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Time & Distance

Chapter Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Convert between km/h and m/s without a calculator.
2. Solve problems on trains crossing poles, platforms, and other trains.
3. Handle boats & streams (upstream/downstream) questions.
4. Crack relative speed, average speed, and circular track problems.
5. Spot the traps examiners set in SSC CGL, IBPS PO, and RRB NTPC papers.

The Hook: The Train That "Shrank"

Imagine you're standing on a platform. A 240-metre-long train zooms past you in 12 seconds. Simple, right?

Now the same train crosses a 360-metre-long platform. How long does it take?

Most students instantly say: "It took 12 seconds for 240 metres, so 360 metres should take 18 seconds."

Wrong. The correct answer is 30 seconds.

Why? Because when a train crosses *you* (a point), it only needs to cover its own length. But when it crosses a *platform*, it must cover its own length PLUS the platform's length — 240 + 360 = 600 metres. At 20 m/s, that's 30 seconds.

This one idea — *"what distance is actually being covered?"* — is the heart of almost every Time & Distance question in competitive exams. Master it, and this chapter becomes free marks.

Let's Be Honest First

If Time & Distance questions make you nervous, you're not alone. Students usually struggle for three reasons:

• Unit confusion — mixing km/h with metres and seconds.
• Relative speed — not knowing when to add and when to subtract speeds.
• Hidden distances — forgetting that a train must "drag its own body" across a platform.

We'll fix all three, one small chunk at a time.

Chunk 1: The One Formula That Rules Them All

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

From this single formula come two more:

• Distance = Speed × Time
• Time = Distance ÷ Speed

Everything in this chapter — trains, boats, races, circular tracks — is just this formula wearing different costumes.

The Golden Triangle trick: Draw a triangle with D on top, S and T at the bottom. Cover the one you want to find:

• Cover D → S × T
• Cover S → D ÷ T
• Cover T → D ÷ S

🧠 Active Recall: Without looking back — if a car covers 150 km in 3 hours, what is its speed? And how long would it take to cover 200 km at that speed? *(Answer: 50 km/h; 4 hours)*

Chunk 2: Unit Conversion — The 5/18 Magic Number

Examiners love giving speed in km/h and distance in metres. You must convert.

$$1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{5}{18} \text{ m/s}$$

So:

• km/h → m/s: multiply by 5/18
• m/s → km/h: multiply by 18/5

Example: 72 km/h = 72 × 5/18 = 20 m/s. (This is why the train in our hook moved 240 m in 12 s.)

Mnemonic 🎭: *"Small guy, big appetite."* m/s is the small unit, so to reach it you multiply by the small fraction 5/18. km/h is the big unit, so you multiply by the big fraction 18/5. Small wants small, big wants big.

🧠 Active Recall: Convert 90 km/h to m/s, and 15 m/s to km/h. *(Answer: 25 m/s; 54 km/h)*

Chunk 3: Average Speed — The Trap Everyone Falls Into

Question: A man goes from A to B at 40 km/h and returns at 60 km/h. What is his average speed for the whole journey?

If you said 50 km/h — you just walked into the examiner's favourite trap. The average of 40 and 60 is 50, but average speed is NOT the average of speeds, because he spends *more time* travelling at the slower speed.

Average Speed = Total Distance ÷ Total Time

For equal distances at speeds *x* and *y*:

$$\text{Average Speed} = \frac{2xy}{x+y}$$

Here: (2 × 40 × 60) / (40 + 60) = 4800/100 = 48 km/h.

For three equal-distance speeds x, y, z:

$$\text{Average Speed} = \frac{3xyz}{xy + yz + zx}$$

🧠 Active Recall: A cyclist covers a certain distance at 30 km/h and returns the same distance at 20 km/h. Average speed? *(Answer: 2×30×20/50 = 24 km/h — not 25!)*

Chunk 4: Relative Speed — When Two Things Move

This is the concept that separates toppers from the rest. The rule is beautifully simple:

• Opposite directions (towards or away from each other): ADD the speeds.
• Same direction (one chasing the other): SUBTRACT the speeds.

Example 1 (opposite): Two trains, 60 km/h and 40 km/h, approach each other. Relative speed = 100 km/h. They close the gap fast.

Example 2 (same direction): A police jeep at 80 km/h chases a thief at 60 km/h, with 10 km between them. Relative speed = 20 km/h. Time to catch = 10/20 = 0.5 hour = 30 minutes.

Memory hook 🎭: *"Enemies add, friends subtract."* Moving opposite (like enemies facing off) → speeds ADD. Moving the same way (like friends walking together) → the faster one only gains by the DIFFERENCE.

🧠 Active Recall: Two runners on a straight road run in the same direction at 12 km/h and 9 km/h. If the faster one starts 6 km behind, when does he catch up? *(Answer: relative speed 3 km/h, time = 2 hours)*

Chunk 5: Trains — The "Drag Your Body" Rule

Remember the hook? Here's the complete rulebook:

Example: Train A (150 m, 54 km/h) meets Train B (200 m, 36 km/h) coming from the opposite direction. Time to cross each other?

• Relative speed = 54 + 36 = 90 km/h = 25 m/s
• Distance = 150 + 200 = 350 m
• Time = 350/25 = 14 seconds

🧠 Active Recall: A 180 m train crosses a pole in 9 seconds. What is its speed in km/h? *(Answer: 20 m/s = 72 km/h)*

Chunk 6: Boats & Streams — The River Helps or Fights

• Speed of boat in still water = B, Speed of stream = S
• Downstream (with the current): B + S — the river helps you.
• Upstream (against the current): B − S — the river fights you.

And the reverse formulas examiners love:

$$B = \frac{\text{Downstream} + \text{Upstream}}{2}, \qquad S = \frac{\text{Downstream} - \text{Upstream}}{2}$$

Example: A boat goes 30 km downstream in 2 hours and returns upstream in 6 hours.

• Downstream speed = 15 km/h, Upstream speed = 5 km/h
• Boat speed B = (15+5)/2 = 10 km/h
• Stream speed S = (15−5)/2 = 5 km/h

🧠 Active Recall: A boat's speed in still water is 12 km/h and the stream flows at 3 km/h. How long to go 45 km downstream? *(Answer: 45/15 = 3 hours)*

Chunk 7: Classic Exam Models

1. The "late/early" model: Walking at 3/4 of his usual speed, a man is 20 minutes late. Usual time?

• Speed becomes 3/4 → time becomes 4/3. Extra time = 1/3 of usual = 20 min → Usual time = 60 minutes.
• Shortcut: Usual time = (late time × new speed fraction) ÷ (1 − fraction) = (20 × 3/4) ÷ (1/4) = 60 min.

2. Circular tracks: Two people run around a circular track. They meet (anywhere on the track) every:

• Same direction: L ÷ (difference of speeds)
• Opposite direction: L ÷ (sum of speeds)
  where L = track length. They meet at the starting point after LCM of their individual lap times.

3. Stoppages: A train travels at 60 km/h without stoppage and 45 km/h with stoppages. Stoppage time per hour = (60−45)/60 × 60 = 15 minutes per hour.

🧠 Active Recall: A bus covers a route at 50 km/h without stops and 40 km/h with stops. How many minutes per hour does it halt? *(Answer: (10/50)×60 = 12 minutes)*

Summary Table

Your Next Step

You've just built the complete Time & Distance toolkit: one master formula, the 5/18 conversion, the add/subtract relative-speed rule, and the "drag your body" train rule. That's 90% of what SSC, Banking, and Railway exams ask.

⏰ Review this chapter after 2 days. Close the book, and from memory write down: the 5/18 rule, the average speed formula, and the train-crossing table. If any one of them feels fuzzy, re-read just that chunk — not the whole chapter.

Then move to the Practice Sets. Speed in this topic comes from solving, not re-reading. You've got this! 💪