A hard question on ratios for P6, with a step-by-step solution, diagrams and the answer. Ratio questions appear in the Hua Xia Cup's P4 to P6 first rounds, qualifying rounds and finals, such as linked ratios and ratios that change.
The question
Every Saturday Yan cycles the same cycle track from start to end.
Last Saturday Yan rode 270 metres a minute; this Saturday, into the wind, only 180 metres a minute, and took 18 minutes longer.
Both times the speed stayed the same the whole way.
How long is the cycle track from start to end?
Show the hints (3)
💡Hint 1: The distance is the same both times. If the speed goes down, what happens to the time? How are the two ratios related?
💡Hint 2: The speeds are 270 : 180 = 3 : 2; with the same distance the times go the other way round: 2 : 3.
💡Hint 3: The times differ by 1 part, which is 18 minutes.
Worked solution
Have a go first, then read the solution step by step.
1.Speeds last Saturday and this Saturday: 270 : 180 = 3 : 2.
2.Distance = speed × time, and the distance is the same both times. The speeds are 3 and 2 parts; with times of 2 and 3 parts, both distances are 3 × 2. So the times go the other way round: last Saturday and this Saturday are 2 : 3.
3.Last Saturday it took 2 parts of time and this Saturday 3: 1 part apart, which is 18 minutes. 1 part = 18 ÷ 1 = 18 minutes.
4.Last Saturday Yan took 2 × 18 = 36 minutes.
5.Distance = 270 × 36 = 9720 metres.
6.Check: this Saturday took 9720 ÷ 180 = 54 minutes, 18 minutes more ✓
⚠️Common mistake: taking the times as 3 : 2 too. Over the same distance, slower means more time, so the time ratio is turned round.
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