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Numerical Aptitude Syllabus / Module 9 · Time, Speed & Distance / Week 9

Week 9: Time, Speed & Distance

Week 8's rate model, moved from a job to a number line. Speed is just "distance covered per unit time" — the same rate idea, and combining two moving objects' speeds is Week 8's combined-rate idea, applied with one new wrinkle: direction matters here in a way it never did for two people painting a wall.

Module 9 of 20 Week 9 of 20 ~2–3 Hours Practice Problems Included

By the end of this week, you'll be able to

  • Compute relative speed correctly for same-direction and opposite-direction motion
  • Solve train problems, including a train crossing a platform vs. a pole
  • Solve boats-and-streams problems using upstream/downstream speed
1 The Core Formula & Unit Conversion 2 Relative Speed 3 Trains — Crossing Platforms, Poles & Each Other 4 Boats & Streams 5 Practice Problems 6 Knowledge Check

1. The Core Formula & Unit Conversion

Everything in this chapter is Distance = Speed × Time, rearranged. The one mechanical trap: test questions mix km/h and m/s freely, and forgetting to convert is the single most common source of wrong answers.

the conversion factor worth memorizing
km/h to m/s: multiply by 5/18
m/s to km/h: multiply by 18/5

Worked: convert 72 km/h to m/s
  72 x 5/18 = 20 m/s

Worked: convert 25 m/s to km/h
  25 x 18/5 = 90 km/h

The fraction 5/18 comes directly from unit definitions: 1 km = 1000m and 1 hour = 3600s, so 1 km/h = 1000/3600 m/s = 5/18 m/s. Re-deriving it once is enough to trust it permanently, rather than risking a flipped fraction under time pressure.

2. Relative Speed

When two objects move, how fast they close the gap (or pull apart) between them depends on their relative speed — and whether you add or subtract their speeds depends entirely on direction.

same direction vs. opposite direction
SAME direction (one chasing another):
  Relative speed = |Speed_A - Speed_B|  (SUBTRACT)

  Two cars, one at 80 km/h and one at 50 km/h, moving the
  same way -- the gap closes at 80-50 = 30 km/h

OPPOSITE directions (moving toward or away from each other):
  Relative speed = Speed_A + Speed_B  (ADD)

  Two trains approaching each other at 60 km/h and 40 km/h --
  the gap closes at 60+40 = 100 km/h

The intuition: moving in the same direction, the faster one only gains ground at the difference in speeds — if both moved at identical speed, the gap would never change at all. Moving toward each other, both speeds contribute fully to closing the distance, so they add.

3. Trains — Crossing Platforms, Poles & Each Other

A train problem is a relative-speed problem with one added subtlety: the train's own length matters, because "crossing" something means the ENTIRE train has passed it, not just the front.

crossing a stationary point (pole/person) vs. a platform
Crossing a POLE (a point, effectively zero length):
  Time = (Train's own length) / Speed

Crossing a PLATFORM (has its own length):
  Time = (Train's length + Platform's length) / Speed

Worked: a 150m train moving at 54 km/h crosses a 250m platform.
Find the time taken.

  Speed = 54 x 5/18 = 15 m/s
  Distance to cover = 150 + 250 = 400m
  Time = 400/15 = 26.67 seconds
two trains crossing each other
Two trains, lengths 120m and 180m, moving TOWARD each other
at 45 km/h and 36 km/h. Find the time to cross each other.

  Relative speed = 45+36 = 81 km/h = 81 x 5/18 = 22.5 m/s
  Total distance = 120+180 = 300m  (BOTH lengths -- crossing
                                     is complete only when the
                                     trains fully clear each other)
  Time = 300/22.5 = 13.33 seconds

The pattern to internalize: distance = sum of both objects' lengths whenever "crossing" involves two extended objects (train-platform, train-train), and speed = relative speed using Section 2's same/opposite-direction rule.

4. Boats & Streams

A current adds to or subtracts from a boat's own speed, depending on direction — this is Section 2's relative-speed idea again, with the stream itself acting as one of the two "objects."

upstream / downstream formulas
Let b = boat's speed in still water, s = stream's speed

Downstream speed (WITH the current) = b + s
Upstream speed (AGAINST the current) = b - s

Given downstream and upstream speeds, recover b and s:
  b = (Downstream + Upstream) / 2
  s = (Downstream - Upstream) / 2

Worked: a boat's downstream speed is 16 km/h, upstream speed
is 10 km/h. Find the boat's speed and the stream's speed.

  b = (16+10)/2 = 13 km/h
  s = (16-10)/2 = 3 km/h

This b/s recovery formula is just Week 6's ratio-combination logic in a new outfit: downstream and upstream speeds are (b+s) and (b−s); their average cancels the ±s term and leaves b, and half their difference isolates s — a direct application of solving two linear equations by adding and subtracting them.

5. Practice Problems

Practice

30 problems, no calculator — Easy, Medium & Tough

Convert every speed to consistent units before combining. Easy problems should take under 30 seconds each; Medium under 60 seconds; Tough under 90 seconds.

Easy (1–10)

  1. Convert 36 km/h to m/s.
  2. Convert 15 m/s to km/h.
  3. Find the distance covered by a car traveling at 60 km/h for 3 hours.
  4. Find the time taken to cover 150km at 50 km/h.
  5. Find the speed of a car covering 240km in 4 hours.
  6. A man walks at 5 km/h. Find the distance covered in 30 minutes.
  7. Convert 90 km/h to m/s.
  8. Find the time to cover 500m at 10 m/s.
  9. A train travels at 72 km/h. Find its speed in m/s.
  10. Find the distance covered in 45 minutes at 40 km/h.
Easy — Answers

1) 36×5/18=10 m/s. 2) 15×18/5=54 km/h. 3) 60×3=180km. 4) 150/50=3 hours. 5) 240/4=60 km/h. 6) 5×0.5=2.5km. 7) 90×5/18=25 m/s. 8) 500/10=50 sec. 9) 72×5/18=20 m/s. 10) 40×0.75=30km.

Medium (11–20)

  1. Two cars start from the same point, moving in the same direction, at 40 km/h and 60 km/h. Find the distance between them after 3 hours.
  2. Two trains move toward each other at 50 km/h and 70 km/h. Find the time to meet if they start 360km apart.
  3. A 100m train crosses a pole in 5 seconds. Find its speed in km/h.
  4. A 150m train crosses a 250m platform in 20 seconds. Find its speed in km/h.
  5. A boat's speed in still water is 10 km/h and the stream's speed is 2 km/h. Find the downstream speed.
  6. A boat's downstream speed is 18 km/h and upstream speed is 10 km/h. Find the boat's speed in still water.
  7. Two trains, 130m and 120m long, moving in opposite directions at 50 km/h and 40 km/h, cross each other. Find the time taken.
  8. A man covers a certain distance at 6 km/h and returns at 4 km/h. Find his average speed for the whole journey.
  9. A car covers 360km, half at 60 km/h and half at 90 km/h. Find the average speed.
  10. A boat's speed in still water is 15 km/h and stream speed is 3 km/h. Find the time to cover 36km downstream.
Medium — Answers

11) Relative speed=20km/h; distance=20×3=60km. 12) Relative speed=120km/h; time=360/120=3 hours. 13) Speed=100/5=20 m/s=72 km/h. 14) Distance=400m; speed=400/20=20 m/s=72 km/h. 15) 10+2=12km/h. 16) (18+10)/2=14km/h. 17) Relative speed=90 km/h=25 m/s; distance=250m; time=250/25=10 sec. 18) Average=2×6×4/10=4.8 km/h. 19) Average=2×60×90/150=72 km/h. 20) Downstream speed=18km/h; time=36/18=2 hours.

Tough (21–30)

  1. A train 240m long crosses a platform in 24 seconds and a pole in 12 seconds. Find the length of the platform.
  2. Two trains start simultaneously from stations A and B, 300km apart, and travel toward each other at 40 km/h and 60 km/h. Find the distance from A where they meet.
  3. A boat covers 15km upstream and 15km downstream in a total of 4 hours. If the boat's speed in still water is 8 km/h, find the stream's speed.
  4. A man can row 10 km/h in still water. If the river runs at 2 km/h, find his total time to row 24km upstream and 24km back downstream.
  5. A car covers 300km. If the speed is increased by 10 km/h, the time taken is reduced by 1 hour. Find the original speed.
  6. Two stations A and B are 340km apart. A train starts from A at 8:00am toward B at 60 km/h. Another starts from B at 9:00am toward A at 80 km/h. Find the time they meet.
  7. A thief runs at 8 km/h. A policeman starts chasing him 30 minutes later at 10 km/h. Find the time it takes the policeman to catch the thief, from when the policeman starts.
  8. A man walking at 5 km/h crosses a bridge in 15 minutes. Find the length of the bridge in meters.
  9. A train overtakes two persons walking in the same direction at 2 km/h and 4 km/h, passing them completely in 9 and 10 seconds respectively. Find the train's speed and length.
  10. Two trains, 400m and 600m long, run on parallel tracks. Moving in the same direction, the faster crosses the slower in 100 seconds; moving in opposite directions, they cross in 20 seconds. Find the speeds of both trains.
Tough — Answers

21) Train speed=240/12=20 m/s; (240+P)/20=24 → P=240m. 22) Relative speed=100km/h; time=3h; distance from A=40×3=120km. 23) 15/(8-s)+15/(8+s)=4 → 15×16=4(64-s²) → s²=4 → s=2 km/h. 24) Up=8,down=12; time=24/8+24/12=3+2=5 hours. 25) 300/x-300/(x+10)=1 → x²+10x-3000=0 → x=50 km/h. 26) Train1 covers 60km before 9am; remaining=280km; combined speed=140km/h; time=2h → meet at 11:00am. 27) Head start=8×0.5=4km; relative gain=2km/h; time=4/2=2 hours. 28) 5×(15/60)=1.25km=1,250m. 29) (v-2)×9=(v-4)×10 → v=22km/h; length=(22-2)×(5/18)×9=50m. 30) x-y=1km÷(100/3600h)=36km/h; x+y=1km÷(20/3600h)=180km/h → x=108km/h, y=72km/h.

6. Knowledge Check

Four quick questions. Expand each to check your answer.

Q1

Why does relative speed use SUBTRACTION for two objects moving in the same direction, but ADDITION for objects moving toward each other?

Moving the same direction, only the SPEED DIFFERENCE actually closes the gap between them — if both moved at the same speed, the gap would never change at all, so equal speeds must cancel to zero, which subtraction does. Moving toward each other, both speeds fully contribute to closing the distance simultaneously, so their full magnitudes add together.

Q2

Why does a train crossing a platform need to cover (train length + platform length), rather than just the platform's length?

"Crossing" means the entire train has fully passed the platform, which requires the train's own length to also travel clear of the platform's far edge — the front of the train has to travel the platform's length, AND then the rest of the train's length has to follow before the whole train has cleared it.

Q3

Why does finding a boat's speed in still water use the AVERAGE of its downstream and upstream speeds?

Downstream speed is (boat speed + stream speed) and upstream speed is (boat speed - stream speed) — adding these two expressions cancels the stream speed term entirely (+s and -s sum to zero), leaving exactly twice the boat's own speed, so dividing that sum by 2 isolates the boat's speed alone.

Q4

Why is converting all speeds to a single consistent unit (km/h or m/s) before combining them a genuinely important step, not just a formality?

Adding or subtracting quantities expressed in different units (like adding a km/h speed directly to an m/s speed) produces a number that doesn't correspond to any real physical quantity — the arithmetic is only meaningful once both values represent the same unit, since relative speed and distance formulas assume a single consistent unit system throughout.

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