Direct proportion means both quantities rise and fall together in the same ratio, written y = kx. Inverse proportion means one quantity rises while the other falls, written y = k / x. To tell them apart, ask what happens to the second quantity as the first one increases: if it rises too, it's direct; if it falls, it's inverse.
Both types of question give you a pair of related quantities and ask you to find a missing value once the relationship changes. The setup looks almost identical on the page, so many students default to whichever formula they used last, without checking that the situation actually matches.
The fix is not to memorise more formulas. It is to read the context sentence carefully and decide, before writing anything, whether the two quantities move the same way or opposite ways. A map scale is a familiar example of direct proportion in this sense: real length and map length always rise together in the same fixed ratio.
Step 1
Name the two quantities, x and y
Step 2
Ask: as x increases, does y increase or decrease?
Step 3
Rises together = direct, y = kx. Opposite ways = inverse, y = k/x
Step 4
Substitute one known pair to find k, then answer the question
The cost of ribbon, C dollars, is directly proportional to its length, L metres. 4 metres of ribbon costs $6.00. Find the cost of 10 metres.
Solution
C = kL.6.00 = k × 4, so k = 1.50.C = 1.50L.C = 1.50 × 10 = $15.00.Quick check: length went from 4 m to 10 m, a factor of 2.5. Cost should also scale by 2.5: 6.00 × 2.5 = 15.00. Matches, so the answer is correct.
The time, T hours, taken to fill a tank is inversely proportional to the number of pumps used, P. 3 pumps take 8 hours to fill the tank. Find the time taken using 6 pumps.
Solution
T = k / P.8 = k / 3, so k = 24.T = 24 / P.T = 24 / 6 = 4 hours.Quick check: the number of pumps doubled from 3 to 6, so the time should halve: 8 ÷ 2 = 4. Matches, so the answer is correct.
Some O-Level questions extend the idea: y directly proportional to x² is written y = kx², and y inversely proportional to √x is written y = k / √x. The same-way test still applies: check what happens to the power of x, not x itself, when deciding direct or inverse. The method for finding k and substituting back is unchanged. If you want to picture what these relationships look like on a grid, see the guide on sketching y = axⁿ graphs.
Example (y is directly proportional to x squared)
y = kx².18 = k × 3² = 9k, so k = 2.y = 2 × 5² = 2 × 25 = 50.The step students get wrong
Reading "y is proportional to x²" and writing y = kx by habit, forgetting the squared term entirely. Always copy the power exactly as stated in the question into your equation before substituting any numbers. The same slip happens in reverse with inverse proportion: "inversely proportional to x²" must go into the denominator as x², not x.
The number of workers, W, needed to build a wall is inversely proportional to the number of days, D, allowed. 12 workers can finish in 15 days. How many workers are needed to finish in 9 days?
Solution
W = k / D.12 = k / 15, so k = 180.W = 180 / 9 = 20 workers.How do I know without a doubt whether a question is direct or inverse?
Read the sentence connecting the two quantities and ask what happens if one increases: if the other also increases in the same ratio, it's direct proportion. If the other decreases in a matching ratio, it's inverse proportion. Do not rely on the word "proportional" alone. Always check whether "directly" or "inversely" is stated, and if neither is stated, reason from the real-world context.
Can a quantity be neither directly nor inversely proportional to another?
Yes. Many real-life relationships are neither, such as a fixed delivery charge plus a per-item cost, which is why rate questions that combine a fixed part with a per-unit part are not pure proportion. O-Level questions that use the word "proportional" are testing direct or inverse specifically, but not every relationship in the syllabus is one of these two types, so always check the wording rather than assuming.
Is direct proportion the same as a straight-line graph through the origin?
Yes. If y is directly proportional to x, plotting y against x gives a straight line through the origin with gradient k. Inverse proportion instead gives a curve, the reciprocal graph shape, since y = k / x is not linear.
How is this different from reverse percentage or average speed questions?
Reverse percentage asks you to find an original value before a percentage change was applied, which is a different skill from spotting a proportional relationship. Average speed mistakes usually come from treating a rate as if it were directly proportional when the correct approach uses total distance over total time. Both are worth reviewing separately once you are confident with direct and inverse proportion.
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Mr. Gan works with students who want a reliable test for every proportion question, not a coin flip between two formulas.
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