Question

A streamer goes downstream from one port to another in 9 hours. It covers the same distance upstream in 10 hours. If the speed of the stream be 1 km/h, find the speed of the streamer in still water and the distance between the ports.

Answer 1

Hint: Focus on the point that while a boat covers in downstream, the speed of the current is added to the speed of the streamer while in case of the upstream, speed of the current is subtracted from that of the streamer.

Complete step-by-step answer:
Let’s start with what is speed. Speed is a scalar quantity defined as the distance travelled by a particle or object per unit time.
Generally, we deal with two kinds of speeds. One is instantaneous, and the other is the average speed. For uniform motion, both are identical.
Average speed is defined as the total distance covered by a body divided by the time taken by the body to cover it.
$\therefore {{v}_{avg}}=\dfrac{\text{distance covered}}{\text{time taken}}$
$\Rightarrow \text{time taken}=\dfrac{\text{distance covered}}{{{v}_{avg}}}$
Now, starting with the solution to the above question. Let the distance between the ports be x km and speed of streamer be y km/h.
It is given in the question that when a boat covers the distance between the ports upstream, it takes a total of 10 hours to complete the journey.
We know $\text{time taken}=\dfrac{\text{distance covered}}{{{v}_{avg}}}$ , so, we get
$\text{time taken}=\dfrac{\text{distance covered upstream }}{{{v}_{avg}}-{{v}_{current}}}$
$\Rightarrow 10=\dfrac{x}{y-1}............(i)$
It is also given in the question that when a boat covers the distance between the ports downstream, it takes a total of 9 hours to complete the journey.
$\text{time taken}=\dfrac{\text{distance covered downstream }}{{{v}_{avg}}+{{v}_{current}}}$
$\Rightarrow 9=\dfrac{x}{y+1}...........(ii)$
Now if we divide equation (i) by equation (ii), we get
$\dfrac{10}{9}=\dfrac{\dfrac{x}{y-1}}{\dfrac{x}{y+1}}$
$\Rightarrow \dfrac{10}{9}=\dfrac{y+1}{y-1}$
On cross-multiplication, we get
$9y+9=10y-10$
$\Rightarrow y=19\text{ km/hr}\text{.}$
Now we will substitute the value of y in equation (i). So, we get
$10=\dfrac{x}{19-1}$
\[\Rightarrow x=180\text{ km}\]
Therefore, the speed of the streamer is 19 km/hr, and the distance between the two ports is 180 km.

Note: Always try to keep the quantities according to a standardised unit system; this helps you to solve the question in an error-free manner. Also, it is prescribed to write each and every statement given in the question in mathematical form as it ensures that you are not missing any information given in the question.