Three students, A, B, and C, start from point P at the same time on their bicycles to go to point Q. A reaches point Q first, turns back, and meets B at a distance of 5 km from Q. When B reaches Q, he also turns back and meets C at a distance of 3 km from Q. If A’s speed is three times C’s speed, what is the distance (in km) between points P and Q?
Question
Three students, A, B, and C, start from point P at the same time on their bicycles to go to point Q. A reaches point Q first, turns back, and meets B at a distance of 5 km from Q. When B reaches Q, he also turns back and meets C at a distance of 3 km from Q. If A’s speed is three times C’s speed, what is the distance (in km) between points P and Q?
Solution
Let's denote the distance between points P and Q as D.
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When A meets B, A has traveled the distance D (from P to Q) plus 5 km (from Q back towards P), while B has only traveled D - 5 km (from P towards Q).
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When B meets C, B has traveled the distance D (from P to Q) plus 3 km (from Q back towards P), while C has only traveled D - 3 km (from P towards Q).
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Given that A's speed is three times C's speed, the time it takes for A to travel D + 5 km is the same as the time it takes for C to travel D - 3 km.
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Similarly, the time it takes for B to travel D + 3 km is the same as the time it takes for C to travel D - 3 km.
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Therefore, we can set up the following equation: D + 5 = 3(D - 3).
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Solving this equation gives D = 10 km.
So, the distance between points P and Q is 10 km.
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