Maths olympiad question · P5–P6★★★★ Expert
Work problems: olympiad question 10
An expert question on work problems for P5–P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P3 to P6 papers; harder ones have someone leaving partway, people taking turns, or cows grazing a field that keeps growing.
The question
At a concert at the Hung Hom Coliseum, fans queue at the entrance.
When the gates open there is already a queue outside, and after that the same number of new fans join the queue every minute.
Each gate lets the same number of fans in every minute.
With 15 gates open, the queue is gone exactly 16 minutes after opening; with only 12 gates, it takes 28 minutes.
Today only 1 gate is open at first;
2 minutes after opening the organisers open more.
To have the queue gone within 12 minutes of opening, at least how many more gates must they open?
Show the hints (3)
- Hint 1: Call what one gate lets in per minute 1 part. Compare the two cases: how many parts arrive a minute, and how many were waiting at opening?
- Hint 2: First find how many parts are still queuing after 2 minutes with 1 gate; people keep arriving after that.
- Hint 3: 8 parts arrive a minute and 112 were waiting. After 2 minutes 126 parts are still queuing, to be cleared in the remaining 10 minutes.
Worked solution
Have a go first, then read the solution step by step.
- 1.Let one gate's minute be 1 part. 15 gates for 16 minutes: 240 parts; 12 gates for 28 minutes: 336.
- 2.The extra 336 − 240 = 96 parts arrived in the extra 12 minutes, so 8 parts arrive a minute; 240 − 8 × 16 = 112 were waiting at opening.
- 3.First 2 minutes: 8 × 2 = 16 parts join, 1 × 2 = 2 go in, leaving 112 + 16 − 2 = 126 (with only 1, the gates can't keep up and the queue grows).
- 4.126 parts must clear in the remaining 10 minutes: 126 ÷ 10 = 12 remainder 6, so each minute the gates must let in at least 13 parts more than arrive (gates come whole): at least 8 + 13 = 21 gates.
- 5.There is 1 already, so at least 21 − 1 = 20 more.
- 6.Check: with 20 more, 210 parts go in and 80 arrive in 10 minutes, clearing 126 ✓; with 19 more, only 120 beyond arrivals, short of 126 ✗
- ⚠️Common mistake: forgetting the queue was still growing in the first 2 minutes and using the 112 parts at opening; or rounding 126 ÷ 10 down instead of up.
Answer: 20 gates
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