Time and Distance — SSC GD Study Notes

Overview

Time and Distance is a high-scoring topic in SSC GD Elementary Mathematics. Every year, expect 2–3 direct questions on speed, average speed, trains, and boats-and-streams. The topic tests your ability to convert units, apply the fundamental formula, and handle relative motion scenarios. Mastering this section gives you quick marks because once you learn the standard patterns, most problems solve in under 60 seconds.

The key to success is understanding the single core relationship: **Distance = Speed × Time**. All variations — average speed, train crossing, boats moving upstream or downstream — simply rearrange or extend this formula. Focus on unit conversions (km/h ↔ m/s) and recognise problem types quickly. With 15–20 practice problems across each sub-topic, you'll develop the speed and accuracy needed for exam conditions.

Common mistakes include forgetting unit conversions, confusing relative speeds, and misapplying average speed formulas. The notes below address these traps directly.

Key Concepts

• **Fundamental Relationship**: Distance = Speed × Time. Every problem boils down to finding one quantity when the other two are known.
• **Unit Conversion**: 1 km/h = 5/18 m/s. Multiply km/h by 5/18 to get m/s; multiply m/s by 18/5 to get km/h. Trains and platform problems almost always need m/s.
• **Average Speed**: When distances are equal, average speed = 2xy/(x+y) where x and y are the two speeds. When times are equal, average speed is the simple arithmetic mean (x+y)/2. Never just average speeds blindly.
• **Relative Speed**: When two objects move in the same direction, relative speed = |Speed₁ − Speed₂|. When moving toward each other (opposite directions), relative speed = Speed₁ + Speed₂.
• **Train Problems**: Length of train matters. Time to cross a pole/man = (Length of train) / (Speed of train). Time to cross a platform = (Length of train + Length of platform) / (Speed of train).
• **Boats and Streams**: Speed in still water = b, stream speed = s. Downstream speed = b + s, upstream speed = b − s. If downstream and upstream speeds are given, still water speed = (downstream + upstream)/2 and stream speed = (downstream − upstream)/2.
• **Meeting and Crossing**: If two objects start simultaneously from opposite ends, time to meet = (Total distance) / (Sum of speeds). If they start from the same point moving in the same direction, time for faster to lap = (Initial gap) / (Difference in speeds).
• **Return Journey**: If a person travels to a place at speed x and returns at speed y, the average speed for the whole journey is 2xy/(x+y), not (x+y)/2.

Formulas / Key Facts

1. **Distance = Speed × Time** — Core formula. Rearrange as Speed = Distance/Time or Time = Distance/Speed. 2. **Conversion: km/h to m/s** — Multiply by 5/18. Example: 72 km/h = 72 × 5/18 = 20 m/s. 3. **Conversion: m/s to km/h** — Multiply by 18/5. Example: 15 m/s = 15 × 18/5 = 54 km/h. 4. **Average Speed (equal distances)** — Average speed = 2xy/(x+y) if distances covered at speeds x and y are equal. 5. **Average Speed (equal times)** — Average speed = (x+y)/2 if time spent at speeds x and y are equal. 6. **Relative Speed (same direction)** — Relative speed = Speed₁ − Speed₂. 7. **Relative Speed (opposite direction)** — Relative speed = Speed₁ + Speed₂. 8. **Train crossing a pole** — Time = Length of train / Speed of train. 9. **Train crossing a platform** — Time = (Length of train + Length of platform) / Speed of train. 10. **Two trains crossing each other** — Time = (Length₁ + Length₂) / Relative speed. 11. **Downstream speed** — b + s (b = boat speed in still water, s = stream speed). 12. **Upstream speed** — b − s. 13. **Still water and stream speed from downstream/upstream** — b = (downstream + upstream)/2 and s = (downstream − upstream)/2.

Worked Examples

**Example 1: Basic Speed-Distance-Time** A car travels 180 km in 3 hours. Find its speed. **Solution:** Speed = Distance / Time = 180/3 = 60 km/h.

**Example 2: Average Speed (Equal Distances)** A person travels 120 km at 40 km/h and returns the same distance at 60 km/h. Find the average speed. **Solution:** Use the formula for equal distances: Average speed = 2xy/(x+y). Average speed = 2 × 40 × 60 / (40 + 60) = 4800/100 = 48 km/h. (Note: Do NOT calculate (40+60)/2 = 50. That is wrong.)

**Example 3: 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 m/s and km/h. **Solution:** Total distance = Length of train + Length of platform = 150 + 250 = 400 m. Speed = Distance / Time = 400/20 = 20 m/s. Convert to km/h: 20 × 18/5 = 72 km/h.

**Example 4: Boats and Streams** A boat travels downstream 60 km in 4 hours and upstream 40 km in 5 hours. Find the speed of the boat in still water and the speed of the stream. **Solution:** Downstream speed = 60/4 = 15 km/h. Upstream speed = 40/5 = 8 km/h. Speed in still water = (15 + 8)/2 = 23/2 = 11.5 km/h. Stream speed = (15 − 8)/2 = 7/2 = 3.5 km/h.

**Example 5: Two Trains Meeting** Two trains, each 100 m long, are moving toward each other at 54 km/h and 36 km/h. In how much time will they cross each other? **Solution:** Convert speeds to m/s: 54 km/h = 54 × 5/18 = 15 m/s, 36 km/h = 36 × 5/18 = 10 m/s. Relative speed = 15 + 10 = 25 m/s (opposite directions). Total length to cross = 100 + 100 = 200 m. Time = 200/25 = 8 seconds.

Common Mistakes

1. **Mistake: Averaging speeds as (x+y)/2 when distances are equal.** **Fix:** Use the harmonic mean formula 2xy/(x+y) for equal distances. Simple arithmetic mean applies only when times are equal.

2. **Mistake: Forgetting to convert km/h to m/s in train problems.** **Fix:** Train lengths are in metres and time in seconds, so always convert speed to m/s using ×5/18. Forgetting this step leads to wrong numerical answers.

3. **Mistake: Adding train lengths when a train crosses a stationary object (pole, man).** **Fix:** When crossing a pole or man, use only the train's length. Add platform/bridge length only when the train crosses a platform or bridge.

4. **Mistake: Using wrong relative speed direction.** **Fix:** Same direction → subtract speeds. Opposite direction → add speeds. Visualise the scenario: are they coming together or chasing?

5. **Mistake: Confusing downstream and upstream formulas.** **Fix:** Downstream is with the current, so speed increases: b + s. Upstream is against the current, so speed decreases: b − s. Remember "down = add, up = subtract."

Quick Reference

• **D = S × T** — Distance equals speed times time. All problems stem from this.
• **km/h to m/s: ×5/18** — Essential for train problems with lengths in metres.
• **Average speed (equal D): 2xy/(x+y)** — Not the arithmetic mean when distances are equal.
• **Relative speed: same direction subtract, opposite direction add.**
• **Train + pole: use train length only; train + platform: add both lengths.**
• **Boats: downstream = b+s, upstream = b−s; derive b and s from these.**