# Problems on Boats & Streams - Time, Speed & Distance

• Authored by Nitin Gupta, Founder, Director at AlphaNumeric.

When a boatman is rowing in still water, say a lake, he would be moving at a speed at which he can row. This speed is called the speed of boat in still water or simply speed of boat. But consider the same boatman in a stream. Because of the current he is either aided (if rowing in the direction of the stream, this is called Downstream) or will be opposed (if rowing against the stream, called Upstream).
If the speed of boatman in still water is B and the speed of the stream is S, we have
Downstream Speed (D) = B + S
Upstream Speed (U) = B – S

Do not confuse with relative speed. In relative speed we saw that when two objects travel in same direction with speeds S1 and S2, the relative speed is S1 – S2. In the case of boats and streams, when boat is travelling in the direction of stream, the downstream speed of the boat is taken as B + S. Since they are in same direction, should we not consider relative speed to be B – S? No.

In case of relative speeds, the two objects are moving independently i.e. say police running to catch thief, the speed of police does not get ‘transferred’ or affects the speed of the thief. Whereas in the case of Boats and Streams, the speed of the Stream gets ‘transferred’ to the speed of the boat, the boat is travelling ‘on’ the stream. This is not a case of relative speed. Here the stream is ‘aiding’ the boat, unlike the case of police and thief where neither ‘aids/ hinders’ the other.

If the speed of boatman is lesser than the speed of the current or stream, the upstream speed will be negative i.e. he is trying to row upstream, but rather than move in that direction, he is taken in opposite direction by the stream. But such situations do not occur in math problems on this topic.

A boat covers a distance of 16 km in 2 hours when rowing downstream and in 4 hours if rowing upstream. What is the speed of the boat in still water?

The rowing speed of a man in still water is 7.5 kmph. In a river flowing at 1.5 kmph, it takes the same boatman 50 minutes to row a certain distance and come back. Find the distance.

Here Speed Downstream = 7.5 + 1.5 = 9 km/hr and Speed
Upstream = 7.5 – 1.5 = 6 km/hr

Approach 1: Formulaic approach
Let the required distance be x. Equating the time taken, x/6 + x/9 = 5/6 hr
Solving we get, x = 3. Hence the place is 3 km away.

Approach 2: Use of proportionality.
In going upstream and downstream, the distance covered is same. Hence time taken are in inverse proportion to their speeds. Ratio of upstream and downstream speeds is 6 : 9 i.e. 2 : 3.
Thus, ratio of time travelling upstream and downstream is 3 : 2.
And we know the total time is 50 minutes. Thus time travelled upstream is 30 minutes and time travelled downstream is 20 minutes.
Now the distance can be found using speed × time i.e. 6 kmph × 30 min = 6 kmph × 1/2 hr = 3 km
(it could also be found using downstream speed and time i.e. 9 kmph × 20 min = 9 kmph × 1/3 hr = 3 km)

A boatman rows to a place 48 km distant and back in 14 hours. He finds that he can row 4 km with the stream in the same time as 3 km against the stream. Find the rate of the stream

Consider the statement, ‘he can row 4 km with the stream in the same time as 3 km against the stream’. Clearly the time is constant and thus, downstream speed and upstream speed are in the ratio of distances covered i.e. 4 : 3.
Next, consider travelling 48 km downstream and 48 km upstream. Again since distance is constant, time taken will be inversely proportional to speed i.e. time taken to travel downstream and that taken to travel upstream are in ratio 3 : 4. And we know that the total time taken is 14 hours.
Thus, 6 hours is taken to travel 48 km downstream (speed = 8 kmph) and 8 hours is taken to travel 48 km upstream (speed = 6 kmph).
Thus, speed of stream = 1 = (d-u)/2 = (8-6)/2

A man travels downstream for 5 hours and again upstream for 5 hours. Yet it is at a distance of 2 kms from the place it started. What is the speed of stream?

Think of this problem as a person in a moving train. If he walks for five minutes in the direction the train is moving and then reverses direction and again walks for 5 minutes, he would come back to his original position in the train. However if he (and the train) was at New Delhi when he started to walk, and now he is at Faridabad (10 kms away from Delhi), is it not obvious that in the 10 minutes the train has taken him from New Delhi to Faridabad and speed of train is 1 km/min. Thus in the above problem also had the stream been stationary, after rowing 5 hours in either direction he would have come back to the original spot. But he is away from the original spot by 2 kms means the stream is moving and has taken him 2 kms downstream in 10 hours.

A swimmer jumps from a bridge over a canal and swims 1 kilometer stream up. After that first kilometer, he passes a floating cork. He continues swimming for half an hour and then turns around and swims back to the bridge. The swimmer and the cork arrive at the bridge at the same time. The swimmer has been swimming with constant speed. How fast does the water in the canal flow?

It is obvious that the cork does not move relatively to the water (i.e. has the same speed as the water). So if the swimmer is swimming away from the cork for half an hour (up stream), it will take him another half hour to swim back to the cork again. Because the swimmer is swimming with constant speed (constant relatively to the speed of the water!) you can look at it as if the water in the river doesn't move, the cork doesn't move, and the swimmer swims a certain time away from the cork and then back. So in that one hour time, the cork has floated from 1 km up stream to the bridge.
Conclusion: The water in the canal flows at a speed of 1 km/h.

S1 and S2 started off from points A and B respectively along course of river at 7m/s and 3m/s resp. towards each other. Distance b/w A and B is 200m. After crossing each other S1 reaches B and S2 reaches A. After that they reverse their direction to comeback to their resp. starting points. When they meet for 1st time, ratio of distance covered by them is 4:1. After how much time they meet for 2nd time?

200 * 4/5 = 160 covered by S1 with speed of 7 + x
200 * 1/5 = 40 covered by S2 with speed of 3 - x
relative speed = 10 m/s
200/10 = 20 sec
so , speed of stream will be 1 m/s
40/8 = 5 sec taken by S1 to reach B
In 5 sec S2 will swim 10 mtr more
now distance between them is 50 mtr
so , they will meet after 50/4 = 12.5
total time taken from starting = 20 + 5 + 12.5 = 37.5

A cyclist drove one km, with the wind in his back, in three minutes and drove the same way back, against the wind in four minutes. If we assume that the cyclist always puts constant force on the pedals, how much time would it take him to drive one km without wind? [SNAP 2008]

1/(v + u) = 3 min and 1/(v - u) = 4 min => v = 7u => 1/u => 24 min => 1/7u = 24/7 minutes.

Practice Questions

1. A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours, it can go 40 km upstream and 55 km down-stream. The speed of the boat in still water is: [IIFT 2008]
a) 3 km/hour
b) 4 km/hour
c) 8 km/hour
d) None of the above

2. At his usual rowing speed, Rahul can travel 12 miles downstream in a certain river in six hours less than it takes him to travel the same distance upstream. But if he could double his usual rowing speed for this 24 mile round trip, the downstream 12 miles will then take only one hour less than the up-stream 12 miles. What is the speed of the current in miles per hour? (CAT 2001)
a. 7/3
b. 4/3
c. 5/3
d. 8/3

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