3.3 Inverse Proportion
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Two quantities are in inverse proportion if an increase in one causes a proportional decrease in the other (their product remains constant). If x and y are inversely proportional, x × y = constant, or x₁y₁ = x₂y₂.
Example 19: 12 men can finish a piece of work in 15 days. In how many days will 18 men finish the same work? Men and days are inversely proportional (more men → fewer days). 12 × 15 = 18 × d → d = 180/18 = 10 days.
Example 20: A car travelling at 60 km/h covers a certain distance in 5 hours. At what speed must it travel to cover the same distance in 4 hours? Speed and time are inversely proportional for a fixed distance: 60 × 5 = s × 4 → s = 300/4 = 75 km/h.
Example 20A: 15 workers can complete a wall in 8 days working 6 hours a day. How many days will 12 workers take to complete the same wall working 10 hours a day? Total work (man-hours) is constant: 15 × 8 × 6 = 12 × d × 10 → 720 = 120d → d = 6 days. (This “product = constant” logic extends the simple two-variable inverse proportion to three variables — workers, days, and hours per day — all inversely linked to the total work.)
The exam-friendly way to distinguish direct from inverse: direct proportion problems keep a ratio constant (x₁/y₁ = x₂/y₂); inverse proportion problems keep a product constant (x₁y₁ = x₂y₂). Reading the question once to identify “does more of this mean more of that, or less of that?” settles which formula applies before you write a single number.
Example 20B (additional): A pump fills a tank in 5 hours running at 800 litres/hour. If a bigger pump works at 1000 litres/hour, how long will it take to fill the same tank? Total tank volume = 5 × 800 = 4000 litres. Time for bigger pump = 4000/1000 = 4 hours.
Example 20C (additional, edge case with rotating wheels): A wheel of radius 20 cm makes 30 revolutions per minute to cover a certain distance. How many revolutions per minute must a wheel of radius 15 cm make to cover the same distance in the same time? Distance covered ∝ radius × number of revolutions, so for a fixed distance, radius and revolutions are inversely proportional: 20 × 30 = 15 × x → 600 = 15x → x = 40 revolutions per minute.