Time, Speed and Distance
Overview
Time, Speed and Distance (TSD) is a foundational arithmetic topic in IBPS PO Prelims that appears in two forms: direct formula-based questions and as part of Data Interpretation sets. Mastery of this topic also unlocks related problems on trains, boats and streams, and races—all of which use the same core relationship.
In the Prelims exam, you can expect 2–4 questions on TSD concepts, often combined with ratio-based shortcuts. The key to scoring here is not just knowing formulas but developing the instinct to convert problems into the basic relationship and apply relative speed concepts quickly. Questions are designed to be solved in under 90 seconds each, so mental math and unit conversion skills are essential.
The entire topic rests on one equation: **Distance = Speed × Time**. Every variation—trains crossing, boats in streams, average speed—is just this equation applied creatively.
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Key Concepts
- **Basic Relationship**: Distance = Speed × Time. Rearrange as needed: Speed = Distance/Time, Time = Distance/Speed.
- **Unit Conversion**: To convert km/hr to m/s, multiply by 5/18. To convert m/s to km/hr, multiply by 18/5. This conversion appears in almost every train problem.
- **Relative Speed (Same Direction)**: When two objects move in the same direction, relative speed = difference of their speeds. The faster object "catches up" at this rate.
- **Relative Speed (Opposite Direction)**: When two objects move toward each other, relative speed = sum of their speeds. They approach each other at this combined rate.
- **Trains Crossing**: When a train crosses a stationary object (pole, person), distance covered = length of train. When crossing a platform or another train, distance = sum of both lengths.
- **Boats and Streams**: Downstream speed = boat speed + stream speed. Upstream speed = boat speed − stream speed. Still water speed and stream speed can be derived from these two.
- **Average Speed**: For equal distances at two different speeds, average speed = 2ab/(a+b), not the simple arithmetic mean.
- **Inverse Proportionality**: When distance is constant, speed and time are inversely proportional. If speed doubles, time halves.
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Formulas / Key Facts
| Concept | Formula | |---------|---------| | Basic relation | D = S × T | | km/hr to m/s | Multiply by 5/18 | | m/s to km/hr | Multiply by 18/5 | | Relative speed (same direction) | S₁ − S₂ | | Relative speed (opposite direction) | S₁ + S₂ | | Train crossing pole | Time = Length of train / Speed of train | | Train crossing platform | Time = (Length of train + Length of platform) / Speed | | Two trains crossing each other | Time = (L₁ + L₂) / Relative speed | | Downstream speed | B + S (boat + stream) | | Upstream speed | B − S | | Speed of boat in still water | (Downstream + Upstream) / 2 | | Speed of stream | (Downstream − Upstream) / 2 | | Average speed (equal distances) | 2ab / (a + b) |
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Worked Examples
**Example 1: Train Crossing a Platform**
A train 150 m long crosses a 250 m platform in 20 seconds. Find the speed of the train in km/hr.
*Solution:*
- Total distance = 150 + 250 = 400 m
- Time = 20 s
- Speed = 400/20 = 20 m/s
- Convert to km/hr: 20 × 18/5 = **72 km/hr**
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**Example 2: Two Trains Crossing Each Other**
Two trains of lengths 120 m and 180 m are running toward each other at 54 km/hr and 36 km/hr. In how many seconds will they cross each other?
*Solution:*
- Total distance = 120 + 180 = 300 m
- Relative speed (opposite direction) = 54 + 36 = 90 km/hr
- Convert to m/s: 90 × 5/18 = 25 m/s
- Time = 300/25 = **12 seconds**
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**Example 3: Boats and Streams**
A boat travels 24 km downstream in 3 hours and returns upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream.
*Solution:*
- Downstream speed = 24/3 = 8 km/hr
- Upstream speed = 24/4 = 6 km/hr
- Speed of boat in still water = (8 + 6)/2 = **7 km/hr**
- Speed of stream = (8 − 6)/2 = **1 km/hr**
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**Example 4: Average Speed**
A car travels from A to B at 40 km/hr and returns at 60 km/hr. What is the average speed for the entire journey?
*Solution:*
- Since distances are equal, use: 2ab/(a+b)
- Average speed = 2 × 40 × 60 / (40 + 60) = 4800/100 = **48 km/hr**
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Common Mistakes
- **Adding speeds for average speed** → Wrong. For equal distances, use 2ab/(a+b). The arithmetic mean (50 km/hr in Example 4) is incorrect.
- **Forgetting to add lengths when trains cross** → When a train crosses a platform or another train, the train must cover its own length plus the other object's length. Missing this gives wrong answers.
- **Using km/hr directly in train problems** → Train lengths are in meters, so always convert speed to m/s first. Mixing units is the #1 calculation error.
- **Confusing relative speed direction** → Same direction means subtract speeds; opposite direction means add. Reversing this logic flips your answer.
- **Ignoring stream direction effect** → Downstream makes the boat faster (add stream speed); upstream makes it slower (subtract). Students sometimes add both ways or subtract both ways.
- **Assuming boat speed equals downstream speed** → Boat speed in still water is the average of downstream and upstream speeds, not either one alone.
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Quick Reference
- D = S × T — the only formula you truly need; everything else derives from it.
- km/hr → m/s: multiply by 5/18 | m/s → km/hr: multiply by 18/5
- Train + platform: distance = sum of both lengths
- Opposite direction: add speeds | Same direction: subtract speeds
- Still water speed = (Downstream + Upstream) / 2
- Average speed for equal distances = 2ab/(a+b), never (a+b)/2