GCSE Maths · Higher · Chapter 21: Variation · Lesson 1

Direct proportion

Grades 5–8About 40 minutesCalculator allowed34 marks

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NameClassDate

Do Now

Grades 4–6

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

A1
Needed today
Share £88 in the ratio $3 : 5$ . Find the larger share.
[1 mark]
Answer
A2
From chapter 17
A parabola crosses the $x$ -axis at $- 1$ and $5$ . Find the $x$ -coordinate of its turning point.
[1 mark]
Answer
A3
From chapter 17
Solve $x^{2} = 9$ .
[1 mark]
Answer
A4
From chapter 17
Work out $b^{2} - 4 a c$ for $3 x^{2} - 4 x - 2 = 0$ .
[1 mark]
Answer

Grades 5–7

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

Example 1

$y$ is directly proportional to $x$ . When $x = 8$ , $y = 52$ . Find $y$ when $x = 13$ .

Hint Write

y = k x

, find

k

, then use the formula.

B1
$W$ is directly proportional to $h$ . When $h = 15$ , $W = 54$ . Find $W$ when $h = 40$ .
[3 marks]
Answer

Example 2

$y$ is directly proportional to $x^{2}$ . When $x = 4$ , $y = 56$ . Find the positive value of $x$ when $y = 126$ .

Hint Find

k

y = k x^{2}

, then solve for

x^{2}

and take the square root.

B2
$E$ is directly proportional to $v^{2}$ . When $v = 6$ , $E = 90$ . Find the positive value of $v$ when $E = 250$ .
[3 marks]
Answer

Grades 6–7

C1
$y$ is directly proportional to $x$ . When $x = 49$ , $y = 21$ . Find $y$ when $x = 121$ .
[2 marks]
Answer
C2
$y$ is directly proportional to $x^{3}$ . $x$ is multiplied by 4. What number is $y$ multiplied by?
[1 mark]
Answer

C3
Three pairs of values are $(2, 7)$ , $(5, 17.5)$ and $(8, 28)$ . Is $y$ directly proportional to $x$ ? Give a reason.
[2 marks]
C4
The graph shows that $y$ is directly proportional to $x$ . Write $y$ in terms of $x$ .
[2 marks]
Answer
C5
$y$ is directly proportional to $x^{2}$ . Which sketch, A, B or C, shows this? Give a reason.
[2 marks]
C6
$y$ is directly proportional to $x^{3}$ . When $x = 2$ , $y = 20$ . Find $y$ when $x = 4$ , and find $x$ when $y = 540$ .
[3 marks]
Answer $y = \dots\dots$ $x = \dots\dots$

Grades 7–8

D1
Ben says, " $y$ is directly proportional to $x^{2}$ , so when $x$ is multiplied by 3, $y$ is multiplied by 6."
Explain why Ben is wrong, and say what $y$ is multiplied by.
[2 marks]

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 a direct proportion as a formula and find the constant $k$ .
I can use $y = k x^{2}$ or $y = k x$ in both directions.
I can tell from a table or a graph whether $y$ is directly proportional to $x$ .

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