Question

A motor boat whose speed is 15${\displaystyle \frac{\text{km}}{\text{hr}}}$ in still water, goes 30km downstream and comes back in a total of 4hours 30minutes. Determine the speed of the stream.

Answer 1

Hint: In downstream, the boat goes with the water-flow so we add both the speeds of boat and water and in upstream, boat goes against the direction of water-flow so we subtract both the speeds. Then we use the formula for speed to find the answer.

Complete step-by-step answer:

Let the speed of the stream be x${\displaystyle \frac{\text{km}}{\text{hr}}}$.
Then,
Speed downstream = (15 + x) ${\displaystyle \frac{\text{km}}{\text{hr}}}$
Since the still water is going downstream. The speeds add up.

Speed upstream = (15 - x) ${\displaystyle \frac{\text{km}}{\text{hr}}}$
Since the still water is going downstream. The speeds are subtracted.

 Speed = ${\displaystyle \frac{\text{total distance covered}}{\text{total time taken}}}$

Total time = ${\displaystyle \frac{\text{total distance covered}}{\text{speed}}}$

Given distance d = 30km
Total time t = 4${\displaystyle \frac{1}{2}}$hours

Total time = Time take to travel downstream + Time taken to travel upstream
⟹${\displaystyle \frac{30}{\text{15 + x}}}+{\displaystyle \frac{30}{\text{15 - x}}}=4{\displaystyle \frac{1}{2}}$
$$\begin{gathered} \Rightarrow {\displaystyle \frac{\left[30\left(15-\text{x}\right)+30\left(\text{15 + x}\right)\right]}{\left(\text{15 + x}\right)\left(\text{15 - x}\right)}}={\displaystyle \frac{9}{2}} \\ \Rightarrow {\displaystyle \frac{\left[450-30\text{x + 450 + }30\text{x}\right]}{225+15\text{x - 15x - }{\text{x}}^2}}={\displaystyle \frac{9}{2}} \end{gathered}$$
$$\begin{gathered} \Rightarrow {\displaystyle \frac{900}{225-{\text{x}}^2}}={\displaystyle \frac{9}{2}} \\ \Rightarrow {\displaystyle \frac{100}{225-{\text{x}}^2}}={\displaystyle \frac{1}{2}} \\ \Rightarrow 200=225-{\text{x}}^2 \\ \Rightarrow {\text{x}}^2=25 \\ \Rightarrow \text{x = 5}{\displaystyle \frac{\text{km}}{\text{hr}}} \end{gathered}$$
Hence, the speed of the stream is 5${\displaystyle \frac{\text{km}}{\text{hr}}}$.

Note: In order to solve this type of questions the key is to identify that in downstream both the speeds are added up and when in upstream both the speeds are subtracted. Using the formula for speed is the next step we proceed with. Then we equate the total time and individual times taken to calculate the answer.