# Trains, Boats and Streams

## Introduction

Trains, boats and streams are some of the most common and important topics asked in online assessments. Almost every competitive examination includes questions on this topic. It is pretty simple to understand as it is associated with Speed, Time, and Distance. Let's begin with some fundamental concepts of speed, time, and distance.

## Speed, Time & Distance

• Distance = Speed x Time
• If an object covers a distance of X km/hr and an equal distance of Y km/hr, its average speed will be [ 2XY / ( X+Y ) ] km/hr.
• If we want to convert the speed of an object from km/hr to m/sec, we multiply by 5 / 18 i.e 1 km/hr = 5 / 18 m/sec.
•  Similarly,  if we want to convert m/sec to km/hr, we multiply by 18 / 5 i.e 1 m/sec = 18 / 5 km/hr.
• If two objects/people move in the same direction with speeds S1 km/hr and S2 km/hr, their relative speed will be | S1– S2 | km/hr.
• Similarly, if two objects/people move in the opposite direction with speeds S1 km/hr and S2 km/hr, their relative speed will be (S1 + S2) km/hr.

## Trains

• Time taken by a T meters long train moving with a speed of K m/sec to pass a stationary object of length L meters would be ( T + L ) / K sec

For example, to pass a 100 m wide building with a train of length 500 m moving at a speed of 60 m / s,  The time taken would be (100 + 500 ) / 60 sec, i.e. 600 / 60 sec = 10 sec.

Note: In the case of a pole or a man, we can neglect its length with respect to the length of the train. The time taken to pass a man or a pole is T / K sec.

• If two trains of lengths T1 and T2 meters are separated by K meters and traveling in the same direction at speeds X and Y meters per second, the time required by the faster train to pass the slower train is (T1 + T2 + K) / |X – Y| sec.
• If two trains of lengths, T1 meters, and T2 meters, are K meters apart and are moving in the opposite/different directions with speeds X m/sec and Y m/sec, the time required to cross each other is (T1 + T2 + K) / |X + Y| sec.

### Questions

Q1: A 400-meter train passes a man standing on a sidewalk near a railway track in 8 seconds. How long will it take to pass an 850-meter long platform?

Solution: Speed = ( 400 / 8 ) m/sec = 50 m/sec.

Required Time = ( 400 + 850 ) / 50 = 1250 / 50 = 25 seconds.

Q2: Two trains each of length 200 m and 300 m are 100 m apart. They start travelling towards each other on parallel tracks, at speeds 5m/s and 7 m/s, respectively. How long will the trains take to pass each other?

Solution: Time required= ( 200 + 300 + 100 ) / ( 5 + 7 ) = 600 / 12 = 50 sec.

Q3: Two trains, each with a length of 100 and 250 meters, are 150 meters apart. They begin traveling in the same direction on parallel tracks with a speed of 36 km/hr and 18 km/hr, respectively. How much time will the faster train take to pass the slower train?

Solution

Speed of the first train = 36 km/hr = 36 x 5/18 = 10 m/s.

Speed of the second train = 18 km/hr = 18 x 5/18 = 5 m/s.

Time required = ( 100 + 250 + 150 ) / | 10 - 5 | =  500 / 5 = 100 sec =1 min 40 sec.

## Boats and streams

General terms

• Still Water: Water of a river or any other source which is not flowing is known as still water.
• Stream:  The water of a river or any other source which is moving at a certain speed is known as a stream.
• Upstream: When a boat is moving against the flow of water, it is going upstream.
• Downstream: When a boat is moving with the flow of water, it is going downstream.

Important Formulas and Concepts

• If the boat or swimmer is moving at X km/hr in still water and the stream is moving at Y km/hr,
• Speed Downstream = ( X + Y ) km/hr.
• Speed Upstream =( X - Y ) km/hr.
• Speed of boat/swimmer in still water = ½ x ( Speed Downstream + Speed Upstream) km/hr.
• Speed of the stream = ½ x ( Speed Downstream - Speed Upstream ) km/hr.

### Questions

Q1. In still water, a boat can travel at 10 km/h. Calculate how long it took the boat to travel 75 kilometers downstream at a speed of 5 km/hr.

Solution: Speed Downstream = ( 10 + 5 ) = 15 km / hr.

Time taken = ( 75 / 15 ) hr = 5 hr.

Q2. A boat travels at 10 km/hr along the stream and 6 km/hr against it in one hour. In still water, the boat's speed (in km/hr) is:

Solution:

Speed Downstream = 10 km/hr.

Speed Upstream = 6 km/hr.

Speed in still water = ½ (10 + 6) km/hr = 8 km/hr.

Q3. In 2 hours, a boatman can travel 4 kilometers against the stream's current, and in 10 minutes, he can travel 3 kilometers with the current. How long does it take to travel 4 kilometers in still water?

Solution:

Speed Upstream = 4 / 2 = 2 km/hr.

Speed Downstream = 3 / ( 10 / 60 ) = 18 km/hr.

Speed of boat in still water = ½ x( 18 + 2 ) = 10 km/hr.

Time taken = 4 / 10 = 0.4 hr = 24 min.

Q4. A boat's speed in standing water is 10 kmph, while the stream's speed is 5 kmph. A man rows to a location 105 kilometers away and then returns to his starting point. The time taken by him is :

Solution: Speed Upstream = 10 + 5 = 15 km/hr.

Speed Downstream = 10 - 5 = 5 km/hr.

Total time taken = ( 105 /15 + 105 / 5) = 28 hrs.

Q1. What is the relationship between Speed, Distance and Time?

Ans: The relation between them is:  Distance = Speed x Time

Q2. Why do we neglect the length of the pole or a man with respect to the length of the train?

Ans: Because the length of a man or a pole will be minimal with respect to the length of the train.

Q3. Speed of a boat going upstream slower or faster than the speed of the boat in still water?

Ans: It will be slower than the speed in still water as:

Speed Downstream = Speed in still water - Speed of the stream

## Key Takeaways

In this blog, we discussed the relation between speed, distance, time, and various terms like relative speed, still water, stream, upstream, downstream, and the key concepts and formulas related to trains, boats, and streams and some example questions for a better understanding.

In case of any comments or suggestions, feel free to post them in the comments section.

You can learn more about Train, Boats, and Streams from here and practice similar problems on the Codestudio. 