Question

A motorboat whose speed is 9 km/hr in still water, goes 15 km downstream and comes back in a total time of 3 hours 45 minutes. Find the speed of the stream.
(a) 3 km/hr
(b) 9 km/hr
(c) 6 km/hr
(d) 2 km/hr

Answer 1

Hint: To solve this problem involving algebraic expressions, we will let the speed of the motorboat be x and the speed of the stream be y. Now, we will use the fact that when the motorboat goes upstream, the speed of the boat is (x-y) km/hr while when the motorboat goes downstream, the speed of the boat is (x+y) km/hr. We will then add up the time of their journeys upstream and downstream and equate it to 3 hours 45 minutes, to get the speed of the stream (y).

Complete step-by-step solution -

To solve the above problem, we understand the conditions given in the above problem. The speed of a motorboat in still water is 9 km/hr (x). Let the speed of the stream be y. Now, the speed of the motorboat downstream would be x+y and the speed of the motorboat upstream would be x-y (where, x = 9). Now, we are given that the motorboat goes 15 km downstream and comes back (in upstream direction) in a total time of 3 hours 45 minutes (which is $\dfrac{15}{4}$ hours). Thus, we use the formula $\text{Time = }\dfrac{\text{Distance}}{\text{Speed}}$
We have,
$\dfrac{15}{x+y}+\dfrac{15}{x-y}=\dfrac{15}{4}$
Now, x = 9, thus,
$\begin{align}
  & \dfrac{15}{9+y}+\dfrac{15}{9-y}=\dfrac{15}{4} \\
 & \dfrac{1}{9+y}+\dfrac{1}{9-y}=\dfrac{1}{4} \\
 & \dfrac{9-y+9+y}{(9+y)(9-y)}=\dfrac{1}{4} \\
 & \dfrac{18}{(9+y)(9-y)}=\dfrac{1}{4} \\
\end{align}$
(9+y) (9-y) = 72
81 - ${{y}^{2}}$ = 72
${{y}^{2}}$ = 9
y = 3 (y is the speed of the stream and thus cannot have negative value)
Thus, the speed of the stream is 3 km/hr.
Hence, the correct answer is (a) 3 km/hr.

Note: In most of the algebraic problems, it is important to convert the problem in English into mathematical equations. Further, while solving the equations, it is important only to pick the useful solutions from the values of variables we get. For example, in this problem, we rejected y = -3 as the solution since the speed of the stream cannot have a negative value.