Multiple choice

A boatman takes his boat in a river against the stream from A to B, where AB = 21 km, and again returns to A. He takes 10 hours in all. The time taken by him to cover 7 km going downstream is equal to the time taken by him to cover 3 km going against the stream. Find the velocity of the river.

  1. 2 kmph

  2. 5 kmph

  3. 3 kmph

  4. 7 kmph

  5. None of these
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A Correct answer
Explanation

Let v be the boat speed and u be the river speed. Time downstream for 7 km is 7/(v+u), time upstream for 3 km is 3/(v-u). Given 7/(v+u) = 3/(v-u), we get 7v-7u = 3v+3u, so 4v = 10u or v = 2.5u. Total time 21/(v-u) + 21/(v+u) = 10. Substituting v=2.5u: 21/(1.5u) + 21/(3.5u) = 10. 14/u + 6/u = 10, so 20/u = 10, u = 2.

AI explanation

The condition that covering 7 km downstream takes the same time as 3 km upstream gives the ratio of downstream speed to upstream speed as 7 to 3. Let the speed of the boat in still water be b and the stream velocity be s, meaning b plus s divided by b minus s equals 7 divided by 3, which simplifies to 3b plus 3s equals 7b minus 7s, yielding b equals 2.5s. The total time for the 21 km round trip is 21 divided by b plus s plus 21 divided by b minus s equals 10 hours, and substituting b equals 2.5s gives 21 divided by 3.5s plus 21 divided by 1.5s equals 10. Solving 6 divided by s plus 14 divided by s equals 10 results in 20 divided by s equals 10, so the velocity of the river is 2 kmph.

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