Boats And Streams: How To Solve 5 Types Of Problems

Dear Reader, problems under boats and streams are not only easy to solve but interesting as well. In this tutorial, you will see 5 important types of problems.

Let us begin the tutorial now.

Type I: Finding Speed Of Boat Using Direct Formula

In this type, you will be finding speed of boat in still water (i.e., when water is not flowing/running). You have to remember a very simple formula as shown below.

Let us see an example to understand this type.

Example Question 1: A boat travels at 9 km/h along the stream and 6 km/h against the stream. Find the speed of the boat in still water.

Solution:
From the question, you can write down the below values.
Downstream speed of the boat = 9 km/h
Upstream speed of the boat = 6 km/h

You have to substitute the above values in the below formula.
Speed of the boat in still water = ½ (Downstream speed + Upstream speed)
= ½ (9 + 6)
=7.5 km/h

Type II: Finding Speed Of Stream Using Direct Formula

This type is similar to type 1. But there is one difference. Here you have to find speed of stream and not the speed of the boat.
You have to use the below formula to find speed of stream.

Below is your example.

Example Question 2: A man rows downstream 30 km and upstream 12 km. If he takes 4 hours to cover each distance, then the velocity of the current is:

Solution:
In this question, downstream and upstream speeds are not given directly. Hence you have to calculate them first.

Step 1: Calculation of downstream speed
You know that the man rows 30 Km in 4 hours downstream
You know the familiar formula that Speed = Distance/Time
Therefore, Downstream speed = Distance travelled downstream / Time taken
= 30/4 Km/h
Downstream speed = Distance travelled in downstream / Time taken in Downstream travel
= 30/4 … value 1

Step 2: Calculation of upstream speed
You know that the man rows 12 Km in 4 hours upstream
So, Upstream speed = Distance travelled in upstream / Time taken
=12/4 … value 2

Step 3: Calculation of speed of stream
You have to substitute values got in steps 1 and 2 in below formula to find the speed of the stream.
Speed of the stream = ½ ( Downstream speed – upstream speed)
= ½ (30/4 – 12/4)
= ½(18/4)
= 2.25 km/h

Type III: Find Distance Of Places

In this type, you have to find distance of places based on given conditions. Below example will help you to understand better.

Example Question 3: A man can row 5 km/h in still water. If in a river running at 2 km an hour, it takes him 40 minutes to row to a place and return back, how far off is the place ?

Solution:
From the question, you can write down the below values.
Speed of the man in still water = 5 km/h
And speed of the river = 2 km/h
Using the above data, you have to first calculate downstream and upstream speeds.
Downstream speed = Speed of man in still water + Speed of the river
= 5 + 2 = 7 km/h … value 1
And, Upstream speed = Speed of man in still water – Speed of the river
= 5- 2 = 3 km/h … value 2

The man rows to a particular place and comes back. You have to calculate the distance of this place. Let this distance be X. See the below diagram to understand clearly. (Man starts from A, travels to B and comes back. Therefore distance between A and B = X)

You have to use the below equation to find the value of X,
Total time to travel from A to B and come back to A= Time taken from A to B (downstream) + Time taken from B to A (upstream)
You know the familiar formula, Speed = Distance/Time. Therefore, Time = Distance / Speed. Therefore, above equation becomes,

Total time to travel from A to B and come back to A = Distance from A to B/downstream speed + Distance from B to A/upstream speed

But, as per our assumption, distance from A to B = distance from B to A = X.
Also we have calculated downstream and upstream speeds at the start (see values 1 and 2).
So the above equation becomes,
Total time to travel from A to B and come back to A = X/7 + X/3

In question, you can see that the man takes 40 minutes to travel to B and come back to A. You have to convert this to hours and apply in above equation. (We are converting from minutes to hours because we are using speed values in km per hour units.)
40 minutes = 40/60 hours = 2/3 hours
Our equation becomes,
2/3 = X/7 + X/3
2/3 = (3X + 7X)/21
(21 X 2) / 3 = 10X
X = 42/30
= 1.4 Km

Type IV: Using Man’s Still Water Speed Calculate Stream’s Speed

In this type, you have to follow two steps.

1. Using man’s still water speed, you have to calculate upstream and downstream speeds.
2. Using upstream and downstream speeds, you have to find the speed of the stream.

Below example will help you understand better.

Example Question 4: A man can row 9 km/h in still water. It takes him twice as long to row up as to row down the river. Find the rate of the stream.

Solution:
Step 1: Calculate upstream and downstream speeds.
Assume that the man’s speed in upstream be X km/h
From the question, you know that his downstream speed is twice of upstream speed.
Then, his downstream speed = 2X km/h
You know the formula that, Man’s speed in still water = ½ (Upstream speed + Downstream speed)
=1/2 (X + 2X)
= 3X/2

But, in question, the man’s speed in still water is given to be 9 km/h
Therefore, 3x/2 = 9
X = 6 km/h.
Based on our assumptions, you can easily calculate upstream and downstream speeds as shown below.
Upstream speed = X = 6 km/h
Downstream speed = 2X = 12 km/h

Step 2: Calculate Speed Of The Stream
You already know the basic formula shown below.
Speed of the stream = ½ (Downstream speed – Upstream speed)
If you substitute the downstream and upstream speeds of step 1 in the above formula, you will get,
Speed of the stream = ½(12 – 6)
= 3 km/h.

Type V: Equations Based Boats And Stream Problems

In this type, you have to form linear equations based on conditions given. You have to solve those equations to find the answer.

Below example will help you to understand this type clearly.

Example Question 5: Kavin can row 10 km upstream and 20 km downstream in 6 hours. Also, he can row 20 km upstream and 15 km downstream in 9 hours. Find the rate of the current and the speed of the man in still water.

Solution:
You have to make below assumptions to form equations.
Let the upstream speed be X km/h
And downstream speed be Y km/h.

You already know the below equation. (If you are not clear about this, refer to the equation in type 3.)
Time for downstream travel + Time for upstream travel = Total Time for upstream and downstream travel

Using the familiar Speed = Distance / Time formula, the above equation can be simplified as shown below.
Distance travelled in downstream/downstream speed + Distance travelled in upstream/upstream speed = Total Time for upstream and downstream travel
If you substitute the values in question in above equation, you will get the below 2 equations.
10/x + 20/y = 6 …equation 1
20/x + 15/y = 9 …equation 2

Assume that 1/x = u and 1/y = v, Now you rewrite the above equations as given below.
10u + 20v = 6 …equation 3
20u + 15v = 9 …equation 4
If you multiply equation 3 by 2 , you will get, 20u + 40v = 12 …equation 5

If you subtract equation 4 from equation 5, you will get
By cancelling out u, we get, v = 3/25
If you substitute v = 3/25 in equation 3, you will get,
10u + 20(3/25) = 6
10u + 12/5 = 6
10u = 18/5
u = 9/25

Note: To solve such linear equations, there is another simple shortcut. Here is the video link to that shortcut.

From the values of u and v, you can find the downstream and upstream speeds as shown below.
Upstream speed = X = 1/u = 25/9 km
and Downstream speed = Y = 1/v = 25/3 km
You can now calculate the speed of the man in still water, using our familiar formula.
Speed of the man in still water = ½ (downstream speed + upstream speed)
= ½(25/3 + 25/9)
=½(100/9) = 50/9 = 5.6 kmph

Also, you know the formula for speed of the current.
Speed of the current = ½(downstream speed – upward stream)
= ½(25/3 – 25/9)
=1/2(50/9) = 25/9 = 2.8 km/h

You can type your doubts in the comments section below. You can also suggest improvements to the above tutorial