac{k}{36} \]
\[ k=540 \]
Find inverse function
Therefore
\[ T=\frac{540}{m} \]

Final answer

$T=\frac{540}{m}$.

4. May 2025 4WM2H · Q18

Medium3 marks

May 2025 4WM2H Question 18 (3 marks): Direct & Inverse Proportion – 18 W is inversely proportional to y2 W = 50 when y = 4 Find a formula for W in terms of y (Total for Question 18 is 3 marks)

Show worked solution

Find inverse function
Since $W$ is inversely proportional to $y^2$,
\[ W=\frac{k}{y^2} \]
Evaluate fraction
Using $W=50$ when $y=4$:
\[ 50=\frac{k}{16} \]
\[ k=800 \]

Final answer

$W=\dfrac{800}{y^2}$.

5. Nov 2024 P1H · Q14

Medium3 marks

Nov 2024 P1H Question 14 (3 marks): Direct & Inverse Proportion – 14 B is inversely proportional to the square of d B = 0 25 . when d = 12 Find a formula for B in terms of d (Total for Question 14 is 3 marks)

Show worked solution

Find inverse function
$B$ is inversely proportional to $d^2$, so
\[ B=\frac{k}{d^2} \]
Use when
Use $B=0.25$ when $d=12$:
\[ 0.25=\frac{k}{12^2} \]
\[ k=0.25\times 144=36 \]
Find inverse function
Therefore
\[ B=\frac{36}{d^2} \]

Final answer

$B=\frac{36}{d^2}$.

6. Jun 2024 P2H · Q17

Medium3 marks

Jun 2024 P2H Question 17 (3 marks): Direct & Inverse Proportion – 17 Q is directly proportional to the square root of d Q = 4.5 when d = 324 Find a formula for Q in terms of d (Total for Question 17 is 3 marks)

Show worked solution

Find inverse function
$Q$ is directly proportional to $\sqrt{d}$, so
\[ Q=k\sqrt{d} \]
Use when
Use $Q=4.5$ when $d=324$:
\[ 4.5=k\sqrt{324} \]
\[ 4.5=18k \]
\[ k=0.25 \]
Find inverse function
Therefore
\[ Q=\frac{1}{4}\sqrt{d} \]

Final answer

$Q=\frac{1}{4}\sqrt{d}$.

7. Jan 2022 P1H · Q15

Medium3 marks

Jan 2022 P1H Question 15 (3 marks): Direct & Inverse Proportion – 15 A is inversely proportional to C2 A = 40 when C = 1.5 Calculate the value of C when A = 1000 C = (Total for Question 15 is 3 marks)

Show worked solution

Find inverse function
$A$ is inversely proportional to $C^2$, so
\[ A=\frac{k}{C^2} \]
Use when
Use $A=40$ when $C=1.5$:
\[ 40=\frac{k}{1.5^2} \]
\[ k=40\times 1.5^2=90 \]
Find inverse function
So
\[ A=\frac{90}{C^2} \]
Find inverse function
When $A=1000$,
\[ 1000=\frac{90}{C^2} \]
\[ C^2=0.09 \]
\[ C=0.3 \]

Final answer

$C=0.3$.

8. May-Nov 2020 P1H · Q14

Medium3 marks

May-Nov 2020 P1H Question 14 (3 marks): Direct & Inverse Proportion – 14 F is inversely proportional to the square of v. Given that F = 6.5 when v = 4 find a formula for F in terms of v. (Total for Question 14 is 3 marks)

Show worked solution

Find inverse function
$F$ is inversely proportional to $v^2$, so
\[ F=\frac{k}{v^2} \]
Use when
Use $F=6.5$ when $v=4$:
\[ 6.5=\frac{k}{4^2} \]
\[ k=6.5\times 16=104 \]
Find inverse function
Therefore
\[ F=\frac{104}{v^2} \]

Final answer

$F=\frac{104}{v^2}$.

9. Jun 2026 P1H · Q19

Medium4 marks

Jun 2026 P1H Question 19 (4 marks): Direct & Inverse Proportion – 19 M is directly proportional to p2 M = 2 when p = 5 (a) Find a formula for M in terms of p (3) (b) Work out the value of M when p = 15 M = (1) (Total for Question 19 is 4 marks)

Show worked solution

Part (a) | Step 1
\[M=kp^2,\qquad2=k(5^2)\implies k=\frac{2}{25}.\]
Introduce a constant of proportionality and use the given pair.
Part (a) | Step 2
\[M=\frac{2p^2}{25}.\]
Substitute the constant into the general relationship.
Part (b) | Step 1
\[M=\frac{2(15)^2}{25}=18.\]
Use the formula from (a).

Final answer

(a) $M=2p^2/25$

(b) $M=18$

10. Jun 2025 P1H · Q18

Medium4 marks

Jun 2025 P1H Question 18 (4 marks): Direct & Inverse Proportion – 18 y is inversely proportional to x 3 y = 4 when x = 3 Work out the value of x when y = 864 x = (Total for Question 18 is 4 marks)

Show worked solution

Find inverse function
$y$ is inversely proportional to $x^3$, so
\[ y=\frac{k}{x^3} \]
Use when
Use $y=4$ when $x=3$:
\[ 4=\frac{k}{3^3} \]
\[ k=4\times 27=108 \]
Find inverse function
So
\[ y=\frac{108}{x^3} \]
Find inverse function
When $y=864$,
\[ 864=\frac{108}{x^3} \]
\[ x^3=\frac{108}{864}=\frac18 \]
\[ x=\frac12 \]

Final answer

$x=\frac12$.

11. May 2023 P1HR · Q17

Medium4 marks

May 2023 P1HR Question 17 (4 marks): Direct & Inverse Proportion – 17 F is inversely proportional to the square of r F = 36 when r = 4 (a) Find a formula for F in terms of r (3) (b) Work out the value of F when r = 48 (1) (Total for Question 17 is 4 marks)

Show worked solution

Find inverse function
$F$ is inversely proportional to $r^2$, so
\[ F=\frac{k}{r^2} \]
Use when
Use $F=36$ when $r=4$:
\[ 36=\frac{k}{4^2} \]
\[ k=36\times 16=576 \]
Find inverse function
So
\[ F=\frac{576}{r^2} \]
Find inverse function
When $r=48$,
\[ F=\frac{576}{48^2}=\frac{576}{2304}=0.25 \]

Final answer

$F=\frac{576}{r^2}$, and $F=0.25$ when $r=48$.

12. Jun 2022 P2H · Q17

Medium4 marks

Jun 2022 P2H Question 17 (4 marks): Direct & Inverse Proportion – 17 M varies directly as the cube of h M = 4 when h = 0.5 Find the value of h when M = 500 (Total for Question 17 is 4 marks)

Show worked solution

Find inverse function
$M$ varies directly as $h^3$, so
\[ M=kh^3 \]
Use when
Use $M=4$ when $h=0.5$:
\[ 4=k(0.5)^3 \]
\[ k=\frac{4}{0.125}=32 \]
Find inverse function
So
\[ M=32h^3 \]
Find inverse function
When $M=500$,
\[ 500=32h^3 \]
\[ h^3=15.625 \]
\[ h=2.5 \]

Final answer

$h=2.5$.

13. Jan 2022 P2HR · Q20

Medium4 marks

Jan 2022 P2HR Question 20 (4 marks): Direct & Inverse Proportion – 20 y is inversely proportional to x x is directly proportional to T 3 Given that y = 8 when T = 25 find the exact value of T when y = 27 T = (Total for Question 20 is 4 marks)

Show worked solution

Find inverse function
$y$ is inversely proportional to $\sqrt{x}$, so
\[ y\propto \frac{1}{\sqrt{x}} \]
Find inverse function
$x$ is directly proportional to $T^3$, so $\sqrt{x}\propto T^{3/2}$.
Find inverse function
Therefore
\[ y\propto \frac{1}{T^{3/2}} \]
Find inverse function
So
\[ yT^{3/2}=k \]
Use when
Use $y=8$ when $T=25$:
\[ k=8\times 25^{3/2}=8\times 125=1000 \]
Find inverse function
When $y=27$,
\[ 27T^{3/2}=1000 \]
\[ T^{3/2}=\frac{1000}{27}=\left(\frac{10}{3}\right)^3 \]
Raise both sides to the
Raise both sides to the power $\frac{2}{3}$:
\[ T=\left(\frac{10}{3}\right)^2=\frac{100}{9} \]

Final answer

$T=\frac{100}{9}$.

14. Specimen 4WM2H · Q16

Medium4 marks

Specimen 4WM2H Question 16 (4 marks): Direct & Inverse Proportion – 16 M is directly proportional to h^3. M = 4 when h = 0.5. Find h when M = 500. (Total for Question 16 is 4 marks)

Show worked solution

Find inverse function
Since $M$ is directly proportional to $h^3$,
\[ M=kh^3 \]
Use when
Use $M=4$ when $h=0.5$:
\[ 4=k(0.5)^3 \]
\[ 4=0.125k \]
\[ k=32 \]
Use
Now use $M=500$:
\[ 500=32h^3 \]
\[ h^3=15.625 \]
\[ h=2.5 \]

Final answer

$h=2.5$.

15. Jun 2026 P1HR · Q20

Hard5 marks

Jun 2026 P1HR Question 20 (5 marks): Direct & Inverse Proportion – 20 y is directly proportional to t2 y = 40 when t = 4 t is directly proportional to the cube root of x t = 12 when x = 8 Work out the value of y when x = 27 y = (Total for Question 20 is 5 marks)

Show worked solution

Solution | Step 1
\[y=kt^2,\qquad40=k(4^2)\quad\Longrightarrow\quad k=\frac52.\]
Direct proportionality to t squared means a constant times t squared.
Solution | Step 2
\[t=c\sqrt[3]{x},\qquad12=c\sqrt[3]{8}=2c\quad\Longrightarrow\quad c=6.\]
Find the second proportionality constant using the given pair of values.
Solution | Step 3
\[x=27\quad\Longrightarrow\quad t=6\sqrt[3]{27}=18.\]
Use the cube-root relationship first.
Solution | Step 4
\[y=\frac52(18)^2=810.\]
Substitute the new value of t into the square relationship.

Final answer

$y=810$

16. Jun 2025 P1HR · Q18

Hard5 marks

Jun 2025 P1HR Question 18 (5 marks): Direct & Inverse Proportion – 18 F is inversely proportional to the cube of r F = 6 when r = 2 (a) Find a formula for F in terms of r (3) (b) Find the value of r when F = 3072 r = (2) (Total for Question 18 is 5 marks)

Show worked solution

Find inverse function
$F$ is inversely proportional to $r^3$, so
\[ F=\frac{k}{r^3} \]
Use when
Use $F=6$ when $r=2$:
\[ 6=\frac{k}{2^3} \]
\[ k=48 \]
Find inverse function
So
\[ F=\frac{48}{r^3} \]
Find inverse function
When $F=3072$,
\[ 3072=\frac{48}{r^3} \]
\[ r^3=\frac{48}{3072}=\frac{1}{64} \]
\[ r=\frac14 \]

Final answer

$F=\frac{48}{r^3}$, and $r=\frac14$.

17. Jun 2023 P2H · Q16

Hard5 marks

Jun 2023 P2H Question 16 (5 marks): Direct & Inverse Proportion – 16 Q is directly proportional to t The graph shows the relationship between Q and t for 0 < t < 8 9 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1 O t Q (a) Find a formula for Q in terms of t (3) Q is increased by 20% (b) Find the pe

Show worked solution

Find inverse function
\[ Q\propto \sqrt t \]
Find inverse function
So
\[ Q=k\sqrt t \]
Read the graph
From the graph, when $t=4$, $Q=6$.
\[ 6=k\sqrt4 \]
\[ 6=2k \]
\[ k=3 \]
Rearrange formula
So the formula is
\[ Q=3\sqrt t \]
Find inverse function
For part (b), $Q$ is increased by $20\%$, so $Q$ is multiplied by $1.2$.
Find inverse function
Since $Q\propto \sqrt t$,
\[ t\propto Q^2 \]
Find inverse function
So $t$ is multiplied by
\[ 1.2^2=1.44 \]
This is a increase
This is a $44\%$ increase.

Final answer

(a) $Q=3\sqrt t$, (b) $44\%$

18. May 2021 P2H · Q16

Hard6 marks

May 2021 P2H Question 16 (6 marks): Direct & Inverse Proportion – 16 A is inversely proportional to the square of r A = 5 when r = 0.3 (a) Find a formula for A in terms of r (3) (b) Find the value of A when r = 7.5A A = (3) (Total for Question 16 is 6 marks)

Show worked solution

Find inverse function
$A$ is inversely proportional to $r^2$, so
\[ A=\frac{k}{r^2} \]
Use when
Use $A=5$ when $r=0.3$:
\[ 5=\frac{k}{0.3^2} \]
\[ k=5\times 0.09=0.45 \]
Find inverse function
So
\[ A=\frac{0.45}{r^2} \]
Rearrange formula
For part (b), $r=7.5A$. Substitute into the formula:
\[ A=\frac{0.45}{(7.5A)^2} \]
\[ A=\frac{0.45}{56.25A^2} \]
\[ 56.25A^3=0.45 \]
\[ A^3=0.008 \]
\[ A=0.2 \]

Final answer

(a) $A=\frac{0.45}{r^2}$, (b) $A=0.2$.

Direct & Inverse Proportion: past paper questions & worked solutions

Questions, easiest first

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