Problems on trains with solutions is a much required one to the students who are getting prepared for competitive exams

Before we look at the problems on trains with solutions, first let us come to know the shortcuts which are required to solve train problems.

__Hint 1 :__

**Speed = Distance / Time**

__Hint 2 :__

**Distance = Speed x Time**

__Hint 3 :__

**Time = Distance / Speed **

__Hint 4 :__

**If the speed is given km per hour and we want to convert it in to meter per second, we have to multiply the given speed by 5/18.**

__Example:__

**90 km/hr = 90x(5/18) = 25 meter/sec**

**Hint 5 :**

**If the speed is given meter per sec and we want to convert it in to km per hour, we have to multiply the given speed by 18/5.**

__Example:__

**25 meter/sec = 25x(18/5) = 90 km/hr**

__Hint 6 :__

**Let the length of the train be "L" meters.**

**Distance traveled to pass a standing man = L meters**

__Hint 7 :__

**Let the length of the train be "L" meters.**

**Distance traveled to pass a pole = L meters**

__Hint 8 :__

**Let the length of the train be "a" meters and the length of the platform be "b" meters.**

**Distance traveled to pass the platform = (a+b) meters**

__Hint 9 :__

**If two trains are moving on the same directions with speed of "p" m/sec and "q" m/sec (here p > q), **

**then their relative speed = (p-q) m/sec.**

__Hint 10 :__

**If two trains are moving opposite to each other in different tracks with speed of "p" m/sec and "q" m/sec, **

**then their relative speed = (p+q) m/sec.**

__Hint 11 :__

**Let "a" and "b" are the lengths of the two trains.**

**They are traveling on the same direction with the speed "p" m/sec and "q" m/sec (here p > q),**

**then the time taken by the faster train to cross the slower train**

**= (a+b) / (p-q) seconds.**

__Hint 12 :__

**Let "a" and "b" are the lengths of the two trains.**

**They are traveling ****opposite to each other in different tracks with the speed "p" m/sec and "q" m/sec,**

**then the time taken by the trains to cross each other**

**= (a+b) / (p+q) seconds.**

__Hint 13 :__

**Two trains leave at the same time from the stations P and Q and moving towards each other.**

**After crossing, they take "p" hours and "q" hours to reach Q and P respectively.**

**Then the ratio of the speeds of two trains **

**= square root (q) : square root (p)**

__Hint 14 :__

**Two trains are running in the same direction/opposite direction. **

**The person in the faster train observes that he crosses the slower train in "m" seconds. **

**Then the distance covered in "m" seconds in the relative speed **

**= Length of the slower train**

__Hint 15 :__

**Two trains are running in the same direction/opposite direction.**

**The person in the slower train observes that the faster train crossed him "m" seconds. **

**Then the distance covered in "m" seconds in the relative speed **

**= Length of the faster train**

Here we are going to look at some Problems on trains with solutions.

**Problem 1 :**

If the speed of a train is 20 m/sec, find the speed the train in kmph.

**Solution :**

Speed = Distance / Time

Speed = 20 m/sec

Speed = 20 x 18/5 m/sec

Speed = 72 kmph

**Hence, the speed of the train is 72 kmph **

**Let us look at the next problem on "**Problems on trains with solutions"

**Problem 2 :**

A trains covers 240 kilometers in 4 hours. Find the speed of the train in meter per second.

**Solution :**

Speed = Distance / Time

Speed = 240 / 4 kmph

Speed = 60 kmph

Speed = 60 x 5/18 m/sec

Speed = 16.67 m/sec

**Hence, the speed of the train is 16.67 m/sec. **

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 3 :**

The length of a train is 300 meter and length of the platform is 500 meter. If the speed of the train is 20 m/sec, find the time taken by the train to cross the platform.

**Solution :**

Distances needs to be covered to cross the platform is

= Sum of the lengths of the train and platform

So, distance traveled to cross the platform = 300 + 500

= 800 meters

Time = Distance / Speed

Time = 800 / 20

Time = 40 seconds

**Hence, time taken by the train to cross the platform is 40 seconds. **

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 4 :**

A train of length 250 meters is running at a speed of 90 kmph. Find the time taken by the train to cross a pole.

**Solution :**

First let us con vert the speed of the train from kmph to m/sec.

Speed = 90 kmph

Speed = 90 x 5/18 m/sec

Speed = 25 m/sec

Distances needs to be covered to cross the pole is

= Length of the train

So, distance traveled to cross the pole = 250 meters

Time = Distance / Speed

Time = 250 / 25

Time = 10 seconds

**Hence, time taken by the train to cross the pole is 40 seconds. **

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 5 :**

A train is running at a speed of 20 m/sec.. If it crosses a pole in 30 seconds, find the length of the train in meters.

**Solution :**

The distance covered by the train to cross the pole

= Length of the train

**Given :** Speed is 20 m/sec and time taken to cross the pole is 30 seconds

We know, Distance = Speed x Time

So, length of the train = Speed x Time

Length of the train = 20 x 30

Length of the train = 600 meters

**Hence, length of the train is 600 meters.**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 6 :**

It takes 20 seconds for a train running at 54 kmph to cross a platform.And it takes 12 seconds for the same train in the same speed to cross a man walking at the rate of 6 kmph in the same direction in which the train is running. What is the length of the train and length of platform (in meters).

**Solution :**

Relative speed of the train to man = 54 - 6 = 48 kmph

= 48 x 5/18 m/sec

= 40/3 m/sec

When the train passes the man, it covers the distance which is equal to its own length in the above relative speed

**Given :** It takes 12 seconds for the train to cross the man

So, the length of the train = Relative Speed x Time

= (40/3) x 12

= 160 m

Speed of the train = 54 kmph

= 54 x 5/18 m/sec

= 15 m/sec

When the train crosses the platform, it covers the distance which is equal to the sum of lengths of the train and platform

**Given :** The train takes 20 seconds to cross the platform.

So, the sum of lengths of train and platform

= Speed of the train x Time

= 15 x 20

= 300 meters

That is,

Length of train + Length of platform = 300

160 + Length of platform = 300

Length of platform = 300 - 160

Therefore, length of platform = 140 meters

**Hence the lengths of the train and platform are 160 m and 140 m respectively**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 7 :**

Two trains running at 60 kmph and 48 kmph cross each other in 15 seconds when they run in opposite direction. When they run in the same direction, a person in the faster train observes that he crossed the slower train in 36 seconds. Find the length of the two trains (in meters).

**Solution :**

When two trains are running in opposite direction,

relative speed = 60 + 48

= 108 kmph

= 108 x 5/18 m/sec

= 30 m/sec

Sum of the lengths of the two trains

= sum of the distances covered by the two trains in the above relative speed

Then, sum of the lengths of two trains = 30 x 15 = 450 m

When two trains are running in the same direction,

relative speed = 60 - 48

= 12 kmph

= 12 x 5/18

= 10/3 m/sec

When the two trains running in the same direction, a person in the faster train observes that he crossed the slower train in 36 seconds.

**The distance he covered in 36 seconds in the relative speed is equal to the length of the slower train. **

Length of the slower train = 36 x 10/3 = 120 m

Length of the faster train = 450 - 120 = 330 m **Hence, the length of the two trains are 330m and 120m**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 8 :**

Two trains of length 250m and 200m run on parallel lines. When they run in the same direction, it will take 30 second to cross each other. When they run in opposite direction, it will take 10 seconds to cross each other. Find the speeds of the two trains (in kmph).

**Solution :**

Let the speeds of the two trains be S1 and S2

Total distance covered to cross each other = 250 + 200 = 450 m

When they run in same direction direction,

relative speed ------> S1 - S2 = 450 /30

S1 - S2 = 15 -------(1)

When they run in same opposite direction

relative speed ------> S1 + S2 = 450 /10

S1 + S2 = 45 -------(2)

Solving the (1) and (2), we get

S1 = 30 m/sec = 30 x 18/5 kmph = 108 kmph

S2 = 15 m/sec = 15 x 18/5 kmph = 54 kmph **Hence, speeds of the two trains are 108 kmph and 54 kmph.**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 9 :**

Find the time taken by a train 100m long running at a speed of 60 kmph to cross another train of length 80 m running at a speed of 48 kmph in the same direction.

**Solution :**

Total distance covered to cross each other = 100 + 80 = 180 m

Relative speed of the two trains (Same direction)

= 60 - 48 = 12 kmph

= 12 x 5/18

= 10/3 m/sec

Time taken to cross = Distance / Speed

= 180 / (10/3) seconds

= 180 x 3/10 seconds

= 54 seconds

**Hence, time taken by the faster train to cross the slower train is 54 seconds.**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 10 :**

Two trains of equal length are running on parallel lines in the same direction at 46 kmph and 36 kmph. The faster train crosses the slower train in 36 seconds. Find the length of each train.

**Solution :**

Let "x" be the length of each train

Total distance covered to cross each other

= Sum of the lengths of the two trains

Total distance covered to cross each other = x + x = 2x meters

Relative speeds of the two trains

= 46 - 36

= 10 kmph

= 10 x 5/18 m/sec

= 25/9 m/sec

Distance = Speed X Time

2x = (25/9) x 36

2x = 100

x = 50

**Hence, the length of each train is 50 meter**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 11 :**

Two stations A and B are 110 km apart on a straight line. One train starts from A at 7 a.m. and travels towards B at 20 kmph. Another train starts from B at 8 a.m. and travels towards A at a speed of 25 kmph. At what time will they meet ?

**Solution :**

Let the trains meet each other "m" hours after 7 a.m.

Distance covered by A in "m" hrs = Speed x Time = 20m km

At a particular time after 8 a.m,

if train A had traveled "m" hours, then train B would have traveled (m-1) hours.

Because it started at 8.00 am. (one hour later)

Distance covered by B in (m - 1) hrs = 25(m - 1)

At the meeting point,

Distance of A + Distance of B = Total distance (from A to B)

20m + 25(m-1) = 110

20m + 25m - 25 = 110

45m = 135

m = 3 hours

So, two train meet each other 3 hrs after 7 a.m

That is, at 10 a.m.

**Hence, the time at which they will meet is 10 a.m.**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 12 :**

Two trains are running at 40 kmph and 20 kmph respectively in the same direction .Faster train completely passes a man who is sitting in the slower train in 9 seconds. What is the length of the faster train?

**Solution :**

Relative speed of two trains = 40 - 20

= 20 kmph

= 20 x 5/18 m/sec

= 50/9 m/sec **Given :** Faster train completely passes a man who is sitting in the slower train in 9 seconds.

Length of the faster train

= The distance covered by the faster train in this 9 seconds

= Speed x Time

= (50/9) x 9

= 50 m

**Hence, the length of the faster train is 50 m.**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 13 :**

Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. Find the ratio of their speeds

**Solution :**

Let "a" m/sec and "b" m/sec be the speeds of two trains respectively

First train crosses the man in 27 seconds with speed "a" m/sec

Length of the first train = Distance covered in 27 seconds

Then, length of the first train = Speed x Times

**length of the first train = 27a**

Second train crosses the man in 17 seconds with speed "b" m/sec

Length of the second train = Distance covered in 17 seconds

Then, length of the second train = Speed x Times

**length of the second train = 17b**

**Given :** They cross each other in 23 seconds

The distance covered by both the trains in this 23 seconds

Relative speed x Time = Sum of the lengths of the two trains

(a + b) x 23 = 27a + 17b

23a + 23b = 27a + 17b

6b = 4a

6/4 = a/b

3/2 = a/b

**Hence, the ratio of their speeds is 3:2**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 14 :**

A train passes a station platform in 36 seconds and a man standing on the platform in 20 seconds. If the speed of the train is 54 km/hr, what is the length of the platform ?

**Solution :**

Speed of the train = 54 kmph

= 54 x 5/18 m/sec

= 15 m/sec

The train passes the man in 20 seconds

The distance covered by the train in this 20 seconds is equal to the length of the train.

And also distance = Speed x Time

**Then, length of the train = 20 x 15 = 300 m **

Let "m" be the length of the platform

**Given :** The train crosses the platform in 36 seconds

The distance covered by the train in this 36 seconds

= Sum of the lengths of the train and platform

The distance covered by the train in 36 seconds = 300 + m

Distance / Speed = Time

(300 + m) / 15 = 36

300 + m = 540

m = 540 - 300

m = 240 meters

**Hence, the length of the platform is 240 meters**

**Let us look at the next problem on ****"**Problems on trains with solutions"

**Problem 15 :**

Two trains are moving in opposite directions at 60 km/hr and 90 km/hr. Their lengths are 1.10 km and 0.9 km respectively. Find the time taken by the two trains to cross each other.

**Solution :**

Relative speed = 60 + 90 = 150 kmphr

= 150 x 5/18 m/sec

= 125/3 m/sec

When they cross each other,

Distance covered by both the trains = sum of the lengths

So, the distance covered by them = 1.1 + 0.9

= 2 km

= 2 x 1000 m

= 2000 m

Time = Distance / Speed

Time = 2000 / (125/3)

Time = 2000 x 3/125

Time = 48 seconds

**Hence, time taken by the two trains to cross each other is 48 seconds**

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**24. Problems on Quadratic Equations**

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