Direct and Inverse Proportion
Two quantities are in direct proportion if they increase or decrease together at the same rate, so their ratio stays constant - for example y = kx. They are in inverse proportion if one increases as the other decreases, so their product stays constant - for example y = k/x.
Method
- Decide whether the quantities are in direct or inverse proportion from the context - do they rise together, or does one fall as the other rises?
- For direct proportion, write y = kx, then use a known pair of values to find the constant k.
- For inverse proportion, write y = k/x, then use a known pair of values to find the constant k.
- Once k is known, substitute the new value you're given into the formula and solve for the unknown.
- For proportion involving squares or cubes, such as y proportional to x^2, use the same method but with y = kx^2 or y = k/x^2.
Worked example
y is directly proportional to x. When x = 5, y = 20. Find y when x = 8.
- Write the proportion statement as a formula: y = kx.
- Substitute the known pair of values: 20 = k x 5.
- Solve for k: k = 20 divided by 5 = 4.
- The formula is y = 4x.
- Substitute x = 8: y = 4 x 8 = 32.
Practice questions
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Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
The pressure, P, of a gas is inversely proportional to its volume, V. When V = 8, P = 15. Find P when V = 12.
The distance a stone falls, d metres, is directly proportional to the square of the time taken, t seconds. The stone falls 45 m in 3 seconds. Work out how far the stone falls in 5 seconds.
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