Episode 8 — Aptitude and Reasoning / 8.12 — Problems on Trains
8.12.c Solved Examples -- Problems on Trains
Problem 1: Train Crossing a Pole (Easy)
A train 150 m long passes a pole in 15 seconds. Find the speed of the train in km/h.
Distance = Length of train = 150 m
Time = 15 seconds
Speed = 150 / 15 = 10 m/s
Convert to km/h: 10 x (18/5) = 36 km/h
Answer: 36 km/h
Problem 2: Train Crossing a Platform (Easy)
A train 200 m long crosses a platform 300 m long in 25 seconds. Find the speed of the train.
Total distance = 200 + 300 = 500 m
Time = 25 seconds
Speed = 500 / 25 = 20 m/s = 20 x (18/5) = 72 km/h
Answer: 72 km/h
Problem 3: Finding Train Length (Easy)
A train running at 54 km/h crosses a pole in 20 seconds. Find the length of the train.
Speed = 54 km/h = 54 x (5/18) = 15 m/s
Length = Speed x Time = 15 x 20 = 300 m
Answer: 300 m
Problem 4: Finding Platform Length (Easy)
A 250 m long train running at 72 km/h crosses a platform in 30 seconds. Find the length of the platform.
Speed = 72 x (5/18) = 20 m/s
Total distance = Speed x Time = 20 x 30 = 600 m
Platform length = 600 - 250 = 350 m
Answer: 350 m
Problem 5: Time to Cross a Bridge (Easy)
A 180 m train at 36 km/h crosses a bridge 270 m long. How long does it take?
Speed = 36 x (5/18) = 10 m/s
Total distance = 180 + 270 = 450 m
Time = 450 / 10 = 45 seconds
Answer: 45 seconds
Problem 6: Two Trains -- Opposite Direction (Moderate)
Two trains, 120 m and 180 m long, run in opposite directions at 54 km/h and 36 km/h. How long do they take to cross each other?
Two trains opposite direction:
+--120 m--+ +--180 m--+
| |-----> <-----| |
+---------+ 54km/h +---------+ 36km/h
Total distance = 120 + 180 = 300 m
Relative speed = 54 + 36 = 90 km/h = 90 x (5/18) = 25 m/s
Time = 300 / 25 = 12 seconds
Answer: 12 seconds
Problem 7: Two Trains -- Same Direction (Moderate)
A train 200 m long running at 72 km/h overtakes a train 300 m long running at 54 km/h. How long does it take?
Same direction:
+--200 m--+ +--300 m--+
| Fast |--> | Slow |-->
+---------+ +---------+
72 km/h 54 km/h
Total distance = 200 + 300 = 500 m
Relative speed = 72 - 54 = 18 km/h = 18 x (5/18) = 5 m/s
Time = 500 / 5 = 100 seconds
Answer: 100 seconds (1 min 40 sec)
Problem 8: Train Crossing a Man Walking -- Same Direction (Moderate)
A train 150 m long passes a man walking at 6 km/h in the same direction in 10 seconds. Find the speed of the train.
Man walking same direction:
+--150 m--+
| TRAIN |-----> S_train
+---------+
o----> 6 km/h (same direction)
Distance = 150 m (only train's length; man is a point)
Relative speed = S_train - 6 km/h
Let train speed = S km/h
Relative speed in m/s = (S - 6) x 5/18
Time = 150 / [(S - 6) x 5/18]
10 = 150 x 18 / [5(S - 6)]
10 = 2700 / [5(S - 6)]
10 = 540 / (S - 6)
S - 6 = 54
S = 60 km/h
Answer: 60 km/h
Problem 9: Train Crossing a Man Walking -- Opposite Direction (Moderate)
A 200 m train at 72 km/h passes a man running at 8 km/h in the opposite direction. How long does it take?
Opposite direction:
+--200 m--+
| TRAIN |-----> 72 km/h
+---------+
<----o 8 km/h
Relative speed = 72 + 8 = 80 km/h = 80 x (5/18) = 200/9 m/s
Time = 200 / (200/9) = 200 x 9/200 = 9 seconds
Answer: 9 seconds
Problem 10: Finding Speed from Pole and Platform (Moderate)
A train crosses a pole in 18 seconds and a 360 m long platform in 36 seconds. Find the length and speed of the train.
Using the shortcut:
Speed = Platform length / (T_platform - T_pole)
= 360 / (36 - 18)
= 360 / 18
= 20 m/s
= 72 km/h
Length of train = Speed x T_pole = 20 x 18 = 360 m
Answer: Speed = 72 km/h, Length = 360 m
Problem 11: Man on a Train Watching a Platform (Moderate)
A man sitting in a train travelling at 60 km/h observes that a platform takes 15 seconds to pass him completely. Find the length of the platform.
The man is a point observer ON the train.
The platform appears to pass him at the train's speed.
Speed = 60 x (5/18) = 50/3 m/s
Length of platform = Speed x Time = (50/3) x 15 = 250 m
Answer: 250 m
NOTE: The train's own length is irrelevant here.
Problem 12: Man on Train Watching Another Train -- Opposite (Moderate)
A man in a train 120 m long at 54 km/h sees a train 180 m long at 36 km/h coming from the opposite direction. How long does the other train take to pass him?
The man is a point. He watches the OTHER train pass.
Distance = Length of OTHER train = 180 m (not 120 + 180)
Relative speed = 54 + 36 = 90 km/h = 25 m/s
Time = 180 / 25 = 7.2 seconds
Answer: 7.2 seconds
CONTRAST: If asked "how long for the two trains to cross each other":
Distance would be 120 + 180 = 300 m
Time = 300 / 25 = 12 seconds (different answer!)
Problem 13: Two Trains Meeting Between Stations (Moderate)
Two trains start at the same time from stations 450 km apart, heading towards each other. Train A goes at 60 km/h and Train B at 90 km/h. After how long do they meet, and where?
A --------450 km---------> B
60 km/h ---> <--- 90 km/h
Relative speed = 60 + 90 = 150 km/h
Time to meet = 450 / 150 = 3 hours
Meeting point from A = 60 x 3 = 180 km
Meeting point from B = 90 x 3 = 270 km
Check: 180 + 270 = 450 km (correct)
Answer: 3 hours, 180 km from A
Problem 14: Train Crossing a Tunnel (Moderate)
A train 250 m long travelling at 90 km/h enters a tunnel 500 m long. How long does it take for the train to completely pass through the tunnel?
"Completely pass through" means from the front entering to the rear exiting.
Total distance = L_train + L_tunnel = 250 + 500 = 750 m
Speed = 90 x (5/18) = 25 m/s
Time = 750 / 25 = 30 seconds
Answer: 30 seconds
Problem 15: Finding Both Speeds of Two Trains (Advanced)
Two trains of lengths 100 m and 200 m cross each other in 10 seconds when moving in opposite directions and in 50 seconds when moving in the same direction. Find the speed of each train.
L1 + L2 = 100 + 200 = 300 m
Opposite direction: (S1 + S2) = 300 / 10 = 30 m/s ...(1)
Same direction: (S1 - S2) = 300 / 50 = 6 m/s ...(2)
(assuming S1 > S2)
Adding (1) and (2): 2.S1 = 36 --> S1 = 18 m/s = 64.8 km/h
Subtracting: 2.S2 = 24 --> S2 = 12 m/s = 43.2 km/h
Verification:
Opposite: 300 / (18+12) = 300/30 = 10 s (correct)
Same: 300 / (18-12) = 300/6 = 50 s (correct)
Answer: S1 = 64.8 km/h, S2 = 43.2 km/h
Problem 16: Train Passing Man Then Platform (Advanced)
A train passes a man standing on a platform in 8 seconds and passes the platform (264 m long) in 20 seconds. Find the length and speed of the train.
Crossing man (like pole): L / S = 8 ...(1)
Crossing platform: (L + 264) / S = 20 ...(2)
Shortcut: S = 264 / (20 - 8) = 264 / 12 = 22 m/s
Length: L = 22 x 8 = 176 m
Speed in km/h = 22 x (18/5) = 79.2 km/h
Answer: Length = 176 m, Speed = 79.2 km/h
Problem 17: Two Trains of Equal Length (Advanced)
Two trains of equal length cross a pole in 5 seconds and 8 seconds respectively. Find the time they take to cross each other when moving in opposite directions.
Let length of each train = L
Speed of Train A: Sa = L/5
Speed of Train B: Sb = L/8
Opposite direction:
Total distance = L + L = 2L
Relative speed = L/5 + L/8 = L(8+5)/40 = 13L/40
Time = 2L / (13L/40) = 2L x 40/(13L) = 80/13 = 6.15 seconds
Answer: 80/13 seconds (approximately 6.15 seconds)
Problem 18: Train and Man Running on Bridge (Advanced)
A 300 m train at 72 km/h approaches a 200 m bridge. A man stands at the center of the bridge. At what minimum speed should the man run to avoid being hit by the train? The man starts running when the train is 100 m from the bridge.
+--300m--+ 100m |===200m===|
| TRAIN |---------> | o |
+--------+ | (man) |
72km/h | center |
Train speed = 72 x (5/18) = 20 m/s
Train is 100 m from the bridge entrance.
Man is at center of 200 m bridge = 100 m from each end.
Option 1: Man runs TOWARDS the train (to exit before train arrives)
Man needs to run 100 m to reach the near end.
Train needs to travel 100 m to reach the near end.
Time for train = 100/20 = 5 seconds
Man's speed = 100/5 = 20 m/s (very fast, not practical)
Option 2: Man runs AWAY from the train (to exit far end before train reaches)
Man needs to run 100 m to reach the far end.
Train needs to travel 100 + 200 + 300 = 600 m to completely cross the bridge.
Wait -- the man just needs to be past the far end before the train reaches
the far end. The train's front reaches the far end after:
Distance = 100 + 200 = 300 m
Time = 300/20 = 15 seconds
Man needs to cover 100 m in 15 seconds.
Man's speed = 100/15 = 20/3 m/s = 6.67 m/s = 24 km/h
Answer: 24 km/h (running away from the train)
Problem 19: Trains Leaving at Different Times (Advanced)
Train A leaves station X at 8:00 AM at 60 km/h towards Y. Train B leaves station Y at 9:00 AM at 80 km/h towards X. Distance XY = 280 km. At what time do the trains meet?
In 1 hour (before B starts), A covers: 60 x 1 = 60 km
Remaining distance when B starts = 280 - 60 = 220 km
Relative speed (towards each other) = 60 + 80 = 140 km/h
Time after 9:00 AM = 220 / 140 = 11/7 hours
= 1 hour + 4/7 hour
= 1 hour + 34.3 minutes
= 1 hour 34 minutes (approximately)
Meeting time = 9:00 AM + 1 hr 34 min = 10:34 AM (approx)
Distance from X = 60 x (1 + 11/7) = 60 x 18/7 = 1080/7 = 154.3 km
Answer: Approximately 10:34 AM, about 154 km from X
Problem 20: Train Crossing Two People at Different Speeds (Advanced)
A train crosses a man walking at 5 km/h in the same direction in 20 seconds, and a man walking at 3 km/h in the opposite direction in 15 seconds. Find the speed and length of the train.
Let train speed = S km/h, length = L m
Same direction (man at 5 km/h):
Relative speed = (S - 5) km/h = (S-5) x 5/18 m/s
L = [(S-5) x 5/18] x 20 = 100(S-5)/18 ...(1)
Opposite direction (man at 3 km/h):
Relative speed = (S + 3) km/h = (S+3) x 5/18 m/s
L = [(S+3) x 5/18] x 15 = 75(S+3)/18 ...(2)
Equating (1) and (2):
100(S-5)/18 = 75(S+3)/18
100(S-5) = 75(S+3)
100S - 500 = 75S + 225
25S = 725
S = 29 km/h
L = 100(29-5)/18 = 100 x 24/18 = 2400/18 = 133.33 m
Answer: Speed = 29 km/h, Length = 400/3 m (approximately 133.33 m)
Problem 21: Longest Time to Cross (Advanced)
A train 400 m long runs at 72 km/h. Find the time to: (a) cross a pole (b) cross a 600 m platform (c) cross a man walking at 4 km/h (same direction) (d) cross a man running at 8 km/h (opposite direction)
Speed = 72 km/h = 20 m/s
(a) Crossing pole:
T = 400/20 = 20 seconds
(b) Crossing platform:
T = (400+600)/20 = 1000/20 = 50 seconds
(c) Crossing man (same direction):
Man speed = 4 x 5/18 = 10/9 m/s
Relative speed = 20 - 10/9 = 170/9 m/s
T = 400 / (170/9) = 3600/170 = 21.18 seconds
(d) Crossing man (opposite):
Man speed = 8 x 5/18 = 20/9 m/s
Relative speed = 20 + 20/9 = 200/9 m/s
T = 400 / (200/9) = 3600/200 = 18 seconds
Answer: (a) 20 s (b) 50 s (c) 21.18 s (d) 18 s
Problem 22: Finding Length of a Faster Train (Advanced)
A train running at 108 km/h completely passes a slower train running at 72 km/h in the same direction in 50 seconds. If the slower train is 250 m long, find the length of the faster train.
Relative speed = 108 - 72 = 36 km/h = 10 m/s
Total distance = L_fast + L_slow = L + 250
Time = (L + 250) / 10 = 50
L + 250 = 500
L = 250 m
Answer: The faster train is 250 m long.
Problem 23: Train Passing a Signal Post Then Overtaking (Advanced)
Train A passes a signal post in 12 seconds. It then overtakes Train B (300 m long, running at 36 km/h in the same direction) in 60 seconds. If Train A's speed is 72 km/h, find its length.
Speed of A = 72 km/h = 20 m/s
Length of A: L = 20 x 12 = 240 m
Overtaking Train B:
Relative speed = 72 - 36 = 36 km/h = 10 m/s
Total distance = L_A + L_B = 240 + 300 = 540 m
Time = 540 / 10 = 54 seconds
Wait -- the problem says 60 seconds. Let me re-read.
Hmm, if the overtaking time is 60 seconds:
(L_A + 300) / 10 = 60
L_A + 300 = 600
L_A = 300 m
But from pole crossing: L_A = 20 x 12 = 240 m
These are inconsistent, which means the problem might intend us to find
the length using the overtaking info:
Let L_A = length of A.
(L_A + 300) / 10 = 60
L_A = 300 m
Speed from pole = 300/12 = 25 m/s = 90 km/h
Since the problem states 72 km/h, let us use the given data:
L_A = 20 x 12 = 240 m
Answer: 240 m (from the signal post data, which is more direct)
Problem 24: Three Trains Problem (Advanced)
Train A (200 m) and Train B (300 m) travel in the same direction at 60 km/h and 40 km/h respectively. Train C (150 m) travels in the opposite direction at 50 km/h. How long does Train C take to cross Train A? How long to cross Train B?
Train C crosses Train A (opposite directions):
Relative speed = 60 + 50 = 110 km/h = 110 x 5/18 = 550/18 m/s
Distance = 200 + 150 = 350 m
Time = 350 / (550/18) = 350 x 18/550 = 6300/550 = 11.45 seconds
Train C crosses Train B (opposite directions):
Relative speed = 40 + 50 = 90 km/h = 90 x 5/18 = 25 m/s
Distance = 300 + 150 = 450 m
Time = 450 / 25 = 18 seconds
Answer: C crosses A in 11.45 s, C crosses B in 18 s
Problem 25: The Classic "Telegraph Post" Spacing Problem (Advanced)
A man in a train observes that he passes 12 telegraph posts in 1 minute. If the distance between two consecutive posts is 120 m, what is the speed of the train in km/h?
12 posts means 11 gaps (fence-post counting).
Total distance = 11 x 120 = 1320 m
Time = 1 minute = 60 seconds
Speed = 1320 / 60 = 22 m/s = 22 x (18/5) = 79.2 km/h
Answer: 79.2 km/h
COMMON MISTAKE: Using 12 gaps instead of 11. Remember:
n posts = (n-1) gaps.
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