Multiple choice

P, Q and R are three towns on a river which flows uniformly. Q is equidistant from P and R. I row from P to Q and back in 10 hours and I can row from P to R in 4 hours. Compare the speed of my boat in still water with that of the river.

  1. 4 : 3

  2. 5 : 3

  3. 6 : 5

  4. 7 : 3
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B Correct answer
Explanation

Let b be boat speed, r be river speed. P to Q is distance d. P to R is 2d. Time P to Q and back: d/(b-r) + d/(b+r) = 10. Time P to R: 2d/(b+r) = 4 => d/(b+r) = 2. Substitute: d/(b-r) + 2 = 10 => d/(b-r) = 8. Ratio (b+r)/(b-r) = 8/2 = 4. b+r = 4b-4r => 3b = 5r. b/r = 5/3.

AI explanation

Let the speed of the boat in still water be b and the speed of the river be s, and let the one-way distance from P to Q be d. Since P to R takes 4 hours, P to Q takes 2 hours downstream, so d divided by (b+s) equals 2. The round trip from P to Q and back takes 10 hours, so d divided by (b+s) plus d divided by (b-s) equals 10, which simplifies to 2 plus d divided by (b-s) equals 10, meaning d divided by (b-s) equals 8. Dividing the equation d equals 2(b+s) by the equation d equals 8(b-s) yields 1 equals one quarter of (b+s) divided by (b-s), resulting in the proportion 4b minus 4s equals b plus s, so 3b equals 5s. Thus, the ratio of the speed of the boat in still water to the speed of the river is 5:3.

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