A train 150 m long running at 54 km/h crosses a pole. How long does it take?

10 s. 54 km/h = 54 518 = 15 m/s. Time = 15015 = 10 s.

A train 120 m long crosses a 180 m platform in 20 seconds. What is its speed?

54 km/h. It covers 120 + 180 = 300 m in 20 s, so speed = 15 m/s = 15 185 = 54 km/h.

Two trains 100 m and 120 m long run in opposite directions at 40 km/h and 32 km/h. How long do they take to completely cross each other?

11 s. Relative speed (opposite directions) = 40 + 32 = 72 km/h = 20 m/s. They cover 100 + 120 = 220 m, taking 22020 = 11 s.

A train running at 60 km/h crosses a man standing on a platform in 9 seconds. What is the length of the train?

150 m. 60 km/h = 503 m/s. Length = 503 9 = 150 m.

A train overtakes two people walking in the same direction at 4 km/h and 8 km/h, passing them in 9 s and 10 s respectively. What is the length of the train?

100 m. With train speed s km/h and length L: L = 9 (s - 4) 518 = 10 (s - 8) 518. So 9(s - 4) = 10(s - 8), giving s = 44. Then L = 9 40 518 = 100 m.

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Quantitative Aptitude · Time, Work & Distance

Problems on Trains — Questions & Answers

Train problems are time-and-distance with the extra idea that the distance covered includes the train's own length. Crossing a pole covers one length; crossing a platform covers train + platform.

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Key formulas

Time to cross a pole $= \dfrac{\text{length of train}}{\text{speed}}$
Time to cross a platform $= \dfrac{\text{train length} + \text{platform length}}{\text{speed}}$
Two trains, opposite directions: relative speed $= u + v$ ; same direction: $|u - v|$

5 solved Problems on Trains questions

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01
A train 150 m long running at 54 km/h crosses a pole. How long does it take?
4 steps · reveal one at a time
1. Step 1: Convert Speed Unit
  The train's speed is given in kilometers per hour, but the length is in meters. To find the time in seconds, we need to convert the speed to meters per second. We use the conversion factor $\frac{5}{18}$ .
2. Step 2: Calculate Speed
  Let's convert the speed: $54 \text{ km/h} = 54 \times \frac{5}{18} = 3 \times 5 = 15 \text{ m/s}$ . So, the train is moving at $15$ meters per second.
3. Step 3: Identify Distance
  When a train crosses a pole, the distance it covers is equal to its own length. The train is $150$ m long, so the distance covered is $150$ m.
4. Step 4: The answer
  So the answer is (C) 10 s. We can now calculate the time taken: Time $=$ Distance $/$ Speed $= 150 \text{ m} / 15 \text{ m/s} = 10 \text{ s}$ . This matches option (C).
02
A train 120 m long crosses a 180 m platform in 20 seconds. What is its speed?
4 steps · reveal one at a time
1. Step 1: Find Total Distance
  When a train crosses a platform, the total distance it travels is the sum of its own length and the length of the platform. So, the total distance covered is $120 \text{ m} + 180 \text{ m} = 300 \text{ m}$ .
2. Step 2: Calculate Speed
  We know the total distance the train covers and the time it takes. Use the formula Speed = Distance / Time. This will give us the speed in meters per second: $300 \text{ m} / 20 \text{ s} = 15 \text{ m/s}$ .
3. Step 3: Convert Units
  The options are in kilometers per hour, so we need to convert the speed. To convert speed from meters per second to kilometers per hour, multiply the value by $\frac{18}{5}$ . So, $15 \text{ m/s} \times \frac{18}{5} = 54 \text{ km/h}$ .
4. Step 4: The answer
  So the answer is (B) 54 km/h. This is because the train covers a total distance of 300 m in 20 seconds, which is 15 m/s, and converting this gives 54 km/h.
03
Two trains 100 m and 120 m long run in opposite directions at 40 km/h and 32 km/h. How long do they take to completely cross each other?
5 steps · reveal one at a time
1. Step 1: Gather Information
  We have two trains. Train 1 is $100$ m long, moving at $40$ km/h. Train 2 is $120$ m long, moving at $32$ km/h. They are moving in opposite directions.
2. Step 2: Total Distance
  To completely cross each other, the trains must cover a distance equal to the sum of their lengths. This total distance is $100 \text{ m} + 120 \text{ m} = 220 \text{ m}$ .
3. Step 3: Relative Speed
  Since the trains move in opposite directions, their speeds add up. The relative speed is $40 \text{ km/h} + 32 \text{ km/h} = 72 \text{ km/h}$ . We convert this to m/s: $72 \times \frac{5}{18} = 20 \text{ m/s}$ .
4. Step 4: Calculate Time
  Now we can find the time taken using the formula: Time = Distance / Speed. So, Time $= \frac{220 \text{ m}}{20 \text{ m/s}} = 11 \text{ s}$ .
5. Step 5: The answer
  So the answer is (A) 11 s. This is the time it takes for the trains to completely cross each other.
04
A train running at 60 km/h crosses a man standing on a platform in 9 seconds. What is the length of the train?
5 steps · reveal one at a time
1. Step 1: Understand the scenario
  When a train crosses a stationary man, the distance it covers is equal to its own length. We are given the train's speed and the time it takes to cross the man. We need to find this length.
2. Step 2: Convert speed units
  The speed is given in kilometers per hour ( $60 \text{ km/h}$ ), but the time is in seconds. To calculate the length in meters, we must convert the speed to meters per second (m/s). Use the conversion factor: $1 \text{ km/h} = \frac{5}{18} \text{ m/s}$ .
3. Step 3: Calculate speed in m/s
  Convert the train's speed: $60 \text{ km/h} = 60 \times \frac{5}{18} \text{ m/s} = \frac{300}{18} \text{ m/s} = \frac{50}{3} \text{ m/s}$ . Now the speed is in units consistent with time and length.
4. Step 4: Calculate train's length
  The formula for distance is Distance = Speed \times Time. Here, the distance is the length of the train. So, Length $= \frac{50}{3} \text{ m/s} \times 9 \text{ s}$ . This calculation will give us the train's length.
5. Step 5: The answer
  So the answer is (C) 150 m. Performing the calculation: Length $= \frac{50}{3} \times 9 = 50 \times 3 = 150 \text{ m}$ . This matches the length of the train.
05
A train overtakes two people walking in the same direction at 4 km/h and 8 km/h, passing them in 9 s and 10 s respectively. What is the length of the train?
5 steps · reveal one at a time
1. Step 1: Define variables
  Let the train's speed be $s$ km/h and its length be $L$ meters. When the train overtakes a person, the relative speed is the difference between their speeds. The distance covered during overtaking is the train's length.
2. Step 2: Formulate equations
  The train passes the first person (4 km/h) in 9 s. So, $L = 9 \times (s - 4) \times \frac{5}{18}$ . It passes the second person (8 km/h) in 10 s. So, $L = 10 \times (s - 8) \times \frac{5}{18}$ . We convert km/h to m/s using $\frac{5}{18}$ .
3. Step 3: Solve for train speed
  Since both expressions equal $L$ , we can set them equal: $9 \times (s - 4) \times \frac{5}{18} = 10 \times (s - 8) \times \frac{5}{18}$ . We can simplify this to $9(s - 4) = 10(s - 8)$ . Solving this gives $9s - 36 = 10s - 80$ , which means $s = 44$ km/h.
4. Step 4: Calculate train length
  Now substitute $s = 44$ km/h into either original equation for $L$ . Using the first one: $L = 9 \times (44 - 4) \times \frac{5}{18}$ . This simplifies to $L = 9 \times 40 \times \frac{5}{18}$ . Calculating this gives $L = 100$ meters.
5. Step 5: The answer
  So the answer is (A) 100 m, because by setting up equations for the train's length based on relative speeds and times, we found the train's speed and then its length.

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