Questions from boats and streams are one of the most frequent questions in the CAT quantitative aptitude section. They are easy to solve and are high scoring.
The problems of boats and streams are also dependent on the basic equation of time, speed and distance :
Speed × Time = Distance.
However, as in the case of trains, the adjustments to be made for solving questions on boats and streams are: The boat has a speed of its own, which is also called the speed of the boat in still water (SB)
Another variable that is used in boat and streams problems is the speed of the stream (SC)
The speed of the movement of the boat is dependent on whether the boat is moving:
(a) In still water the speed of movement is given by →SB
(b) While moving upstream (or against the flow of the water), the speed of movement is given by -> SU = SB - SS
(c) While moving downstream (or with the flow of the water), the speed of movement is given by -> SV = SB + SS
The time of movement and the distance to be covered are to be judged by the content of the problem.

Important Terms and Formulas in Boats and Streams:

Let, u= Speed of the boat in still water
v= Speed of the stream
Upstream: It means that you are moving in the opposite direction from that in which the river flows.
Upstream Speed = ( u – v ) km/hr
Downstream: It means moving along in the direction of the flow of the stream.
Downstream Speed = ( u + v ) km/hr
Still water: When the water is still and not moving and there’s no flow like that in the case of ponds then it’s called still water.
Speed in still water = ½ ( Downstream Speed + Upstream speed )

Stream:- A stream simply refers to the flowing of either water or river at a certain speed. The formula to calculate the speed of the stream is given below.

Speed of stream = ½ ( Downstream Speed - Upstream speed

Average Speed
The average speed of a boat can be calculated using its speed in still water and the speed of the stream. The formula is:
Average speed = {(u-v) * ( u+v)} / u
Case 1: If a boat takes “t” hours to reach a point in still water and comes back to the same point then, the distance between that point and the starting point can be calculated as:
Distance = (u²-v²) * t / 2u km
Case 2: If a boat takes “t” hours more to go to a point upstream than downstream for the same distance, the distance will be
Distance = (u²-v²) * t / 2v km
Speed When Time is Given
If a boat travels a distance downstream in “t1” hours and returns the same distance upstream in “t2” hours, then the speed of the man in still water will be:
Speed = v * { (t2+t1) / (t2-t1) } km/hr

Types of problems in Boats and Streams:

Time-Related Questions: The speed of the stream and the speed of the boat in still water will be given and the question will be to find the time taken by a boat to go downstream or upstream or both.
Example
The speed of a motor boat is that of the stream as 36:5. The boat goes along with the stream in 5 hours and 10 minutes. How much time will it take to come back?
Solution
Let the speed of the motor boat and that of the stream be 36x km/hr. and 5km/hr. respectively.
Then, the speed downstream = (36x + 5x) = 41km/hr.
Speed upstream = (36x – 5x) = 31 km/hr.
Let d the distance.
Then, d/41 km = 5 10/60 = 31/6
d = 1271/6
Time taken while coming back = distance/speed = d/31 = 1271/ (31* 6) hrs. = 6 5 /6

Speed Related Questions: The speed of a boat upstream and downstream will be given and candidates will be asked to find the speed of the boat in still water or the speed of the stream.
Example
The speed of the boat when traveling downstream is 42 km/hr. whereas when traveling upstream it is 18 km/hr. What is the speed of the boat in still water and the speed of the stream?
Solution
This question requires direct use of the formula mentioned earlier.
Speed of the boat in still water = ½ (42 + 18) km/hr. = 30 km/hr.
Speed of the stream = ½ (42 – 18) hm/hr. = 12 km/hr.

Average Speed Questions: The average speed of the boat can be asked. Here, the speed of the boat upstream and downstream will be provided in the question. Average Speed = (upstream speed * downstream speed)/ Boat’s speed in still water
Example
A boat, while going downstream in a river covered a distance of 50 km at an average speed of 60 km per hour. While returning because of water resistance, it took 1 hour 15 minutes to cover the same distance. What was the average speed during the whole journey?
Solution
In this problem we cannot directly use the formula first we need to find the upstream speed and speed in still water.
Upstream speed = distance covered in 1 hour 15 minutes / Time taken to travel 50
miles = 50/5 /4 = 40 km/hr.
Boat’s speed in still water = ½ (Upstream speed + downstream speed) = ½ (40 + 60) = 50 km/ hr.
Now we are in the stage of using the average speed formula.
Average speed during whole journey = (40 * 60)/ 50 = 48 km/ hr.

Distance Related Questions: In this, the distance traveled will be asked. The time taken by the boat to reach a point upstream and downstream will be given.
Example
A man can row 7 ½ km/hr. in still water. If in a river running at 1.5 km an hour, it takes him 50 minutes to row to a place and back, how far off is this place?
Solution
To solve this question we will simply use the formula given above. In this case, t is 50 minutes b is 1.5 km/hr. and x is 7 ½ km/hr.
Distance = [ 50/60 * {(7.5)2 – (1.5)2 }]/ 2* 7.5 = (5 /6 *54)/ 2* 7.5 = 3 km

Solved Example

Question 1
A boat can travel with a speed of 13 km/hr in still water. If the speed of the stream is 4 km/hr, find the time taken by the boat to go 68 km downstream.
1. 2 hours
2. 3 hours
3. 4 hours
4. 5 hours
Answer – Option 3
Solution –
Speed downstream = (13 + 4) km/hr = 17 km/hr
Time taken to travel 68 km downstream = 68/17 hrs = 4 hrs

Question 2.
A man’s speed with the current is 15 km/hr and the speed of the current is 2.5 km/hr. The
man’s speed against the current is:
1. 8.5 km/hr
2. 9 km/hr
3. 10 km/hr
4. 12.5 km/hr
Answer - Option 3
Man’s rate in still water = (15 – 2.5) km/hr = 12.5 km/hr.
Man’s rate against the current = (12.5 – 2.5) km/hr = 10 km/hr.

Question 3.
What would be the time taken by a boat to go 80 km downstream if the speed of the stream is it 5 km/hr and the boat’s speed in still water is 15km/hr?
Solution
Downstream speed of the boat= (15 + 5)= 20 km/hr.
Time is taken by the boat to go 80 km downstream= (80/20) hours= 4 hrs.

Question 4.
What would be the speed of a boat in still water if it covers a distance of 40 km in 4 hours in upstream and covers 40 km in 2 hours while going downstream?

Solution
From the data, upstream speed= (40/4)= 10 km/hr.
And, downstream speed= (40/2)= 20 km/hr.
So, the speed of the boat= ½ (10 + 20)= 15 km/hr.

Question 5.
A boat can travel 20 km downstream in 24 min. The ratio of the speed of the boat in still water to the speed of the stream is 4 : 1. How much time will the boat take to cover 15 km upstream?
1. 20 mins
2. 22 mins
3. 25 mins
4. 30 mins
Answer Option 4-
Down speed =20/24*60=50km/hr
4:1 =4x:x
Downstream speed = 4x+x=5x
Upstream speed = 4x-x=3x 5x= 50; x=10
so up speed 3*10=30
Time = 15/30*60= 30 mins