Maths olympiad question · P6★★★★ Expert
Ratios: olympiad question 9
An expert question on ratios for P6, with a step-by-step solution 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
Ka-lok and Tsz-ching set out at the same time from A and B, walking towards each other on the same road.
Ka-lok covers 30 m a minute and Tsz-ching 24 m.
After they meet, Ka-lok speeds up to 80 m a minute and Tsz-ching to 60 m, and both carry on.
When Ka-lok reaches B, Tsz-ching is still 960 m from A.
How far apart are A and B, in metres?
Show the hints (3)
- Hint 1: Before they meet they walk for the same time, so their distances are in the ratio of their speeds. In what ratio does the meeting point split the road?
- Hint 2: After meeting, Ka-lok has to cover the stretch Tsz-ching walked; in that time Tsz-ching moves on at the new speed ratio.
- Hint 3: The meeting point splits the road 5 : 4 (9 parts). After it the speeds are 4 : 3: while Ka-lok covers 4 parts, Tsz-ching covers 3.
Worked solution
Have a go first, then read the solution step by step.
- 1.Before meeting the speeds are 30 : 24 = 5 : 4, so call the road 9 parts: Ka-lok walked 5, Tsz-ching 4.
- 2.After meeting the speeds are 80 : 60 = 4 : 3. Ka-lok has 4 parts to B; meanwhile Tsz-ching covers 4 × 3 ÷ 4 = 3 parts.
- 3.Tsz-ching has 5 parts to A after meeting and is 5 − 3 = 2 parts short, which is 960 m: 1 part = 960 ÷ 2 = 480 m.
- 4.The road = 9 × 480 = 4320 m.
- 5.Check: they meet after 4320 ÷ (30 + 24) = 80 minutes, Ka-lok having walked 2400 m; the remaining 1920 m take 24 minutes, in which Tsz-ching walks 60 × 24 = 1440 m, leaving 2400 − 1440 = 960 m to A ✓
- ⚠️Common mistake: keeping the old ratio 5 : 4 after the meeting, or taking Tsz-ching's remaining way as 4 parts. After meeting, each must cover the stretch the other already walked.
Answer: 4320 m
Play more like this in Numsprite (unlimited questions, instant marking) →A free maths game: every level free from Easy to Expert (★ to ★★★★), unlimited questions, no download