Question

A man can row a boat in still water at the rate of 6km per hour. If the stream flows at the rate of 2km/hour, it takes half the time going downstream than going upstream the same distance. His average speed for an upstream and downstream trip is

A) 6km/hour

B) \[\dfrac{16}{3}\] km/hour

C) Insufficient data to arrive at the answer

D) None of the above

Answer 1

Hint: Assume the distance that the boat covers be $x$. The speed of the boat while going upstream will be 4km/hr and the speed of the boat while going downstream will be 8km/hr. Now, calculate the time for the distance x while going upstream using the formula, \[\text{time=}\dfrac{\text{distance}}{\text{speed}}\] . Similarly, calculate the time for distance x while going downstream. We know the formula of average speed, 

\[\text{Average}\,\text{speed=}\dfrac{\text{Total distance covered}}{\text{Total time taken}}\] . Here, the total distance is 2x and the total time is the summation of time while going upstream and time while going downstream. Now, solve further and get the answer.

Complete step-by-step answer:

According to the question, it is given that the boat covers the same distance while going upstream and while going downstream.

The speed of the boat = 6km/hr.

The speed of the stream = 2km/hr.

Let us assume the distance while going upstream and while going downstream to be x km.

We know the formula, \[\text{time=}\dfrac{\text{distance}}{\text{speed}}\] .

The speed of the boat while going upstream = (6 – 2) = 4km/hr.

Time taken while going upstream = \[\dfrac{\text{distance}}{\text{speed}}\] = \[\dfrac{x}{4}\] hr ……………………(1) 

The speed of the boat while going upstream = (6 + 2) = 8km/hr.

Time taken while going upstream = \[\dfrac{\text{distance}}{\text{speed}}\] = \[\dfrac{x}{8}\] hr ……………………(2) 

The boat covers the distance x km while going upstream and the distance x while going downstream. So, for a whole trip, the boat covers the distance 2x km.

Total time taken by the boat = Time taken while going upstream + Time taken while going upstream ………………….(3)

From equation (1), equation (2), and equation (3), we have

Total time taken by the boat = \[\dfrac{x}{4}+\dfrac{x}{8}\] = \[\dfrac{3x}{8}\] hr…………………..(4) 

Total distance covered by the boat = 2x km…………………..(5)

We know the formula, \[\text{Average}\,\text{speed=}\dfrac{\text{Total distance covered}}{\text{Total time taken}}\] 

Using equation (4) and equation (5) in the above formula, we get

\[\text{Average}\,\text{speed=}\dfrac{\text{2x}\,\text{km}}{\dfrac{\text{3x}}{\text{8}}\,\text{hr}}\text{=}\dfrac{\text{2x}}{\text{3x}}\text{ }\!\!\times\!\!\text{ 8}\,\text{km/hr=}\dfrac{\text{16}}{\text{3}}\,\text{km/hr}\] .

So, the average speed of the boat is  \[\dfrac{\text{16}}{\text{3}}\,\text{km/hr}\] .

Hence, the correct option (B) is the correct option. 

Note: In this question, one may write 4km/hr as downstream speed and 6km/hr as upstream speed. This is wrong. To overcome this mistake, one needs to know that upstream the boat moves in the opposite direction of the stream. Due to this, the net speed of the boat is given as the subtraction of speed of boat and speed of the stream. But downstream, the boat moves in the same direction of the stream. So the net speed of the boat is given as the addition of speed of stream and speed of the boat.