Question

A ship travels downstream from A to B in two hours and upstream in three hours. Then the time that it will take a log of wood to cover the distance from A to B is:
(A) 12 hr
(B) 7 hr
(C) 5 hr
(D) 1 hr

Answer 1

Hint
Simply use the formula distance = speed $\times$time for upstream as well as downstream journey to calculate the relationship between the velocity of ship and wood. Once this is done, apply the formula distance = speed $\times$time again to get the answer.

Complete step by step answer
Given, When the ship covers the downward stream of journey, time taken = 2 hours
When the ship covers the upward stream of journey, time taken = 3 hours.
Let the distance be x.
Hence for upward journey, using the formula distance = speed $\times$time
$\Rightarrow \dfrac{x}{{{V_s} + {V_w}}} = 2$
And for downward journey, using the formula distance = speed $\times$time we have,
$\Rightarrow \dfrac{x}{{{V_s} - {V_w}}} = 3$
Dividing both the equations we have,
$\Rightarrow \dfrac{{\dfrac{x}{{{V_s} + {V_w}}}}}{{\dfrac{x}{{{V_s} - {V_w}}}}} = \dfrac{2}{3}$
$\Rightarrow \dfrac{{{V_s} - {V_w}}}{{{V_s} + {V_w}}} = \dfrac{2}{3}$
On further solving we have,
$\Rightarrow 2{V_s} + 2{V_w} = 3{V_s} - 3{V_w}$
$\Rightarrow {V_s} = 5{V_w}$
Hence,
Putting the value of ${V_s} = 5{V_w}$ in $\dfrac{x}{{{V_s} + {V_w}}} = 2$ we have,
$\Rightarrow \dfrac{x}{{5{V_w} + {V_w}}} = 2$
$\Rightarrow \dfrac{x}{{{V_w}}} = 12{\text{hr}}{\text{.}}$
Hence the speed of the wood is 12hr.
And the correct answer is option (A).

Note
i) One alternative way to solve this problem is by calculating the distance velocity relationship. Although we somewhat did the same but replacing $\dfrac{x}{{{V_w}}}$ with time could have been better for the underlying purposes. Nevertheless, the solution is still correct.
ii) If complex equations like $\dfrac{{{V_s} - {V_w}}}{{{V_s} + {V_w}}} = \dfrac{2}{3}$ arises and let’s say it can’t be solved directly, then we use componendo and dividendo to solve this kind of equations.