Multiple choice

A boat covers a certain distance between two spots in a river taking $t_1$ hrs going downstream and $t_2$ hrs going upstream. What time will be taken by boat to cover same distance in still water ?

  1. $\dfrac{t_1+t_2}{2}$
  2. $2(t_2=t_1)$
  3. $\dfrac{2 t_1 t_2}{t_1+t_2}$
  4. $\sqrt{t_1 t_2}$
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C Correct answer
Explanation

Let d be the distance, v be boat speed, u be stream speed. t1 = d/(v+u) and t2 = d/(v-u). Then 1/t1 = (v+u)/d and 1/t2 = (v-u)/d. Adding these: 1/t1 + 1/t2 = 2v/d. So v/d = (t1+t2)/(2*t1*t2). The time in still water is d/v = (2*t1*t2)/(t1+t2).

AI explanation

Let the distance be D, the boat's speed in still water be B, and the stream's speed be S, so the downstream speed is D/t1 = B + S and the upstream speed is D/t2 = B - S. By adding these two equations, we get (D/t1) + (D/t2) = 2B, which simplifies to B = (D/2)(1/t1 + 1/t2). The time taken by the boat to cover distance D in still water is T = D/B, which becomes 2 / (1/t1 + 1/t2) after substituting B. Simplifying this fraction yields the harmonic mean formula, 2 * t1 * t2 / (t1 + t2). The result is \dfrac{2 t_1 t_2}{t_1+t_2}.

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