Two quantities can change together in two ways: both grow, or one grows while the other shrinks. Deciding which is the whole battle.
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Ask one question: if the first quantity doubles, does the second double or halve?
↔️ Telling them apart
| DIRECT | INVERSE | |
|---|---|---|
| Behaviour | both rise together | one rises, the other falls |
| Rule | a/b stays constant | a × b stays constant |
| Method | x₁/y₁ = x₂/y₂ | x₁y₁ = x₂y₂ |
| Examples | items and cost; distance and petrol | workers and days; speed and time |
The trap: "more workers, fewer days" is inverse, not direct. Students multiply when they should divide. Say the sentence aloud — if more of one gives less of the other, it is inverse.
Work problems are inverse proportion with one extra idea: think in work done per day.
👷 The one-day method
If A finishes a job in 12 days, in one day A does 1/12 of it.
If B takes 6 days, in one day B does 1/6.
Together in one day they do 1/12 + 1/6 = 3/12 = 1/4, so together they take 4 days.
Sense check: two people working together must take less time than either alone. If your answer is bigger than the faster person’s time, you have made a mistake.
One triangle of formulas covers every question here.
🚗 The three forms
| To find | Use |
|---|---|
| Speed | distance ÷ time |
| Distance | speed × time |
| Time | distance ÷ speed |
Match your units. To turn km/h into m/s, multiply by 5/18. To go the other way, multiply by 18/5. So 72 km/h = 72 × 5/18 = 20 m/s.
Average speed is not the average of the speeds. It is total distance ÷ total time. Going 60 km/h one way and 40 km/h back does not average 50.
12 questions on proportion, work and speed. Ask "does more mean more?" first. 🍀
