Time, Speed and Distance
CSAT · Quantitative Aptitude
Introduction
Speed questions – trains, boats, races and journeys – are a CSAT staple. Most become one-line calculations once you fix units and identify the relative speed.
Key concepts
Units
km/h to m/s: multiply by 5/18; m/s to km/h: multiply by 18/5. 72 km/h = 20 m/s.
Average speed
Average speed = total distance ÷ total time. For equal distances at x and y: 2xy/(x + y).
Relative speed
- Same direction: subtract speeds; opposite directions: add speeds.
- A train crossing a pole or a person covers its own length; crossing a platform or another train covers the sum of lengths.
Boats and streams
Downstream = boat + stream; upstream = boat − stream. Boat speed = (down + up)/2; stream = (down − up)/2.
Early and late
If a person is late at speed v₁ and early at v₂, the time difference (late + early) = d/v₁ − d/v₂, so d = (difference in hours) × v₁v₂/(v₂ − v₁).
Ratios
At constant distance, time is inversely proportional to speed: if speed becomes 4/5, time becomes 5/4 (25% more).
Races
'A beats B by 20 m in a 100 m race' means when A finishes 100 m, B has run 80 m: speed ratio 5 : 4.
- Speed = distance ÷ time
- km/h × 5/18 = m/s
- Equal distances: average = 2xy/(x + y)
- Crossing time = (L₁ + L₂)/relative speed
- Down = b + s; up = b − s
Solved examples
Example 1
A car goes from P to Q at 72 km/h and returns at 36 km/h. What is the average speed for the whole journey?
(a) 48 km/h (b) 54 km/h (c) 50 km/h (d) 24 km/h
- For equal distances, average speed = 2xy/(x + y) (not the simple average).
- = 2 × 72 × 36 / 108 = 48 km/h.
Answer: (a) 48 km/h
Example 2
A boat goes 48 km upstream and comes back. Its speed in still water is 18 km/h and the stream flows at 2 km/h. What is the total time taken?
(a) 3 hours (b) 5 1/3 hours (c) 6 2/5 hours (d) 5 2/5 hours
- Upstream speed = 18 − 2 = 16 km/h; downstream = 18 + 2 = 20 km/h.
- Time = 48/16 + 48/20 = 3 + 2 2/5 = 5 2/5 hours.
Answer: (d) 5 2/5 hours
Example 3
A train 180 m long runs at 108 km/h. How long does it take to cross a platform 150 m long?
(a) 16 seconds (b) 3 1/18 seconds (c) 11 seconds (d) 6 seconds
- Speed = 108 × 5/18 = 30 m/s.
- Distance = train length + platform length = 330 m.
- Time = 330 ÷ 30 = 11 s.
Answer: (c) 11 seconds
Example 4
Two trains 150 m and 80 m long run on parallel tracks at 54 km/h and 18 km/h in the same direction. How long do they take to cross each other?
(a) 46 seconds (b) 23 seconds (c) 11 1/2 seconds (d) 26 seconds
- Relative speed = 54 − 18 = 36 km/h = 10 m/s.
- Distance = 150 + 80 = 230 m.
- Time = 230 ÷ 10 = 23 s.
Answer: (b) 23 seconds
Example 5
Walking at 6 km/h a student reaches school 15 minutes late; at 8 km/h he is 15 minutes early. How far is the school?
(a) 7 km (b) 13 km (c) 6 km (d) 12 km
- Time difference = 15 + 15 = 30 min = 1/2 h.
- d/6 − d/8 = 1/2 → d = 1/2 × 6 × 8/(8 − 6) = 12 km.
Answer: (d) 12 km
Example 6
Two trains 200 m and 80 m long run on parallel tracks at 36 km/h and 54 km/h in opposite directions. How long do they take to cross each other?
(a) 22 2/5 seconds (b) 11 1/5 seconds (c) 14 1/5 seconds (d) 56 seconds
- Relative speed = 36 + 54 = 90 km/h = 25 m/s.
- Distance = 200 + 80 = 280 m.
- Time = 280 ÷ 25 = 11 1/5 s.
Answer: (b) 11 1/5 seconds
Example 7
Two trains 150 m and 80 m long run on parallel tracks at 72 km/h and 36 km/h in the same direction. How long do they take to cross each other?
(a) 7 2/3 seconds (b) 26 seconds (c) 23 seconds (d) 46 seconds
- Relative speed = 72 − 36 = 36 km/h = 10 m/s.
- Distance = 150 + 80 = 230 m.
- Time = 230 ÷ 10 = 23 s.
Answer: (c) 23 seconds
Example 8
Two trains 150 m and 130 m long run on parallel tracks at 72 km/h and 18 km/h in opposite directions. How long do they take to cross each other?
(a) 14 1/5 seconds (b) 11 1/5 seconds (c) 18 2/3 seconds (d) 22 2/5 seconds
- Relative speed = 72 + 18 = 90 km/h = 25 m/s.
- Distance = 150 + 130 = 280 m.
- Time = 280 ÷ 25 = 11 1/5 s.
Answer: (b) 11 1/5 seconds
- Convert to m/s only when lengths are in metres; otherwise stay in km/h.
- Use speed ratios: same distance → inverse time ratio.
- For trains in the same direction, the faster one 'gains' at the difference of speeds.
- Forgetting the train's own length.
- Mixing m/s and km/h.
- Taking the simple average of two speeds.
Practice set
Attempt all 12 questions (about 90 seconds each), then check the worked solutions below.
Q1. A boat goes 60 km upstream and comes back. Its speed in still water is 12 km/h and the stream flows at 3 km/h. What is the total time taken?
(a) 10 hours (b) 10 2/3 hours (c) 6 2/3 hours (d) 11 2/3 hours
Q2. Walking at 4 km/h a student reaches school 15 minutes late; at 5 km/h he is 10 minutes early. How far is the school?
(a) 8 1/3 km (b) 4 1/6 km (c) 3 3/4 km (d) 9 1/3 km
Q3. Walking at 6 km/h a student reaches school 5 minutes late; at 10 km/h he is 5 minutes early. How far is the school?
(a) 2 1/2 km (b) 1 1/4 km (c) 2 2/3 km (d) 3 1/2 km
Q4. A boat goes 48 km upstream and comes back. Its speed in still water is 10 km/h and the stream flows at 5 km/h. What is the total time taken?
(a) 12 4/5 hours (b) 13 4/5 hours (c) 9 3/5 hours (d) 14 4/5 hours
Q5. Walking at 3 km/h a student reaches school 5 minutes late; at 6 km/h he is 15 minutes early. How far is the school?
(a) 1 km (b) 3 km (c) 2 km (d) 4 km
Q6. A boat goes 30 km upstream and comes back. Its speed in still water is 18 km/h and the stream flows at 2 km/h. What is the total time taken?
(a) 3 1/3 hours (b) 1 7/8 hours (c) 4 3/8 hours (d) 3 3/8 hours
Q7. Walking at 5 km/h a student reaches school 15 minutes late; at 10 km/h he is 5 minutes early. How far is the school?
(a) 5 km (b) 4 1/3 km (c) 1 2/3 km (d) 3 1/3 km
Q8. A car goes from P to Q at 90 km/h and returns at 72 km/h. What is the average speed for the whole journey?
(a) 82 km/h (b) 80 km/h (c) 81 km/h (d) 40 km/h
Q9. A boat goes 60 km upstream and comes back. Its speed in still water is 10 km/h and the stream flows at 5 km/h. What is the total time taken?
(a) 12 hours (b) 16 hours (c) 17 hours (d) 18 hours
Q10. Two trains 200 m and 80 m long run on parallel tracks at 36 km/h and 18 km/h in the same direction. How long do they take to cross each other?
(a) 59 seconds (b) 18 2/3 seconds (c) 56 seconds (d) 112 seconds
Q11. A boat goes 48 km upstream and comes back. Its speed in still water is 10 km/h and the stream flows at 2 km/h. What is the total time taken?
(a) 6 hours (b) 9 3/5 hours (c) 11 hours (d) 10 hours
Q12. Walking at 3 km/h a student reaches school 15 minutes late; at 12 km/h he is 15 minutes early. How far is the school?
(a) 7 1/2 km (b) 3 km (c) 2 km (d) 1 km
Q1 – (b) 10 2/3 hours. Upstream speed = 12 − 3 = 9 km/h; downstream = 12 + 3 = 15 km/h. Time = 60/9 + 60/15 = 6 2/3 + 4 = 10 2/3 hours.
Q2 – (a) 8 1/3 km. Time difference = 15 + 10 = 25 min = 5/12 h. d/4 − d/5 = 5/12 → d = 5/12 × 4 × 5/(5 − 4) = 8 1/3 km.
Q3 – (a) 2 1/2 km. Time difference = 5 + 5 = 10 min = 1/6 h. d/6 − d/10 = 1/6 → d = 1/6 × 6 × 10/(10 − 6) = 2 1/2 km.
Q4 – (a) 12 4/5 hours. Upstream speed = 10 − 5 = 5 km/h; downstream = 10 + 5 = 15 km/h. Time = 48/5 + 48/15 = 9 3/5 + 3 1/5 = 12 4/5 hours.
Q5 – (c) 2 km. Time difference = 5 + 15 = 20 min = 1/3 h. d/3 − d/6 = 1/3 → d = 1/3 × 3 × 6/(6 − 3) = 2 km.
Q6 – (d) 3 3/8 hours. Upstream speed = 18 − 2 = 16 km/h; downstream = 18 + 2 = 20 km/h. Time = 30/16 + 30/20 = 1 7/8 + 1 1/2 = 3 3/8 hours.
Q7 – (d) 3 1/3 km. Time difference = 15 + 5 = 20 min = 1/3 h. d/5 − d/10 = 1/3 → d = 1/3 × 5 × 10/(10 − 5) = 3 1/3 km.
Q8 – (b) 80 km/h. For equal distances, average speed = 2xy/(x + y) (not the simple average). = 2 × 90 × 72 / 162 = 80 km/h.
Q9 – (b) 16 hours. Upstream speed = 10 − 5 = 5 km/h; downstream = 10 + 5 = 15 km/h. Time = 60/5 + 60/15 = 12 + 4 = 16 hours.
Q10 – (c) 56 seconds. Relative speed = 36 − 18 = 18 km/h = 5 m/s. Distance = 200 + 80 = 280 m. Time = 280 ÷ 5 = 56 s.
Q11 – (d) 10 hours. Upstream speed = 10 − 2 = 8 km/h; downstream = 10 + 2 = 12 km/h. Time = 48/8 + 48/12 = 6 + 4 = 10 hours.
Q12 – (c) 2 km. Time difference = 15 + 15 = 30 min = 1/2 h. d/3 − d/12 = 1/2 → d = 1/2 × 3 × 12/(12 − 3) = 2 km.
- ×5/18 to m/s
- Relative: same direction −, opposite +
- Train + platform lengths
- Boat: (d + u)/2, stream: (d − u)/2
