Time, Speed and Distance — Study Notes

Overview

Time, Speed and Distance (TSD) forms a cornerstone of the Numerical Ability section in SSC MTS Paper 1, typically accounting for 3–5 questions per exam. This topic tests your ability to connect three fundamental quantities: the distance traveled, the speed of movement, and the time taken. While the basic relationship is straightforward, SSC MTS questions add layers of complexity through scenarios involving trains crossing platforms, boats moving in rivers with currents, and problems involving relative motion between two objects.

Mastery of this topic requires comfort with unit conversions (km/h to m/s), fractional and percentage-based speed changes, and the discipline to draw quick diagrams for train and boat problems. The good news: once you internalize the core formulas and practice 20–30 problems, pattern recognition kicks in and most questions become routine applications. The topic also overlaps with ratio-proportion thinking, so a solid grasp here strengthens your overall quantitative reasoning.

Expect direct calculation questions (find speed given distance and time), train problems (crossing poles, platforms, or other trains), boat-stream problems (upstream/downstream speeds), and relative speed scenarios (two objects moving toward or away from each other). Every question is solvable in under 90 seconds if you know the right formula and avoid unit-conversion mistakes.

Key Concepts

• **The fundamental relationship**: Distance = Speed × Time. Any problem in this topic is a manipulation of these three variables. If two are given, the third can always be found.

• **Unit consistency is non-negotiable**: Speed in km/h with time in hours gives distance in km. Speed in m/s with time in seconds gives distance in meters. The most common error is mixing units without conversion.

• **Conversion between km/h and m/s**: To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5. Memorize this — it appears in nearly every train problem.

• **Relative speed depends on direction**: When two objects move in the same direction, relative speed = difference of speeds. When moving toward each other (opposite directions), relative speed = sum of speeds.

• **Boats and streams introduce effective speeds**: A boat's speed in still water is its own speed. Downstream speed = boat speed + stream speed. Upstream speed = boat speed − stream speed. The stream helps downstream, opposes upstream.

• **Train problems require visualizing lengths**: A train crossing a stationary point (pole, man) covers its own length. A train crossing a platform covers train length + platform length. Two trains crossing each other cover the sum of both lengths.

• **Average speed ≠ average of speeds**: If an object covers equal distances at two different speeds, the average speed is the harmonic mean: (2×s₁×s₂)/(s₁+s₂), not the arithmetic mean (s₁+s₂)/2.

• **Time saved or lost problems use percentage changes**: If speed increases by x%, time decreases by [x/(100+x)]×100%. If speed decreases by x%, time increases by [x/(100−x)]×100%.

Formulas / Key Facts

1. **Distance = Speed × Time**; Speed = Distance/Time; Time = Distance/Speed 2. **km/h to m/s**: Multiply by 5/18 — example: 72 km/h = 72 × 5/18 = 20 m/s 3. **m/s to km/h**: Multiply by 18/5 — example: 15 m/s = 15 × 18/5 = 54 km/h 4. **Relative speed (opposite directions)**: s₁ + s₂ 5. **Relative speed (same direction)**: |s₁ − s₂| 6. **Downstream speed**: Boat speed + Stream speed 7. **Upstream speed**: Boat speed − Stream speed 8. **Boat speed in still water**: (Downstream + Upstream)/2 9. **Stream speed**: (Downstream − Upstream)/2 10. **Time for train to cross a pole/man**: Time = (Length of train)/Speed of train 11. **Time for train to cross a platform/bridge**: Time = (Length of train + Length of platform)/Speed 12. **Average speed for two equal distances at speeds s₁ and s₂**: (2s₁s₂)/(s₁+s₂)

Worked Examples

**Example 1: Basic Distance Calculation** A car travels at 60 km/h for 3.5 hours. What distance does it cover?

*Solution*: Distance = Speed × Time = 60 × 3.5 = 210 km. **Answer: 210 km**

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**Example 2: Train Crossing a Platform** A 150 m long train crosses a 250 m long platform in 20 seconds. Find the speed of the train in km/h.

*Solution*: Total distance covered = Length of train + Length of platform = 150 + 250 = 400 m Time = 20 seconds Speed = Distance/Time = 400/20 = 20 m/s Convert to km/h: 20 × 18/5 = 72 km/h **Answer: 72 km/h**

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**Example 3: Boat and Stream** A boat can row at 8 km/h in still water. The stream flows at 2 km/h. How long does it take the boat to travel 30 km downstream?

*Solution*: Downstream speed = Boat speed + Stream speed = 8 + 2 = 10 km/h Time = Distance/Speed = 30/10 = 3 hours **Answer: 3 hours**

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**Example 4: Relative Speed — Opposite Directions** Two trains, 120 m and 180 m long, are traveling toward each other at 45 km/h and 36 km/h respectively. In how many seconds will they cross each other?

*Solution*: Relative speed = 45 + 36 = 81 km/h = 81 × 5/18 = 22.5 m/s Total distance to cover = 120 + 180 = 300 m Time = 300/22.5 = 13.33 seconds (or 40/3 seconds) **Answer: 13.33 seconds**

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**Example 5: Average Speed** A person travels 120 km at 40 km/h and returns the same distance at 60 km/h. What is the average speed for the entire journey?

*Solution*: Do not average the speeds directly. Average speed = (2 × 40 × 60)/(40 + 60) = 4800/100 = 48 km/h **Answer: 48 km/h**

Common Mistakes

• **Mixing units without conversion** → Always convert km/h to m/s (or vice versa) before plugging into formulas. For train problems with lengths in meters and time in seconds, use m/s. For distances in km and time in hours, use km/h.

• **Using arithmetic mean for average speed** → The average of 40 km/h and 60 km/h is not 50 km/h when equal distances are traveled. Always use the harmonic mean formula for equal-distance scenarios.

• **Forgetting to add lengths in train problems** → When a train crosses a platform, the distance covered equals train length + platform length, not just the platform. When two trains cross, add both train lengths.

• **Confusing upstream and downstream** → Downstream means going with the current (add stream speed to boat speed). Upstream means against the current (subtract stream speed). A common slip: reversing these operations.

• **Incorrectly applying relative speed direction rule** → If two objects move in the same direction, subtract speeds (the faster one "catches up" at the difference). If opposite, add speeds (they "approach" at the sum). Reversing this leads to wrong answers.

Quick Reference

• Distance = Speed × Time; always maintain unit consistency.
• Convert km/h ↔ m/s: multiply by 5/18 or 18/5 respectively.
• Relative speed: add for opposite directions, subtract for same direction.
• Downstream = boat + stream; Upstream = boat − stream.
• Train crossing pole: distance = train length; platform: train + platform length.
• Average speed over equal distances at s₁, s₂: use (2s₁s₂)/(s₁+s₂), not (s₁+s₂)/2.