Questions on speed, time, and distance are common in quantitative aptitude section of most government job exams and bank exams. Among these, problems on trains, particularly concerning the speed of train, the distance traveled by the train, and the time taken by the train to traverse a particular distance are quite popular. Other common types of questions asked include ones when two trains are moving in the same direction or in opposite directions. The problems on trains must be considered as a special case because trains are considerably long structures. We solve these problems using the concept of linear equations in two variables. Let us discuss the problems related to trains and their formulas one by one.

Introduction to Speed, Time, Distance Formula

Speed is defined as the time taken by an object to cover a unit distance (1 meter or 1 kilometre). It can be measured as the ratio of the distance travelled by a body in a given period.

The SI unit of speed is \(\rm{meter/second}\); however, the commonly used unit of speed in daliy life is \(\rm{kilometre/hour}\).

\( {\text{Speed=}}\frac{ { {\text{Distance}}}}{ { {\text{Time}}}}\)

To calculate the distance covered by the object or the time taken to cover the distance, we can use the speed formula by substituting the values of the known quantities in it.

\( {\text{Distance=Speed} \times \text{Time}}\)

\( {\text{Time=}}\frac{ { {\text{Distance}}}}{ { {\text{Speed}}}}\)

We will use the same concept to solve the problems on trains. We also calculate the relative speed of two trains moving in the opposite direction and same direction.

Application of Equations in Problem on Trains

Consider two trains moving in the same or opposite direction. To find the distance travelled by the trains and the time is taken, we need to know the relative speed. In these types of problems, there will be some known and unknown quantities. To determine the unknown quantity such as time, distance and speed, we need to form an equation. Solving the equation, we can get the value of an unknown quantity.

Formulas for Problem on Trains

We know that,

\( {\text{Speed=}}\frac{ { {\text{Distance}}}}{ { {\text{Time}}}}\)

Then,

\( {\text{Speed of the train}} = \frac{ { {\text{Total distance covered by the train}}}}{ { {\text{Time taken}}}}\)

Before we talk about the formulas of problems on trains, some related terms are needed to discuss.

We can convert a \(\rm{kilometre/hour}\) to a \(\rm{meter/second}\) by multiplying it with \(\frac{5}{ {18}}.\)

We know that \(1\,\rm{kilometer} = 1000\,\rm{meter}\) and \(1\,\rm{hour} = 60 × 60 = 3600\,\rm{second}\)

Now, \(\frac{ {1\,{\text{kilometre}}}}{ {1\,{\text{hour}}}} = \frac{ {1000\,{\text{metre}}}}{ {3600\,{\text{second}}}} = \frac{5}{ {18}}\,{\text{metre/second}}\)

The Relative Speed of Two Trains moving in the Same Direction

The relative speed will be the difference between their speed if two trains move in the same direction.

The Relative Speed of Two Trains moving in the Opposite Direction

The relative speed will be the sum of their speed if two trains move in the opposite direction.

Distance Travelled when Two Trains are moving in the Same Direction/Opposite Direction

For both cases, the trains will cover the distance that is equal to the sum of the lengths of two trains.

Distance Travelled when a Train crosses a Stationary Object

When a train crosses a stationary object such as a pole, a standing person, it will cover the distance that is equal to its length.

Distance Travelled when a Train crosses a Platform/a Bridge

When a train crosses a platform or a bridge, it will cover the distance that is equal to the sum of the length of the train and bridge or platform.

Formula of Time when \(2\) Trains moving in the Opposite Direction

If the two trains are moving in opposite directions with a speed of \(x\) and \(y\) and the length of two trains are, \(a\) and \(b\) respectively.

Then, the time taken by the trains to cross each other \( = \frac{{a + b}}{{x + y}}\)

Note: As the trains are running in the opposite direction, the relative speed is \(x + y\).

Formula of Time when \(2\) Trains moving in the Same Direction

If the two trains are moving in the same direction with a speed of \(x\) and \(y\) and the length of two trains are \(a\) and \(b\) respectively.

Then, the time taken by the trains to cross each other \( = \frac{{a + b}}{{x – y}}\)

Note: As the trains are running in the opposite direction, the relative speed is \(x – y\).

Formula of Time When a Train Crosses a Pole

If a train of length \(a\) moving with a speed \(x\), crosses a pole or a person, then the time taken to cross the pole \( = \frac{a}{x}\)

Formula of Time When a Train Crosses a Bridge or a Platform

If a train of length \(a\) moving with a speed \(x\), crosses a platform/bridge of length \(b\) then the time taken to cross the platform/bridge \( = \frac{a + b}{x}\)

Solved Example Probelms on Problem on Trains

Q.1. A train running at the speed of \(36\,\rm{km/hr}\) crosses a stationary object in . Find the length of the train.
Ans: Given, speed of the train is \(36\,\rm{km/hr}\).
We know that when a train crosses a stationary, it covers the distance equal to its length. Now, \(\rm{Distance = Speed} \times \rm{Time}\)
Therefore, the length of the train \( = \frac{ {36 \times 10}}{ {60 \times 60}} = 0.1\, {\text{km}} = 0.1 \times 1000\, {\text{m}} = 100\, {\text{m}}\)

Q.2. A train moving at \(90\,\rm{kmph}\) crosses another train moving in the same direction at \(180\,\rm{kmph}\) in \(30\,\rm{seconds}\). Find the sum of the lengths of both the trains.
Ans: Speed of train \(A = 90\, {\text{kmph}} = 90 \times \frac{ {1000}}{ {3600}} = 90 \times \frac{5}{ {18}} = 25\, {\text{m}}/ {\text{sec}}\)
Speed of train \(B = 180\, {\text{kmph}} = 180 \times \frac{5}{{18}} = 50\, {\text{m}}/ {\text{sec}}\)
The relative speed will be the difference between their speed if two trains move in the same direction
The relative speed \( =( 50-25)\,\rm{m/sec} = 25\,\rm{m/sec}\)
The time taken by train \(A\) to cross the train \(B = 30\,\rm{secs}\)
When the trains are moving in the same direction, the trains will cover the distance that is equal to the sum of the lengths of two trains.
\(\rm{Distance = the}\,\rm{sum}\,\rm{of}\,\rm{the}\,\rm{length}\,\rm{of}\,\rm{two}\,\rm{trains} = \rm{Speed} \times \rm{Time} = 25 × 30\,\rm{m}= 750\,\rm{m}\)
Hence, the sum of the lengths of both the trains is \(750\,\rm{m}\).

Q.3. Two trains of equal length are running on parallel lines in the same direction at \(40\,\rm{km/hr}\) and \(30\,\rm{km/hr}\). The faster train passes, the slower train in \(36\,\rm{seconds}\). Find the length of each train.
Ans: Let us say the length of the trains is \(x\,\rm{m}\), and the distance covered by it is \((x + x) = 2x\,\rm{m}\)
The relative speed of the trains \( = (40 – 30)\, {\text{kmph}} = 10\, {\text{kmph}} = 10 \times \frac{5}{ {18}} = \frac{ {25}}{9}\, {\text{m}}/ {\text{sec}}\)
Speed of the faster train\( = \frac{2x}{36}\)
So, \( \frac{2x}{36} = \frac{25}{9} \)
or, \( x = \frac{25 \times 36}{18} = 50\,\rm{m}\)
Hence, the length of each train is \(50\,\rm{m}\).

Q.4. It takes \(12\,\rm{seconds}\) for a \(360\,\rm{m}\) long train to pass a pole. How long will it take to pass a platform of \(540\,\rm{m}\) long?
Ans: We know that,
Speed of the train \(= \frac{ { {\text{Total distance covered by the train}}}}{ { {\text{Time taken}}}}\)
Speed of the train \(=\frac{ {360}}{ {12}}\,\rm{m/sec} = 30\,\rm{m/sec}\)
When a train crosses a platform, it will cover the distance that is equal to the sum of the length of the train and the platform.
The time taken to pass a platform is \( = \frac{ {360 + 540}}{ {30}} = \frac{ {900}}{ {30}} = 30 {\text{}}\, {\text{seconds}}\)

Q.5. A train \(350\,\rm{m}\) long is running at a speed of \(54\,\rm{km/hr}\). In how much time will it pass a bridge \(100\,\rm{m}\) long?
Ans: Speed of the train \( = 54\, {\text{km}}/ {\text{hr}} = 54 \times \frac{5}{ {18}} = 15\, {\text{m}}/ {\text{sec}}\)
The total distance that needs to be covered by the train \(= (350 + 100)\,\rm{m} = 450\,\rm{m}\)
\( {\text{Time=}}\frac{ { {\text{Distance}}}}{ { {\text{Speed}}}}\)
Hence,the time is taken by the train to pass the bridge \( = \frac{ {450}}{ {15}} = 30\, {\text{seconds}}\)

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Summary

In this article, we have covered how to solve the problems on trains when two trains move in opposite direction/ same direction when a train crosses a stationary object/bridge/platform, the relative speed of two trains, and the relative speed of two trains.

Frequently Asked Questions (FAQ) – Problem on Trains

The frequently asked questions about problems on trains are answered here:

Q.1. How do you solve the problem with \(2\) trains?Ans: For solving the problems related to trains, it is essential to know the relative speed of two trains. If two trains move in the same direction, then the relative speed will be the difference between their speed. If two trains move in the opposite direction, then the relative speed will be the sum of their speed. For both cases, the trains will cover the distance equal to the sum of the lengths of two trains.

Q.2. What are the formulas for the problems on trains? Ans:If the two trains run in opposite directions with a speed of \(x\) and \(y\) and the length of two trains are \(a\) and \(b\) respectively. So, the time taken by the trains to cross each other \( = \frac{{a + b}}{{x + y}}\) If the two trains run in the same direction with a speed of \(x\) and \(y\) and the length of two trains are, \(a\) and \(b\) respectively. So, the time taken by the trains to cross each other \( = \frac{{a + b}}{{x – y}}\) If a train of length \(a\) moving with a speed \(x\), crosses a pole or a person then the time is taken to cross the pole \( = \frac{{a}}{{x}}\) If a train of length \(a\) moving with a speed \(x\), crosses a platform/bridge of length \(b\) then the time is taken to cross the platform/bridge \( = \frac{{a + b}}{{x}}\)

Q.3. How do you solve problems on trains? Ans: The problems on trains are similar to the speed, time, and distance problems. If we know the relative speed of the trains and the length of the trains, we can easily find out the time taken to cover the distance and vice versa.

Q.4. How to convert a kilometre per hour into meters per second? Ans: We can convert \(1\,\rm{km/hr}\) into \(\rm{m/sec}\) by multiplying it by \( = \frac{{5}}{{18}}.\)

Q.5. How do I know the speed of the train? Ans: If the distance covered by the train and the time taken to cover the distance is known, we can find the train’s speed as \({\text{Speed of the train}} = \frac{{{\text{Total distance covered by the train}}}}{{{\text{Time taken}}}}\)

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Maths Practice Questions for Class 8	Maths Mock Test for Class 8
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