KS3 Maths · Year 7 · Chapter 13: Proportion · Lesson 3

Inverse proportion

Year 7About 70 minutesCalculator allowed47 marks

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Quick questions to start.

A1
Needed today
Work out $4.5 \times 16$ .
[1 mark]
Answer
A2
Needed today
Work out $90 \div 2.5$ .
[1 mark]
Answer
A3
Last lesson
$y$ is in direct proportion to $x$ . $y = 15$ when $x = 6$ . Work out $y$ when $x = 10$ .
[1 mark]
Answer
A4
From chapter 12
Work out $12 \div \frac{3}{4}$ .
[1 mark]
Answer

Example, then your turn

Year 7

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

Example 1

A car journey is $180$ km. The table shows the speed of the car and the time the journey takes. Complete the table. Show that the speed and the time are in inverse proportion.

Hint Time

=

distance

\div

speed. Then multiply each speed by its time.

B1
A tank holds $240$ litres. A pump fills it at a steady rate, in litres a minute. Complete the table. Show that the rate and the time are in inverse proportion.
[3 marks]

Example 2

$y$ is inversely proportional to $x$ . When $x = 4$ , $y = 9$ . Write a formula connecting $x$ and $y$ . Work out $y$ when $x = 6$ , and $x$ when $y = 2.5$ .

Hint Inverse proportion means

x y = k

. Find

k

from the pair you are given.

B2
$y$ is inversely proportional to $x$ . When $x = 8$ , $y = 15$ . Write a formula connecting $x$ and $y$ . Work out $y$ when $x = 12$ , and $x$ when $y = 48$ .
[4 marks]
Answer $x y = \dots$ $y = \dots$ $x = \dots$

Example 3

$6$ builders take $10$ days to build a wall. They all work at the same rate. How long would $4$ builders take? How many builders are needed to build it in $5$ days?

Hint Work out the total number of builder-days. Fewer builders take longer.

B3
$8$ pumps empty a pond in $9$ hours. Each pump works at the same rate. How long would $6$ pumps take? What is the smallest number of pumps that can empty the pond in $5$ hours?
[3 marks]
Answer $\dots$ hours $\dots$ pumps

Example 4

$x y = 24$ . Work out $y$ when $x = 2$ , $3$ , $4$ , $6$ , $8$ and $12$ . Plot the points on the grid and join them with a smooth curve.

Hint

y = 24 \div x

. The points do not lie on a straight line, so draw a curve, not straight pieces.

B4
$x y = 36$ . Work out $y$ when $x = 2$ , $3$ , $4$ , $6$ , $9$ and $12$ . Plot the points on the grid and join them with a smooth curve.
[3 marks]

Example 5

Ben rides $60$ km. The graph shows the time, $y$ hours, at a speed of $x$ km/h. Find the time at $12$ km/h and the speed for $2.5$ hours. Write a formula connecting $x$ and $y$ . Find the time at $40$ km/h.

Hint Go up from

12

to the curve, then across. For the formula, multiply the two numbers at any point on the curve.

B5
A prize of £ $360$ is shared equally by $n$ winners. Each gets £ $c$ . Use the graph to find $c$ when there are $6$ winners, and $n$ when each gets £ $40$ . Write a formula connecting $n$ and $c$ .
[3 marks]
Answer£ $\dots$ $\dots$ winners $n c = \dots$

Practice

Year 7

C1
A rectangle has an area of $54$ cm². Its base is $b$ cm and its height is $h$ cm. Write a formula connecting $b$ and $h$ . Work out $h$ when $b = 7.2$ .
[3 marks]
Not drawn accurately
Answer $bh = \dots$ $h = \dots$
C2
Is $y$ inversely proportional to $x$ ? Give a reason.
[2 marks]

C3
$x y = 50$ . Find the value of $x$ when $x$ and $y$ are equal. Give your answer to $2$ decimal places.
[2 marks]
Answer $x = \dots\dots$
C4
£ $600$ is shared equally between $n$ people. Each person gets £ $p$ . How many people share the money when each gets £ $37.50$ ?
[2 marks]
Answer $\dots$ people

Exam-style questions

GCSE Foundation

D1
$5$ people take $12$ hours to pick the apples in an orchard. They all work at the same rate.
(a)
How long would $8$ people take? Give your answer in hours and minutes.
[2 marks]
Answer $\dots$ hours $\dots$ minutes
(b)
The farmer wants the apples picked in $4$ hours. How many people does she need?
[2 marks]
Answer $\dots$ people
Total for D1: 4 marks

D2
Mia says, "When $y = 20 - x$ , $y$ is inversely proportional to $x$ , because $y$ goes down as $x$ goes up."
Is Mia correct? Give a reason for your answer.
[2 marks]
D3
A coach for a school trip costs £ $780$ . The cost is shared equally between the pupils who go.
(a)
Complete the table.
[2 marks]
(b)
Write a formula connecting the number of pupils, $n$ , and the amount each pays, £ $c$ .
[1 mark]
Answer $n c = \dots$
(c)
Each pupil can pay at most £ $18$ . What is the smallest number of pupils that must go?
[2 marks]
Answer $\dots$ pupils
Total for D3: 5 marks
D4
A coach journey takes $3$ hours at an average speed of $64$ km/h.
(a)
How long would the journey take at an average speed of $48$ km/h?
[2 marks]
Answer $\dots$ hours
(b)
The coach must do the journey in $2$ hours $30$ minutes. What average speed must it travel at?
[2 marks]
Answer $\dots$ km/h
Total for D4: 4 marks

Extension

Stretch

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

E1
$y$ is inversely proportional to $x$ . When $x$ goes up by $2$ , $y$ goes down from $15$ to $10$ . Find the value of $k$ in the formula $x y = k$ .
[3 marks]
Hint 1 · What to try
Call the first value of $x$ just $x$ . The second value is $x + 2$ .
Hint 2 · The first line
Both pairs give the same product: $15 x = 10 (x + 2)$ .
Hint 3 · The full method
$15 x = 10 x + 20$ , so $5 x = 20$ and $x = 4$ . Then $k = 4 \times 15 = 60$ . Check: $6 \times 10 = 60$ .
Answer $k = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can tell inverse proportion: the product $x y$ stays the same.
I can find a formula $x y = k$ and use it both ways round.
I can draw and read a graph of inverse proportion.

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