Problem solving with direct and inverse proportion

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Lesson details

Learning outcome

I can use my knowledge of direct and inverse proportion to solve problems.

Key learning points

Algebraic manipulation is needed to solve proportion problems.
Proportional relationships can be modelled graphically and algebraically.

Keywords

Inversely proportional - Two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other.
Direct proportion - Two variables are in direct proportion if they have a constant multiplicative relationship.

Common misconception

Directly proportional graphs all start from (0,0) and are above the y axis.

Directly proportional means there is a multiplier between y and x. This graph is always y=kx and is linear passing through (0,0). A proportionality graphs show the multiplicative relationship between y and x^n and can be linear and non linear.

Teacher tip

A Desmos lesson is an excellent opportunity for pupils to create their own graphs and identify if their graph is directly proportional or inversely proportion. They will also see if their graph has the required number of roots and turning points to complete Q3 in Task C.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0

except where otherwise stated. See Oak's terms & conditions

(Collection 2).

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Worksheet

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Prior knowledge starter quiz

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6 Questions

Q1.
Two variables are in proportion if they have a constant multiplicative relationship.

Correct answer: direct

inverse

opposite

exact

proper

Q2.
Which graph shows no proportional relationship?

Correct Answer: F, f

Q3.
$$y \propto x^2$$, when $$y=192$$, $$x=4$$. Work out the value of $$y$$ when $$x=10$$.

Correct Answer: 1200, 1 200

Q4.
Which of these graphs show $$y \propto \frac{1}{x^n}$$?

Correct answer: A and B

B and C

C and D

D and E

E and F

Q5.
$$y \propto \sqrt{x}$$, when $$y=40$$, $$x=4$$. Work out the value of $$y$$ when $$x=100$$.

Correct Answer: 200

Q6.
$$y \propto \frac{1}{x^2}$$, when $$y=9.6$$, $$x=5$$. Work out the value of $$y$$ when $$x=10$$. Give your answer as a decimal.

Correct Answer: 2.4

Assessment exit quiz

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6 Questions

Q1.
Two variables are inversely proportional if there is a constant relationship between one variable and the reciprocal of the other.

Correct Answer: multiplicative

Q2.
When $$y\propto \sqrt{x}$$, when $$x$$, increases by 96%, what proportion does $$y$$ increase by?

96%

Correct answer: 40%

80%

12%

Q3.
Given the following proportions, select which are the correct proportional relationships when : $$a \propto c$$, $$c \propto b$$, $$b \propto \frac{1}{e}$$ and $$a \propto \frac{1}{f}$$.

Correct answer: $$a \propto b$$

Correct answer: $$a \propto \frac{1}{e}$$

$$b \propto f$$

$$c \propto f$$

Q4.
When $$y \propto x$$ and $$x \propto z^2$$, when $$y=75$$, $$x=3$$ and when $$x=128$$, $$z=4$$. Write $$y$$ in terms of $$z$$.

Correct answer: $$y=200z^2$$

$$y=8z^2$$

$$y=25z^2$$

$$y=2z^2$$

Q5.
When $$y\propto x^2$$, when $$x$$ increases by 20%, what proportion does $$y$$ increase by?

20%

40%

Correct answer: 44%

12%

Q6.
Match the graph with the equations.

Correct Answer:A,$$y=x(x-5)(x+5)$$

Correct Answer:B,$$y=4x^3$$

Correct Answer:C,$$y=x^2-10x$$

Correct Answer:D,$$y=9x^2$$

Correct Answer:E,$$y=4^x$$

Correct Answer:F,$$y=\frac{2}{x}$$

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Problem solving with direct and inverse proportion

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Two variables are in proportion if they have a constant multiplicative relationship.

Q2.Which graph shows no proportional relationship?

Q3.$$y \propto x^2$$, when $$y=192$$, $$x=4$$. Work out the value of $$y$$ when $$x=10$$.

Q4.Which of these graphs show $$y \propto \frac{1}{x^n}$$?

Q5.$$y \propto \sqrt{x}$$, when $$y=40$$, $$x=4$$. Work out the value of $$y$$ when $$x=100$$.

Q6.$$y \propto \frac{1}{x^2}$$, when $$y=9.6$$, $$x=5$$. Work out the value of $$y$$ when $$x=10$$. Give your answer as a decimal.

Assessment exit quiz

6 Questions

Q1.Two variables are inversely proportional if there is a constant relationship between one variable and the reciprocal of the other.

Q2.When $$y\propto \sqrt{x}$$, when $$x$$, increases by 96%, what proportion does $$y$$ increase by?

Q3.Given the following proportions, select which are the correct proportional relationships when : $$a \propto c$$, $$c \propto b$$, $$b \propto \frac{1}{e}$$ and $$a \propto \frac{1}{f}$$.

Q4.When $$y \propto x$$ and $$x \propto z^2$$, when $$y=75$$, $$x=3$$ and when $$x=128$$, $$z=4$$. Write $$y$$ in terms of $$z$$.

Q5.When $$y\propto x^2$$, when $$x$$ increases by 20%, what proportion does $$y$$ increase by?

Q6.Match the graph with the equations.

How to plan a lesson using our resources

Q1.
Two variables are in proportion if they have a constant multiplicative relationship.

Q2.
Which graph shows no proportional relationship?

Q3.
$$y \propto x^2$$, when $$y=192$$, $$x=4$$. Work out the value of $$y$$ when $$x=10$$.

Q4.
Which of these graphs show $$y \propto \frac{1}{x^n}$$?

Q5.
$$y \propto \sqrt{x}$$, when $$y=40$$, $$x=4$$. Work out the value of $$y$$ when $$x=100$$.

Q6.
$$y \propto \frac{1}{x^2}$$, when $$y=9.6$$, $$x=5$$. Work out the value of $$y$$ when $$x=10$$. Give your answer as a decimal.

Q1.
Two variables are inversely proportional if there is a constant relationship between one variable and the reciprocal of the other.

Q2.
When $$y\propto \sqrt{x}$$, when $$x$$, increases by 96%, what proportion does $$y$$ increase by?

Q3.
Given the following proportions, select which are the correct proportional relationships when : $$a \propto c$$, $$c \propto b$$, $$b \propto \frac{1}{e}$$ and $$a \propto \frac{1}{f}$$.

Q4.
When $$y \propto x$$ and $$x \propto z^2$$, when $$y=75$$, $$x=3$$ and when $$x=128$$, $$z=4$$. Write $$y$$ in terms of $$z$$.

Q5.
When $$y\propto x^2$$, when $$x$$ increases by 20%, what proportion does $$y$$ increase by?

Q6.
Match the graph with the equations.