GCSE Maths · Higher · Chapter 21: Variation

Direct and inverse proportion: formulae, graphs and changes

Grades 4–9About 75 minutesCalculator allowed88 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
$y = 4 x$ . Find the value of $y$ when $x = 7.5$ .
[1 mark]
Answer

A2
Find the value of $k$ when $45 = k \times 3^{2}$ .
[1 mark]
Answer
A3
Work out $6 \div 0.25$ .
[1 mark]
Answer
A4
Make $x$ the subject of $y = k x^{2}$ , where $x$ and $k$ are positive.
[2 marks]
Answer
A5
Work out $3343$ .
[1 mark]
Answer

Examples, then your turn

Grades 5–6

Your teacher works through each example with you. Then try the one beside it on your own.

Example 1

$y$ is directly proportional to $x$ . When $x = 6$ , $y = 42$ . Find $y$ when $x = 11$ .

Hint Write

y = k x

. Use the pair of values you are given to find

k

B1
$P$ is directly proportional to $Q$ . When $Q = 12$ , $P = 54$ .
Find a formula for $P$ in terms of $Q$ . Then find $P$ when $Q = 20$ .
[3 marks]
Answer $P = \dots\dots$ $P = \dots\dots$

Example 2

$y$ is directly proportional to $x^{2}$ . When $x = 3$ , $y = 18$ . Find the positive value of $x$ when $y = 50$ .

Hint Write

y = k x^{2}

and find

k

. Then put

y = 50

in and solve for

x

B2
$A$ is directly proportional to the square of $r$ . When $r = 4$ , $A = 48$ . Find the positive value of $r$ when $A = 147$ .
[3 marks]
Answer

Example 3

$y$ is inversely proportional to $x$ . When $x = 5$ , $y = 12$ . Find $y$ when $x = 4$ .

Hint Write

y = \frac{k}{x}

. Then

k = x y

B3
The time, $T$ hours, it takes to paint a fence is inversely proportional to the number of painters, $n$ . 6 painters take 8 hours.
How long do 4 painters take?
[3 marks]
Answer

Example 4

$y$ is inversely proportional to $x^{2}$ . When $x = 4$ , $y = 5$ . Find $y$ when $x = 10$ .

Hint Write

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

, so

k = x^{2} y

B4
$F$ is inversely proportional to the square of $d$ . When $d = 5$ , $F = 36$ . Find the positive value of $d$ when $F = 4$ .
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
In the table, $y$ is directly proportional to $x$ . Find the value of $a$ .
[2 marks]
Answer

C2
$y$ is directly proportional to the square root of $x$ . When $x = 9$ , $y = 15$ . Find $y$ when $x = 64$ .
[3 marks]
Answer
C3
$y$ is inversely proportional to $x$ . On the axes, sketch the graph of $y$ against $x$ for $x > 0$ .
[2 marks]
C4
Here are four sketch graphs.
Not drawn accurately
(a)
Which graph shows $y$ directly proportional to $x$ ?
[1 mark]
Answer
(b)
Which graph shows $y$ directly proportional to $x^{2}$ ?
[1 mark]
Answer
(c)
Which graph shows $y$ inversely proportional to $x$ ?
[1 mark]
Answer
Total for C4: 3 marks
C5
Each part is about what happens to $y$ when $x$ changes.
(a)
$y$ is directly proportional to $x^{3}$ . $x$ is doubled. What number is $y$ multiplied by?
[1 mark]
Answer
(b)
$y$ is inversely proportional to $x^{2}$ . $x$ is multiplied by 4. What number is $y$ multiplied by?
[1 mark]
Answer
Total for C5: 2 marks
C6
The frequency, $f$ hertz, of a vibrating string is inversely proportional to its length, $L$ metres. When $L = 0.6$ , $f = 330$ .
Find $f$ when $L = 0.45$ .
[3 marks]
Answer
C7
The table shows some values of $x$ and $y$ .
Is $y$ inversely proportional to $x$ , or to $x^{2}$ ? Give a reason for your answer.
[2 marks]
C8
$V$ is directly proportional to the cube of $h$ . When $h = 4$ , $V = 160$ .
(a)
Find a formula for $V$ in terms of $h$ .
[2 marks]
Answer
(b)
Find $V$ when $h = 6$ .
[1 mark]
Answer
(c)
Find $h$ when $V = 20$ .
[2 marks]
Answer
Total for C8: 5 marks
C9
The cost, $C$ pounds, of sand is directly proportional to its mass, $m$ kg. The graph shows this.
Work out the cost of 15 kg of sand.
[2 marks]
Answer

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
The cost of a round table top is directly proportional to the square of its diameter. A table top of diameter 75 cm costs £90.
Find the cost of a table top of diameter 1.2 m.
[3 marks]
Answer

D2
Mia says, " $y$ is inversely proportional to $x^{2}$ , so when $x$ is doubled, $y$ is halved."
Mia is wrong. Explain why, and say what happens to $y$ when $x$ is doubled.
[2 marks]
Answer
D3
$y$ is directly proportional to $x^{3}$ .
Show that when $x$ is increased by 20%, $y$ is increased by 72.8%.
[3 marks]
D4
The pressure, $P$ , of a gas is inversely proportional to its volume, $V$ . When $V = 1.5$ , $P = 240$ .
(a)
Find $P$ when $V = 2.4$ .
[2 marks]
Answer
(b)
Find $V$ when $P = 320$ .
[2 marks]
Answer
Total for D4: 4 marks
D5
$a$ is directly proportional to $b^{2}$ . $b$ is inversely proportional to $c$ .
When $c = 2$ , $b = 6$ and $a = 72$ .
Find the value of $a$ when $c = 3$ .
[4 marks]
Answer
D6
The graph shows part of the curve $y = k x^{2}$ . The curve passes through the point $(4, 40)$ .
Find $y$ when $x = 10$ .
[3 marks]
Not drawn accurately
Answer
D7
A ball rolls down a slope from rest. The distance, $d$ metres, it has rolled is directly proportional to the square of the time, $t$ seconds, since it started. In the first 2 seconds it rolls 1.8 metres.
(a)
Find a formula for $d$ in terms of $t$ .
[2 marks]
Answer
(b)
How far does the ball roll during the third second?
[2 marks]
Answer
(c)
The slope is 20 metres long. How long does the ball take to reach the bottom? Give your answer to 3 significant figures.
[2 marks]
Answer
Total for D7: 6 marks
D8
The time, $T$ seconds, for one swing of a pendulum is directly proportional to the square root of its length, $L$ cm. A pendulum of length 64 cm takes 1.6 seconds.
Find the length of a pendulum that takes 2.2 seconds.
[3 marks]
Answer
D9
The table shows some values of $x$ and $y$ .
Find a formula for $y$ in terms of $x$ . You must show your working.
[3 marks]
Answer
D10
$y$ is directly proportional to $x^{n}$ . When $x = 2$ , $y = 24$ . When $x = 4$ , $y = 192$ .
Find the value of $n$ .
[3 marks]
Answer

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
$p$ is directly proportional to the square of $q$ . $q$ is inversely proportional to the cube of $r$ .
$r$ is doubled. What happens to $p$ ?
[3 marks]
Hint 1 · What to try
Write each relationship with its own constant, then put one into the other.
Hint 2 · The first line
$q = \frac{k}{r ^{3}}$ , so $p = m q^{2} = \frac{m k ^{2}}{r ^{6}}$ .
Hint 3 · The full method
$p$ is inversely proportional to $r^{6}$ , and $2^{6} = 64$ , so $p$ is divided by $64$ .
Answer

E2
The surface area of a sphere is proportional to the square of its radius. Its volume is proportional to the cube of its radius.
A balloon is a sphere. Its volume increases by 33.1%. By what percentage does its surface area increase?
[3 marks]
Hint 1 · What to try
Find the number the radius is multiplied by first.
Hint 2 · The first line
The volume is multiplied by $1.331$ , so the radius is multiplied by $3 1.331$ .
Hint 3 · The full method
$3 1.331 = 1.1$ , so the area is multiplied by $1. 1^{2} = 1.21$ . That is an increase of $21$ %.
Answer
E3
The graph of $y = \frac{k}{x}$ passes through the points $(a, 18)$ and $(3 a, a)$ , where $a > 0$ .
Find the value of $a$ and the value of $k$ .
[3 marks]
Hint 1 · What to try
Each point gives an expression for $k$ .
Hint 2 · The first line
$k = 18 a$ and $k = 3 a \times a = 3 a^{2}$ .
Hint 3 · The full method
$3 a^{2} = 18 a$ and $a > 0$ , so $a = 6$ and $k = 108$ .
Answer $a = \dots\dots$ $k = \dots\dots$
E4
$a$ is inversely proportional to $b$ . $b$ is increased by 44%.
Find the percentage decrease in $a$ . Give your answer to 3 significant figures.
[3 marks]
Hint 1 · What to try
Find the number $b$ is multiplied by.
Hint 2 · The first line
$1.44 = 1.2$ , so $a$ is multiplied by $\frac{1}{1.2}$ .
Hint 3 · The full method
$\frac{1}{1.2} = 0.8333 \dots$ , which is a decrease of $16.7$ %.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write a direct proportion as a formula, including $y = k x^{2}$ , $y = k x^{3}$ and $y = k x$ , find $k$ and use it.
I can do the same for inverse proportion, $y = \frac{k}{x}$ and $y = \frac{k}{x ^{2}}$ , and recognise each graph.
I can work out what a change in one quantity does to the other.

Answers and mark scheme

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