Episode 8 — Aptitude and Reasoning / 8.12 — Problems on Trains
8.12.b Tips, Tricks, and Shortcuts -- Problems on Trains
Tip 1: Always Convert to m/s First
The number one mistake in train problems is a unit mismatch. Train lengths are in metres, so convert speed to m/s immediately.
STEP 1 of every train problem: Convert km/h to m/s
Speed (m/s) = Speed (km/h) x 5/18
Quick table:
36 km/h = 10 m/s 72 km/h = 20 m/s
45 km/h = 12.5 m/s 90 km/h = 25 m/s
54 km/h = 15 m/s 108 km/h = 30 m/s
60 km/h = 50/3 m/s 120 km/h = 100/3 m/s
Tip 2: The "What Crosses What" Decision Framework
Before writing any equation, answer these two questions:
Question 1: What is the TOTAL DISTANCE to cover?
+----------------------------------------------------+
| Scenario | Distance |
|------------------------|---------------------------|
| Train passes pole | L_train |
| Train passes platform | L_train + L_platform |
| Train passes train | L_train1 + L_train2 |
| Train passes man on | |
| another train | L_passing_train ONLY |
+----------------------------------------------------+
Question 2: What is the RELATIVE SPEED?
+----------------------------------------------------+
| Scenario | Relative Speed |
|------------------------|---------------------------|
| Object is stationary | Speed of train |
| Same direction | Difference of speeds |
| Opposite direction | Sum of speeds |
+----------------------------------------------------+
Then: Time = Distance / Relative Speed
Tip 3: The Two-Equation Method
When a train crosses a pole in T1 seconds and a platform of length P in T2 seconds:
Speed = P / (T2 - T1) [Shortcut -- no need for two equations]
Then: Length of train = Speed x T1
WHY THIS WORKS:
The EXTRA time (T2 - T1) is spent covering the platform length P.
So P / (T2 - T1) gives the speed directly.
Example:
A train crosses a pole in 12 seconds and a 240 m platform in 24 seconds.
Speed = 240 / (24 - 12) = 240 / 12 = 20 m/s
Length = 20 x 12 = 240 m
Tip 4: Same Train, Two Platforms -- Direct Formula
If a train crosses Platform 1 (length P1) in T1 seconds
and Platform 2 (length P2) in T2 seconds:
Speed = (P2 - P1) / (T2 - T1) [if P2 > P1]
Length = Speed x T1 - P1
Tip 5: Two Trains Crossing Each Other -- Speed Unknowns
If two trains cross each other in opposite directions in T_opp seconds and in the same direction in T_same seconds:
S1 + S2 = (L1 + L2) / T_opp
S1 - S2 = (L1 + L2) / T_same
Adding: 2.S1 = (L1+L2)(1/T_opp + 1/T_same)
Subtracting: 2.S2 = (L1+L2)(1/T_opp - 1/T_same)
Tip 6: "Man on Train" vs "Man on Platform" Trap
This is the most common trap in train problems.
"A train crosses a man standing on a platform"
--> Man is stationary. Distance = L_train. Speed = S_train.
"A train crosses a man sitting in another train (opposite direction)"
--> Man is a point. Distance = L_passing_train ONLY.
Speed = S1 + S2 (opposite).
"Two trains cross each other"
--> Both have length. Distance = L1 + L2.
Speed = S1 + S2 (opposite) or |S1 - S2| (same).
The key distinction:
- "crosses a man" --> only the train's length matters
- "crosses another train" --> both lengths matter
Tip 7: Ratio Shortcut for Same Train Crossing Different Objects
If a train crosses a pole in T1 seconds and a platform in T2 seconds:
L_train / (L_train + L_platform) = T1 / T2
This gives: L_platform / L_train = (T2 - T1) / T1
L_platform = L_train x (T2 - T1) / T1
Example:
Train length = 200 m, crosses pole in 10 s, platform in 25 s.
Platform length = 200 x (25-10)/10 = 200 x 1.5 = 300 m
Tip 8: Speed of Train When Crossing a Walking Man
Man walking SAME direction as train:
Speed of train = L / T + Speed of man (in same units)
Man walking OPPOSITE to train:
Speed of train = L / T - Speed of man
NOTE: L/T gives the relative speed. Add/subtract the man's speed
to get the train's actual speed.
Tip 9: Finding Train Length from Crossing Times
If a train crosses two objects of known lengths in known times:
L + P1 L + P2
------ = ------ (same speed, so equate S)
T1 T2
Cross multiply: T2(L + P1) = T1(L + P2)
T2.L + T2.P1 = T1.L + T1.P2
L(T2 - T1) = T1.P2 - T2.P1
L = (T1.P2 - T2.P1) / (T2 - T1)
Tip 10: Visualization Technique
Draw the scenario BEFORE writing equations. Use these ASCII templates:
Pole problem:
+---L---+
| |---->
+-------+ |
POLE
Platform problem:
+---L---+
| |====P====|
+-------+ |
Two trains (opposite):
+---L1---+ +---L2---+
| |----> |<-------|
+--------+ +--------+
Two trains (same direction):
+---L1---+ +---L2---+
| |-->| |-->
+--------+ +--------+
(faster) (slower)
Tip 11: The "Passenger on the Train" Perspective
When a passenger ON a moving train watches something:
Watching a pole: Time = 0 (pole is a point, passes instantly)
Actually, the pole appears to zip past.
Watching a platform: Time = L_platform / S_train
(Only platform length matters)
Watching another train (opposite): Time = L_other / (S1 + S2)
(Only OTHER train's length matters)
Watching another train (same dir): Time = L_other / |S1 - S2|
(Only OTHER train's length matters)
Tip 12: Common Problem Patterns and Quick Approaches
Pattern 1: "Find the speed of the train"
Usually given: Length + Time to cross something
Approach: S = Total distance / Time
Pattern 2: "Find the length of the train"
Usually given: Speed + Time to cross a pole
Approach: L = S x T
Pattern 3: "Find the length of the platform"
Usually given: Train length + Speed + Time to cross platform
Approach: L_p = S x T - L_train
Pattern 4: "Two trains crossing, find time"
Given: Both lengths + Both speeds + Direction
Approach: T = (L1 + L2) / Relative speed
Pattern 5: "Find when trains meet between two stations"
Given: Distance between stations + Both speeds
Approach: T = D / (S1 + S2) [opposite direction]
Tip 13: Mental Math for Common Train Speeds
Railway problems often use these standard speeds:
Express train: 72 km/h = 20 m/s
Passenger train: 54 km/h = 15 m/s
Freight train: 36 km/h = 10 m/s
Man walking: 5 km/h = 25/18 m/s (approx 1.39 m/s)
Man running: 10 km/h = 25/9 m/s (approx 2.78 m/s)
Tip 14: Double-Check With Reasonableness
A train of 200 m at 72 km/h (20 m/s) crossing a pole:
T = 200/20 = 10 seconds (reasonable)
If your answer is 200 seconds for the same problem,
something is wrong (likely a unit error).
Rule of thumb:
- Crossing a pole: 5-30 seconds typically
- Crossing a platform: 15-60 seconds typically
- Crossing another train (opposite): 5-15 seconds
- Crossing another train (same dir): 30-120 seconds
Tip 15: Working Backwards from Time
If the time is given and you need speed or length:
From T = (L1 + L2) / S_rel:
S_rel = (L1 + L2) / T --> to find speed
L1 = S_rel x T - L2 --> to find length of one train
Summary of Key Shortcuts
1. ALWAYS convert to m/s first
2. Pole: T = L/S
3. Platform: T = (L+P)/S
4. Two equations: Speed = P/(T2-T1)
5. Same direction: use DIFFERENCE of speeds
6. Opposite direction: use SUM of speeds
7. "Man on train" = point observer = only other object's length
8. "Two trains cross" = both lengths added
9. Draw before you solve
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