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Direct and Inverse Proportion

A self-marking exercise in three levels on solving direct and inverse variation problems.

Level 1 Level 2 Level 3 Unitary Method Description Help Exam-Style More Ratio

This is level 1; Direct proportion. You can earn a trophy if you get at least 9 correct.

1. If \(a\) varies directly with \(b\) and \(a=18\) when \(b=6\)

      find \(a\) when \(b=7\)
Correct Wrong

      and find \(b\) when \(a=27\)
Correct Wrong

2. If \(c\) varies directly with \(d\) and \(c=18\) when \(d=3\)

      find \(c\) when \(d=6\)
Correct Wrong

      and find \(d\) when \(c=42\)
Correct Wrong

3. If \(e\) varies directly with \(f\) and \(e=63\) when \(f=7\)

      find \(e\) when \(f=11\)
Correct Wrong

      and find \(f\) when \(e=99\)
Correct Wrong

4. If \(g\) varies directly with \(h\) and \(g=42.7\) when \(h=7\)

      find \(g\) when \(h=10\)
Correct Wrong

      and find \(h\) when \(g=54.9\)
Correct Wrong

5. The distance travelled by a train is directly proportional to the time taken.
If the train can travel 504 kilometers in 7 hours find how many kilometers it can travel in 9 hours.
Correct Wrong

6. The cost of a drain pipe is directly proportional to its length.
A pipe costing £16.50 is 55 centimetres long. How many centimetres long would a drain pipe costing £23.70 be?
Correct Wrong

7. If the pressure of the water on a diver at any point below the surface of the sea varies as the depth of the diver below the surface. If the pressure is 15 psi at a depth of 10 metres, calculate the pressure (in psi) at a depth of 18 metres.
Correct Wrong

8. Click or tap on the graph which could represent direct proportion.
AGraph A BGraph B CGraph C DGraph D EGraph E
Correct Wrong

This is Direct and Inverse Proportion level 1. You can also try:
Level 2 Level 3


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Description of Levels



Level 1 - Direct proportion

Level 2 - Inverse proportion

Level 3 - Mixed non-linear questions

Unitary Method - Test your understanding of the Unitary Method for solving real life proportion problems with this online, self-marking quiz.

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Direct and Inverse Proportion

This video is from Mannel's Maths Music.

Level 1 Example

If \(a\) varies directly with \(b\) and \(a=24\) when \(b=8\) find \(a\) when \(b=9\)

$$a \propto b$$ $$a = kb$$

Where \(k\) is some constant. If \(a=24\) when \(b=8\) then

$$24 = 8k$$ $$k = 3$$

so the equation is

$$a = 3b$$

If \(b = 9\) then

$$a = 3 \times 9 = 27$$

Level 2 Example

If \(a\) is inversely proportional to \(b\) and \(a=4\) when \(b=6\) find \(a\) when \(b=8\)

$$a \propto \frac{1}{b}$$ $$a = \frac{k}{b}$$

Where \(k\) is some constant. If \(a=4\) when \(b=6\) then

$$4 = \frac{k}{6}$$ $$k = 24$$

so the equation is

$$a = \frac{24}{b}$$

If \(b = 8\) then

$$a = 24 \div 8 = 3$$

Level 3 Example

If \(a\) is directly proportional to the square of \(b\) and \(a=24\) when \(b=2\) find \(a\) when \(b=3\)

$$a \propto b^2$$ $$a = kb^2$$

Where \(k\) is some constant. If \(a=24\) when \(b=2\) then

$$24 = 2^2 \times k$$ $$k = 6$$

so the equation is

$$a = 6b^2$$

If \(b = 3\) then

$$a = 6 \times 3^2 = 54$$

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