Ratio, proportion and rates of change: understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y; construct and interpret equations that describe direct and inverse proportion (Higher tier)

The idea

Higher tier only. The key equivalence is that X inversely proportional to Y means X is proportional to 1/Y, so X = k/Y, equivalently XY = k. Direct proportion gives X = kY; inverse proportion gives X = k/Y. For example, if y is inversely proportional to x and y = 5 when x = 2, then k = 5 × 2 = 10, so y = 10/x; when x = 5, y = 2. Powers and roots extend this: y = kx², y = k√x, y = k/x². For instance, if y is inversely proportional to x² and y = 5 when x = 2, then k = 5 × 2² = 20, so y = 20/x²; when x = 5, y = 0.8. Always write the finished formula down, as later parts often need it, sometimes rearranged to find x from a given y.

What earns the marks

• A correct proportion statement or equation containing k, matching the power or root named, such as y = k/x for simple inverse proportion.
• Using the equivalence X inversely proportional to Y ⇔ X proportional to 1/Y to write X = k/Y.
• Substituting the given pair of values to form an equation in k.
• The value of k, even if the answer that follows is wrong.
• Using the completed formula on the part asked, including rearranging it when the other variable is the unknown.

Common mistakes and how to put them right

• Writing y = kx for inverse proportion instead of y = k/x, so the variables move in the same direction rather than opposite directions.

• Squaring the whole product instead of the variable, writing y = (kx)² where the question says y is proportional to the square of x.

• Finding k and stopping there, without going back for the value the question actually wanted.

• Writing y = k/x where the relationship is inverse to the square of x, so dividing by x rather than by x².

• Rounding k partway through, which pushes the final answer outside the accepted range.

Exam tips

• Read the wording closely: the square of x means x², while the square root of x means √x, and the two give different formulas.
• Keep k exact, as a fraction if it will not come out neatly, and round only the final answer.
• Check the formula by putting the original pair of values back in; it should return the value you started with.