Problem 24 · 2017 Math Kangaroo
Stretch
Ratios, Rates & Proportions
distance-speed-time
Every three minutes a bus leaves the airport to drive to the city centre. A car leaves the airport at the same time as a bus and travels the same route to the city centre. Every bus takes 60 minutes for the journey from the airport to the city centre; the car takes only 35 minutes. How many buses does the car overtake on its way to the city centre? (The bus that starts at the same time as the car does not count.)
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Answer: A — 8
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Hint 1 of 2
Send the car and a bus from the airport together and find when the car draws level with each earlier bus.
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Hint 2 of 2
A bus counts only if the car catches it before either reaches the city.
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Approach: catch-up time for each earlier bus
- Let the car leave at time 0; a bus that left 3k minutes earlier is caught when the car has run for 4.2k minutes.
- This catch must happen before the bus arrives: 4.2k + 3k ≤ 60 gives k ≤ 8.
- So the car overtakes buses for k = 1..8, that is 8 buses.
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