Chapter 10: Direct and Inverse Proportions

Two quantities are in direct proportion when they rise and fall together at the same rate; they're in inverse proportion when one rises as the other falls, keeping their product constant. Direct and Inverse Proportions is about recognising which relationship a situation actually involves before solving it — more workers finishing a job faster is inverse; more hours worked earning more pay is direct.

Applying direct-proportion cross-multiplication to a genuinely inverse situation — or the reverse — is the single most common mistake, and checking first whether increasing one quantity should increase or decrease the other avoids it entirely. This chapter builds directly on the ratio work from Comparing Quantities Using Proportion, and reappears constantly in speed-time-distance and work-and-workers problems later on.