Question

A motorboat whose speed is 15km/hr in still water, goes 30km downstream and comes back in a total of 4 hours and 30 minutes. Determine the speed of the stream.

Answer 1

Hint: We solve this problem by assuming the speed of stream as $u$and the speed of boat during the upstream and downstream. Then we use the formula for time, $\text{Time}=\dfrac{\text{Distance}}{\text{Speed}}$ to find the time taken by boat to travel 30kms in both the cases and add them to get the total time taken. Then we equate it to the given time and solve the obtained equation to solve and find the value of $u$, that is the speed of the stream.

Complete step by step answer:
Before solving this problem let us go through the variation of the speed of the boat during the upstream and downstream.
If a boat of speed $v$ is going in the upstream of speed $u$, it means that the boat is going in the opposite direction to the flow of the water. So, the stream opposes the moving of boats. So, the speed of the boat decreases. In that case the speed of boat changes to
$\Rightarrow \left( v-u \right)$
If a boat of speed $v$ is going in the downstream of speed $u$, it means that the boat is going in the same direction as the flow of the water. So, the stream helps the moving of the boat. So, the speed of the boat increases. In that case the speed of boat changes to
$\Rightarrow \left( v+u \right)$
Now we are given that the speed of the boat in still water is 15km/hr. It means that speed of the boat is
$v=15km/hr$
Let the speed of the stream be $u\text{ }km/hr$.
Then from above we can say that speed of the boat in upstream is
$\Rightarrow \left( v-u \right)=\left( 15-u \right)\text{ }km/hr$
Similarly, speed of boat in the downstream is
$\Rightarrow \left( v+u \right)=\left( 15+u \right)\text{ }km/hr$
We are given that the boat goes upstream and downstream to travel 30km and come back in 4 hours and 30 minutes.
So, now let us find the time taken by boat to travel 30kms downstream.
Let us now consider the formula of time.
$\text{Time}=\dfrac{\text{Distance}}{\text{Speed}}$
Using the above formula, we get
$\text{Time}=\dfrac{\text{30}}{15+u}$
Similarly, let us find the time taken by boat to travel 30kms upstream.
Using the above formula for time, we get
$\text{Time}=\dfrac{\text{30}}{15-u}$
As we are given that total time taken is 4 hours 30 minutes, we get
$\text{4hrs 30 min}=\dfrac{\text{30}}{15-u}+\dfrac{\text{30}}{15+u}................\left( 1 \right)$
Now let us convert the time given into hours. As 60 minutes is equal to 1 hour,
$\text{4hrs 30 min}=4+\dfrac{30}{60}=4+\dfrac{1}{2}=\dfrac{9}{2}hrs$
Now let substitute in the equation (1)
$\begin{align}
  & \Rightarrow \dfrac{9}{2}=\dfrac{\text{30}}{15-u}+\dfrac{\text{30}}{15+u} \\
 & \Rightarrow \dfrac{9}{2}=30\times \left( \dfrac{1}{15-u}+\dfrac{\text{1}}{15+u} \right) \\
 & \Rightarrow \dfrac{\text{15+}u+15-u}{\left( 15-u \right)\left( 15+u \right)}=\dfrac{9}{2\times 30} \\
 & \Rightarrow \dfrac{30}{\left( 15-u \right)\left( 15+u \right)}=\dfrac{3}{20} \\
\end{align}$
$\begin{align}
  & \Rightarrow 225-{{u}^{2}}=\dfrac{20\times 30}{3} \\
 & \Rightarrow 225-{{u}^{2}}=200 \\
 & \Rightarrow {{u}^{2}}=25 \\
 & \Rightarrow u=5 \\
\end{align}$

So, we get that speed of the stream is 5km/hr.

Note: The common mistake one makes while solving this type of problems is one might convert 4 hours 30 minutes to minutes and write it as 270 minutes and substitute it in equation (1). But we must remember that we have taken distance in km and speed in km/hr, so time should be in hours.