Direct and Inverse Proportion, Set Up Without Guesswork

Two quantities are in direct proportion when one grows as the other grows, at a fixed rate, and in inverse proportion when one grows as the other shrinks. The questions are short, but students lose marks by guessing the relationship instead of writing it down. The fix is to turn the words into an equation with a constant, find the constant from the given pair of values, then use it.

Direct proportion

If y is directly proportional to x, then y = kx for some constant k. The graph is a straight line through the origin.

Worked example. y is directly proportional to x, and y = 12 when x = 3. Find y when x = 7.

First find k: 12 = k x 3, so k = 4. Then y = 4x, so when x = 7, y = 28. Two steps every time: find the constant from the known pair, then use the equation.

Inverse proportion

If y is inversely proportional to x, then y = k/x, so their product xy = k stays constant. As one doubles, the other halves.

Worked example. y is inversely proportional to x, and y = 6 when x = 4. Find y when x = 8.

Find k: k = xy = 4 x 6 = 24. Then y = 24/x, so when x = 8, y = 24/8 = 3. Notice x doubled from 4 to 8 and y halved from 6 to 3, which is the signature of inverse proportion and a quick sense-check.

Telling them apart

The wording is the clue, but the surer test is the behaviour: if the two quantities move the same way, more with more, it is direct, so y = kx; if they move opposite ways, more with less, it is inverse, so y = k/x. Some questions involve a square or a square root, for example y proportional to x2, and the same method holds: write y = kx2, find k, then use it. The relationship can change shape, but the two-step method does not.

The teaching point

Proportion is never guesswork. Decide direct or inverse from whether the quantities move together or apart, write the equation with a constant k, find k from the given values, and only then answer the question. The single most common error is mixing the two up, using y = kx where the quantities actually move in opposite directions. Naming the relationship before computing is the same diagnose-first discipline GPA carries from PSLE proportion into O-Level.

If your child can do the arithmetic but sets up the wrong relationship, that is a precise, fixable gap. GPA's programmes work on naming the structure first, from PSLE through O-Level. A short diagnostic consult will show where the setup goes wrong.

Every worked value on this page was checked independently before publishing.