KS3 Maths · Year 7 · Chapter 13: Proportion · Lesson 4

Comparing direct and inverse proportion

Year 7About 70 minutesCalculator allowed47 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $21 \div 6$ .
[1 mark]
Answer
A2
Needed today
Work out $2.5 \times 3.2$ .
[1 mark]
Answer
A3
Last lesson
$x y = 48$ . Work out $y$ when $x = 3.2$ .
[1 mark]
Answer
A4
From chapter 12
Work out $0.6 \times 0.04$ .
[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

The table shows two sets of values of $y$ , P and Q, for the same values of $x$ . For each one, say whether it shows direct proportion, inverse proportion or neither. Write the formula.

Hint Work out

y \div x

for each pair, and

x y

for each pair. Look for the one that stays the same.

B1
For each set, R and S, say whether it shows direct proportion, inverse proportion or neither. Give a reason. Write the formula if there is one.
[4 marks]

Example 2

When $p = 15$ , $q = 24$ . Find $q$ when $p = 40$ if (a) $q$ is directly proportional to $p$ , (b) $q$ is inversely proportional to $p$ .

Hint For direct, find

q \div p

. For inverse, find

pq

B2
When $r = 12$ , $s = 7.5$ . Find $s$ when $r = 18$ if (a) $s$ is directly proportional to $r$ , (b) $s$ is inversely proportional to $r$ .
[4 marks]
Answer(a) $s = \dots$ (b) $s = \dots$

Example 3

Does this graph show direct proportion or inverse proportion? Write a formula connecting $x$ and $y$ .

Hint A curve that falls is not direct proportion. Read a point on the curve where it crosses a grid corner.

B3
Does this graph show direct proportion or inverse proportion? Give a reason. Write a formula for $y$ in terms of $x$ .
[3 marks]
Answer $y = \dots\dots$

Example 4

A $300$ g bag of rice is shared into $n$ equal portions, each of $w$ g. Explain why $w$ is inversely proportional to $n$ . Write a formula and find $w$ when $n = 8$ .

Hint Multiply the number of portions by the size of each one. What do you always get?

B4
Ella cycles $36$ km at a steady speed of $v$ km/h. It takes her $t$ hours. Explain why $t$ is inversely proportional to $v$ . Write a formula, and find $t$ when $v = 16$ .
[4 marks]
Answer $v t = \dots$ $t = \dots$ hours

Practice

Year 7

C1
When $x = 40$ , $y = 0.5$ . Write a formula for $y$ in terms of $x$ if they are in direct proportion. Write a formula connecting $x$ and $y$ if they are in inverse proportion.
[2 marks]
Answer $y = \dots$ $x y = \dots$
C2
Which of these formulas show that $y$ is inversely proportional to $x$ ? Write down the letters. $A: y = \frac{12}{x} B: y = \frac{x}{12}$ $C: y = 12 - x D: x y = 5$
[2 marks]
Answer $\dots\dots$

C3
A rectangle has a perimeter of $24$ cm. Its sides are $a$ cm and $b$ cm. Use $a = 4$ and $a = 6$ to show that $b$ is neither directly nor inversely proportional to $a$ .
[3 marks]
Not drawn accurately
C4
Show that $w$ is inversely proportional to $u$ . Write the formula.
[2 marks]
Answer $u w = \dots$

Exam-style questions

GCSE Foundation

D1
Sana reads at a steady rate. The time she takes is directly proportional to the number of pages.
(a)
Work out the two missing times.
[2 marks]
Answer $\dots$ and $\dots$ minutes
(b)
Sana reads a book of $260$ pages. How long does it take her? Give your answer in hours and minutes.
[3 marks]
Answer $\dots$ hours $\dots$ minutes
Total for D1: 5 marks

D2
Theo says, "The table shows inverse proportion, because when $x$ goes up, $y$ goes down."
Is Theo correct? Give a reason for your answer.
[2 marks]
D3
A fence $150$ m long is made from panels that are all the same length. Panels come in three lengths.
(a)
Explain why the number of panels needed is inversely proportional to the length of each panel.
[1 mark]
(b)
Which length of panel gives the cheapest fence? You must show your working.
[4 marks]
Answer $\dots$ m
Total for D3: 5 marks
D4
$4$ taps fill a tank in $45$ minutes. All the taps run at the same rate.
(a)
How long would $3$ taps take to fill the tank?
[2 marks]
Answer $\dots$ minutes
(b)
Jack says, "The time is directly proportional to the number of taps." Is Jack correct? Explain your answer.
[2 marks]
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 $x$ . The value of $x$ goes up by $25%$ . By what percentage does $y$ go down?
[3 marks]
Hint 1 · What to try
Try numbers: start with $x = 100$ and $y = 1$ , so $x y = 100$ .
Hint 2 · The first line
$x$ goes up by $25%$ to $125$ . Then $125 y = 100$ , so $y = 0.8$ .
Hint 3 · The full method
$y$ went from $1$ to $0.8$ , which is down by $0.2$ , or $20%$ . It is not $25%$ .
Answer $\dots %$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can tell direct proportion ( $y \div x$ the same) from inverse proportion ( $x y$ the same).
I can find a missing value if two quantities are directly or inversely proportional.
I can explain why a real situation is direct or inverse proportion.

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