KS3 Maths · Year 7 · Chapter 13: Proportion · Lesson 2

Graphs and direct proportion

Year 7About 70 minutesCalculator allowed47 marks

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Quick questions to start.

A1
Needed today
$y = 3 x$ . Work out $y$ when $x = 2.5$ .
[1 mark]
Answer
A2
Needed today
Solve $6 x = 15$ .
[1 mark]
Answer
A3
Last lesson
$5$ pens cost £ $3.60$ . Work out the cost of $8$ pens.
[1 mark]
Answer
A4
From chapter 7
Write down the gradient of the line $y = 4 x - 3$ .
[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

Fabric is sold by the metre. $4$ m of fabric cost £ $9.20$ . Find a formula for the cost, $y$ pence, of $x$ metres of the fabric.

Hint The formula is

y = m x

. Put the cost in pence:

x = 4

and

y = 920

B1
$2.5$ kg of apples cost £ $4.50$ . The cost, $y$ pence, of $x$ kg of apples is in direct proportion to the mass. Find a formula for $y$ in terms of $x$ . Use it to work out the cost of $7$ kg of apples, in pounds.
[3 marks]
Answer $y = \dots\dots$ £ $\dots\dots$

Example 2

$x$ US dollars are worth $y$ pounds, and $50$ US dollars are worth £ $40$ . Write a formula for $y$ in terms of $x$ . Use it to change $335$ US dollars into pounds and £ $148$ into US dollars.

Hint

m = 40 \div 50

. For the second change, divide by

m

B2
$5$ kg is the same as $11$ pounds (lb). A mass of $x$ kg is $y$ lb. Write a formula for $y$ in terms of $x$ . Use it to change $64$ kg into lb and $99$ lb into kg.
[4 marks]
Answer $y = \dots\dots$ $\dots\dots$ lb $\dots\dots$ kg

Example 3

Theatre tickets cost £ $6$ each. The cost, £ $y$ , of $x$ tickets is $y = 6 x$ . Work out $y$ when $x = 0$ , $2$ , $4$ and $6$ . Plot the points on the grid and join them.

Hint Multiply each number of tickets by

6

. The points should lie on a straight line through the origin.

B3
$y = 2.5 x$ . Work out $y$ when $x = 0$ , $2$ , $4$ and $6$ . Plot the points on the grid and join them with a straight line.
[3 marks]

Example 4

The graph shows the area a tin of paint covers. Use the graph to find the area covered by $3$ litres, and the paint needed for $42$ m². Write a formula for the area, $y$ m², covered by $x$ litres.

Hint Go up from

3

on the bottom axis to the line, then across. For the formula, read one point:

5

litres cover

60

m².

B4
The graph shows the diesel a van uses. Use the graph to find the diesel used for $75$ km, and the distance travelled on $2$ litres. Write a formula for the diesel, $y$ litres, used for $x$ km.
[3 marks]
Answer $\dots$ litres $\dots$ km $y = \dots\dots$

Example 5

Which of these graphs shows that $y$ is in direct proportion to $x$ ? Give a reason.

Hint A direct proportion graph is a straight line that goes through the origin,

(0, 0)

B5
Which of these graphs shows that $y$ is in direct proportion to $x$ ? Give a reason why each of the other two does not.
[3 marks]
Answer

Practice

Year 7

C1
A $150$ g pot of yoghurt contains $21$ g of sugar. $x$ g of the yoghurt contain $y$ g of sugar. Show that $y = 0.14 x$ .
[2 marks]
C2
$y = 1.25 x$ . Write a formula for $x$ in terms of $y$ .
[2 marks]
Answer $x = \dots\dots$

C3
Which of these formulas show that $y$ is in direct proportion to $x$ ? Write down the letters. $A: y = 4 x B: y = x + 4$ $C: y = 0.4 x D: y = 4 x - 1$
[2 marks]
Answer $\dots\dots$
C4
A regular hexagon has sides of $x$ cm. Its perimeter is $y$ cm. Write a formula for $y$ in terms of $x$ . Work out the side of a regular hexagon with a perimeter of $45$ cm.
[3 marks]
Answer $y = \dots\dots$ $\dots\dots$ cm

Exam-style questions

GCSE Foundation

D1
You can use this graph to change between pounds (£) and euros (€).
(a)
Use the graph to change £ $40$ into euros.
[1 mark]
Answer€ $\dots\dots$
(b)
Use the graph to change € $36$ into pounds.
[1 mark]
Answer£ $\dots\dots$
(c)
Rhian changes £ $360$ into euros. How many euros does she get?
[2 marks]
Answer€ $\dots\dots$
Total for D1: 4 marks

D2
Kai says, "The cost of hiring the hall is in direct proportion to the time, because the graph is a straight line."
The graph shows the cost of hiring a hall. Is Kai correct? Give a reason for your answer.
[2 marks]
D3
A machine fills bottles at a steady rate. In $8$ minutes it fills $300$ bottles. The number of bottles, $y$ , is in direct proportion to the time, $x$ minutes.
(a)
Write a formula for $y$ in terms of $x$ .
[2 marks]
Answer $y = \dots\dots$
(b)
How long does the machine take to fill $1000$ bottles? Give your answer in minutes and seconds.
[2 marks]
Answer $\dots$ minutes $\dots$ seconds
Total for D3: 4 marks
D4
Two shops sell the same rope. The graph shows the cost of rope at shop A. At shop B the cost, £ $y$ , of $x$ metres is $y = 2.35 x$ .
(a)
Use the graph to write a formula for the cost, £ $y$ , of $x$ metres of rope at shop A.
[2 marks]
Answer $y = \dots\dots$
(b)
Jo needs $30$ m of rope. Which shop is cheaper for $30$ m, and by how much? You must show your working.
[3 marks]
Answershop $\dots$ by £ $\dots\dots$
Total for D4: 5 marks

Extension

Stretch

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

E1
$y$ is in direct proportion to $x$ . The graph of $y$ against $x$ passes through the points $(6, a)$ and $(a, 24)$ , where $a$ is a positive number. Work out the value of $a$ .
[3 marks]
Hint 1 · What to try
Write $y = m x$ . Put each point into the formula.
Hint 2 · The first line
$a = 6 m$ and $24 = m \times a$ , so $24 = m \times 6 m = 6 m^{2}$ .
Hint 3 · The full method
$m^{2} = 4$ , so $m = 2$ (the line goes up), and $a = 6 \times 2 = 12$ . Check: $2 \times 12 = 24$ .
Answer $a = \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find a formula $y = m x$ from one pair of values.
I can draw and read a graph of direct proportion.
I can tell a direct proportion graph: a straight line through the origin.

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