Ratio, proportion and rates of change Study Guide

Two quantities are in direct proportion when multiplying one by a constant multiplies the other by the same constant. The relationship can be written y = kx, where k is the constant of proportionality. Find k from a known pair, then use it to calculate missing values.

If 5 notebooks cost £12, the cost per notebook is k = 12 ÷ 5 = £2.40. Therefore C = 2.4n, and 8 notebooks cost 2.4 × 8 = £19.20. A direct-proportion graph is a straight line through the origin because zero of one quantity corresponds to zero of the other.

A strong solution states the relevant rule, substitutes or transforms values carefully, shows intermediate reasoning, and checks that the result is reasonable and in the required form. Common errors come from omitted terms, mishandled signs, premature rounding or using a rule that does not match the structure of the problem.

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Higher direct-proportion problems may involve powers, roots or several variables. Translate the wording accurately: y directly proportional to x² means y = kx², while y directly proportional to √x means y = k√x. Determine k from known values before solving or comparing cases.

If y ∝ x² and y = 45 when x = 3, then 45 = 9k, so k = 5 and y = 5x². When x = 7, y = 245. For percentage changes, if x is multiplied by a factor r, then x² and therefore y are multiplied by r².

A strong solution states the relevant rule, substitutes or transforms values carefully, shows intermediate reasoning, and checks that the result is reasonable and in the required form. Common errors come from omitted terms, mishandled signs, premature rounding or using a rule that does not match the structure of the problem.