Question

# A man's speed with the current is 15 km/hr. and the speed of the current is 2.5 km/hr. The mans speed against the current is:\begin{align} & \text{A}.\text{ 8}.\text{5 km}/\text{hr}. \\ & \text{B}.\text{ 9 km}/\text{hr}. \\ & \text{C}.\text{ 1}0\text{ km}/\text{hr}. \\ & \text{D}.\text{ 12}.\text{5 km}/\text{hr}. \\ \end{align}

Hint: Take manâ€™s speed in still water as u km/hr and speed of stream as v km/hr. From given relative speed, find the speed of man in still water by using relation u+v=15 km/hr. Then find the speed of man against the current using the relation u-v.

Complete step-by-step solution:
Let us first look into some terms we will be using in the solution. In water, the direction along the stream is called downstream or speed with the current/water and the direction against the stream is called upstream or against the current/water.
If the speed of a man in still water is u km/hr. and the speed of the stream is v km/hr., then we can formulate the below relations:
Speed with current = (u+v) km/hr. or downstream
Speed against current = (u-v) km/hr. or upstream
We have been given man's speed with the current is 15 km/hr. and the speed of the current is 2.5 km/hr.
Therefore, we will substitute the known values from the question in relation as follows,
Man's speed (with current) = u+v = 15 km/hr. and v = 2.5 km/hr.
Putting the value of v speed of current which is given in question in the above formula, we get the value of manâ€™s actual speed or speed of man in still water i.e. u
\begin{align} & 15=u+v\Rightarrow 15=u+2.5 \\ & \Rightarrow u=15-2.5 \\ & \Rightarrow u=12.5km/hr \\ \end{align}
Solution is not complex here, we actually need to find the speed of man against the current of water. We have both values u and v, so we can put these values in above stated formula for speed of man against the current which is
Speed against the current of water = u-v
Now we know u = 12.5 km/hr and v = 2.5 km/hr
Putting these values in the formula u-v
\begin{align} & \Rightarrow u-v \\ & \Rightarrow \text{12}\text{.5-2}\text{.5} \\ & \Rightarrow \text{10km/hr} \\ \end{align}
Therefore, the speed of man while going against the current/against the stream is 10 km/hr.
Relative speed against current = 10km/hr.
Therefore, option C is correct.

Note: The speed of man with the flow of water or with the current and against the current are relative, these speeds must not be taken as the actual speed of man. If given the man's speed is still water then, we can take this speed as manâ€™s actual speed. In going with the current, speed always increases but going against the current, speed always decreases.