GCSE Maths · Higher · Chapter 21: Variation · Lesson 2

Inverse proportion

Grades 6–8About 50 minutesCalculator allowed50 marks

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Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
$y \propto x$ , and $y = 24$ when $x = 4$ . Find $y$ when $x = 6$ .
[1 mark]
Answer
A2
From chapter 11
A ladder 8 m long makes $6 0^{\circ}$ with the ground. How far is its foot from the wall? ( $cos 6 0^{\circ} = 0.5$ )
[1 mark]
Answer
A3
From chapter 18
A bar has frequency density 3 and class width 15. Find its frequency.
[1 mark]
Answer
A4
From chapter 20
A tangent meets a radius at $P$ . In triangle $O P T$ the angle at $O$ is $6 0^{\circ}$ . Find angle $O T P$ .
[1 mark]
Answer

Grades 6–7

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

Example 1

$y$ is inversely proportional to $x$ . When $x = 9$ , $y = 20$ . Find $y$ when $x = 12$ .

Hint Write

y = \frac{k}{x}

and find

k

first.

B1
$y$ is inversely proportional to $x$ . When $x = 14$ , $y = 30$ . Find $y$ when $x = 35$ .
[3 marks]
Answer

Example 2

$y$ is inversely proportional to $x^{2}$ . When $x = 5$ , $y = 8$ . Find the positive value of $x$ when $y = 2$ .

Hint

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

, so

k = y \times x^{2}

B2
$y$ is inversely proportional to $x^{2}$ . When $x = 3$ , $y = 48$ . Find the positive value of $x$ when $y = 12$ .
[3 marks]
Answer

Grades 6–8

C1
$y$ is inversely proportional to $x$ . When $x = 25$ , $y = 18$ . Find $y$ when $x = 81$ .
[3 marks]
Answer
C2
$y$ is inversely proportional to $x^{3}$ . The value of $x$ is doubled. What number is $y$ divided by?
[2 marks]
Answer

C3
The sketch shows the graph of $y = \frac{k}{x}$ . It passes through the point $(4, 7.5)$ . Find $y$ when $x = 10$ .
[3 marks]
Not drawn accurately
Answer
C4
Values of $x$ and $y$ are $(2, 36)$ , $(3, 24)$ and $(8, 9)$ . Show that $y$ is inversely proportional to $x$ , and find $y$ when $x = 12$ .
[3 marks]
Answer
C5
The time to empty a tank is inversely proportional to the number of pumps. $6$ pumps empty the tank in $50$ minutes. How many pumps are needed to empty it in $20$ minutes?
[3 marks]
Answer
C6
$F$ is inversely proportional to the square of $d$ . When $d = 3$ , $F = 20$ . Write a formula for $F$ in terms of $d$ .
[3 marks]
Answer
C7
$y$ is inversely proportional to $x$ . The value of $x$ is increased by $60%$ . Work out the percentage decrease in $y$ .
[3 marks]
Answer
C8
Which equation shows $y$ inversely proportional to $x^{2}$ ? Write the letter.
$A y = 5 x^{2} B y = \frac{5}{x}$
$C y = \frac{5}{x ^{2}} D y = x^{2} - 5$
[1 mark]
Answer

Grades 7–8

D1
$P$ is inversely proportional to $V$ . When $V = 8$ , $P = 15$ . Leo says, "When $V = 12$ , $P = 11$ , because $V$ has gone up by $4$ , so $P$ goes down by $4$ ."
Leo is wrong. Explain why, and work out the correct value of $P$ when $V = 12$ .
[3 marks]
Answer

D2
The force, $F$ newtons, between two magnets is inversely proportional to the square of the distance, $d$ cm, between them. When $d = 3$ , $F = 80$ .
(a)
Find a formula for $F$ in terms of $d$ .
[3 marks]
Answer
(b)
Work out the distance between the magnets when the force is $5$ newtons.
[2 marks]
Answer
(c)
The distance is multiplied by $4$ . What happens to the force?
[2 marks]
Answer
(d)
Explain why the force can never be $0$ .
[1 mark]
Total for D2: 8 marks
D3
$y$ is inversely proportional to $x$ , and $x$ is directly proportional to $z^{2}$ . When $z = 2$ , $x = 12$ and $y = 5$ . Find $y$ when $z = 4$ . You must show your working.
[4 marks]
Answer

Grade 8 and beyond

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can write $y = \frac{k}{x}$ or $y = \frac{k}{x ^{2}}$ and find $k$ .
I can say what a change in $x$ does to $y$ .
I can use inverse proportion in context.

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