Question

: If a person is rowing at the rate of $5km/h$ in still water , takes thrice as much time in going $40km$upstream as going $40km$ downstream, then the speed of the stream is :
$\begin{gathered}
A. 7.5km/h \\
B. 2.5km/h \\
C. 3.5km/h \\
D. 1.5km/h \\
\end{gathered} $

Answer 1

Hint: This sum is from the chapter Time, Speed, Distance. It is the application of distance formula. Students should be thorough with the formula to be used while calculating speed upstream and downstream . Also keep in mind that Speed Upstream is $u - v$ and speed downstream is $u + v$. To There is only one way to solve this sum and that is equating with either time, speed or distance

Complete step-by-step answer:
Let the speed of boat in still water be $ukm/h$
Let the speed of the stream be $vkm/h$
Given that the speed of the Boat in still water is $5km/h$.
$\therefore u = 5km/h$
Time take for boat to go upstream is given by the formula $\dfrac{{Distance\ Travelled}}{{\operatorname{Re} lative\ Speed}}$
When the boat goes upstream, it is moving in the same direction as the current. Thus the relative speed would become subtraction of Speed of Boat in Still water and Speed of stream i.e. $u - v$
When the boat goes downstream, it is moving in the opposite direction as the current. Thus the relative speed would become addition of Speed of Boat in Still water and Speed of stream i.e. $u + v$

As per the given numerical, time taken for boat to travel $40km$ upstream is given by
\[ \Rightarrow \] ${T_u} = \dfrac{{40}}{{5 - v}}.............(1)$
time taken for boat to travel $40km$ downstream is given by
\[ \Rightarrow \] ${T_u} = \dfrac{{40}}{{5 + v}}.........(2)$
It is given that time taken to travel upstream is $3$hours more than time taken to travel downstream. Thus we can form the following relation between Equation $1$ & Equation $2$
\[ \Rightarrow \] $\dfrac{{40}}{{5 - v}} = 3 \times \dfrac{{40}}{{5 + v}}.............(3)$
Further simplifying Equation $3$
\[ \Rightarrow \] $\dfrac{1}{{5 - v}} = \dfrac{3}{{5 + v}}.............(4)$
Cross-multiplying the above equation we get
\[ \Rightarrow \] $5 + v = 3 \times (5 - v).............(5)$
\[ \Rightarrow \] $4v = 10........(6)$
\[ \Rightarrow \] $v = 2.5km/h$
Speed of the stream is. $2.5km/h$

Answer is Option $B - 2.5km/h$.

Note: This numerical is extremely easy to solve with only one formula involved. The student should keep In mind that it is always speed of boat minus speed of current and not vice versa. Also carefully read the statements before forming the equations.