Problem 20 · 2024 Math Kangaroo
Hard
Ratios, Rates & Proportions
distance-speed-timework-backward
Two candles of equal length are lit at the same time. One candle will burn down completely in 4 hours, the other in 5 hours. Both burn at a constant rate. How many hours do they have to burn until one candle is exactly 3 times as long as the other?
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Answer: A — 4011
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Hint 1 of 2
Write each candle's remaining length as a fraction of time, then set one equal to three times the other.
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Hint 2 of 2
Solve (1 − t/5) = 3(1 − t/4) for the time t in hours.
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Approach: equation for remaining lengths
- After t hours the slow candle has fraction 1 − t/5 left, the fast one 1 − t/4.
- Set the longer equal to 3 times the shorter: 1 − t/5 = 3(1 − t/4).
- This gives t · (3/4 − 1/5) = 2, i.e. t · 11/20 = 2, so t = 40/11 hours.
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