19September 2026
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Quantitative Aptitude · 21 min read
Time, Speed & Distance
Time, Speed and Distance is one formula wearing a dozen costumes. Trains, boats, races and "he walks slower and arrives late" questions all reduce to distance = speed × time, and none of them is difficult once you can see which of the three quantities the question has held constant.
Two ideas do most of the work. The first is that when distance is fixed, speed and time are inversely proportional — which turns a whole family of questions into a one-line ratio. The second is relative speed, which is what makes trains, boats and races the same problem.
Current affairs · 19 September 2026
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Why it matters
Time, Speed & Distance in the exam
| Exam | Expected questions | How it usually appears |
|---|---|---|
| SSC CGL / CHSL Tier 1 | 3–5 questions | Usually one train, one boat and one straight speed question. |
| Banking Prelims & Mains | 2–4 questions | Boats and trains recur; sometimes inside a caselet. |
| RRB NTPC / Group D | 4–6 questions | The highest weight of any exam — trains especially. |
| CAT / MBA entrances | 2–4 questions | Circular tracks, meeting points and multi-leg journeys. |
Start here
The formula and units
Everything on this page comes from one equation. The marks are lost in unit conversion, not in the algebra.
The core relationships
- Distance = Speed × TimeAnd therefore Speed = Distance ÷ Time and Time = Distance ÷ Speed.
- km/h → m/s: multiply by 5/1872 km/h = 72 × 5/18 = 20 m/s. Every train question needs this.
- m/s → km/h: multiply by 18/515 m/s = 15 × 18/5 = 54 km/h.
- Keep all three quantities in one system before computingMetres with seconds, or kilometres with hours. Mixing them is the single most common error in the topic.
Which conversion, and when
Train and race questions are in metres and seconds, so convert speeds to m/s. Journey questions are in kilometres and hours, so leave them alone. A quick sanity check: 5/18 is less than 1, so converting km/h to m/s must make the number smaller.
Guaranteed question
Average speed
Average speed is total distance divided by total time. It is almost never the average of the two speeds, and the question is set precisely because candidates assume it is.
The formulas
- Average speed = total distance ÷ total timeThe definition. When in doubt, compute both totals and divide — it is never wrong.
- Equal DISTANCES at u and v: average = 2uv ÷ (u + v)The harmonic mean. Going 30 km at 30 km/h and 30 km at 60 km/h averages 40 km/h, not 45.
- Equal TIMES at u and v: average = (u + v) ÷ 2Only here is the plain average correct. Read carefully which is held equal.
- Three equal distances at u, v, w: average = 3uvw ÷ (uv + vw + wu)The same harmonic pattern extended.
The average is always closer to the slower speed
You spend more time at the slower speed, so it carries more weight. For 30 and 60 km/h over equal distances the answer is 40 — below the midpoint of 45. If your answer lands above the midpoint, you have used the wrong formula.
The big one
The inverse-ratio shortcut
When the distance is fixed, speed and time are inversely proportional. A whole family of "late by ten minutes" questions collapses to one line with this.
The relationship
- Fixed distance ⇒ Speed ∝ 1 ÷ TimeIf speeds are in the ratio a : b, the times are in the ratio b : a. Simply flip it.
- Walking at a/b of the usual speed ⇒ time becomes b/a of usualAt 3/4 speed the journey takes 4/3 of the usual time — an extra 1/3.
- The extra time equals the difference in the time ratioTime ratio 4 : 3 means the difference is 1 part, and that part is the lateness the question gives you.
Walking at 3/4 of his usual speed, a man reaches his office 10 minutes late. What is his usual time?
- Speeds are in the ratio 3 : 4 (new : usual).
- Distance is fixed, so times are in the inverse ratio 4 : 3 (new : usual).
- The difference is 4 − 3 = 1 part, and that part is the 10 minutes of lateness.
- The usual time is 3 parts.
Answer: 30 minutes.
Read which ratio is which
At 3/4 of the usual speed, the NEW speed is 3 and the usual is 4. Getting these the wrong way round gives 40 minutes instead of 30, and both appear in the options. The slower speed must give the longer time.
Core skill
Relative speed
When two bodies move, what matters is how fast the gap between them changes. That is the relative speed, and it is what makes trains, boats and races one topic rather than three.
The two cases
- Opposite directions (approaching or crossing): relative speed = u + vThe gap closes at the sum, so crossing happens quickly.
- Same direction (overtaking): relative speed = u − vThe gap closes at the difference, so overtaking takes much longer.
- Time to meet = initial gap ÷ relative speedTwo cars 300 km apart approaching at 40 and 60 km/h meet after 300 ÷ 100 = 3 hours.
The tell in the question
Words like "towards each other", "cross" and "meet" mean add. Words like "overtake", "same direction" and "catch up" mean subtract. If a same-direction answer comes out smaller than an opposite-direction one for the same pair, you have swapped them.
Highest yield
Trains
A train is not a point — it has length, and that length is part of the distance covered. Deciding what distance is involved is the whole question.
| Situation | Distance covered | Speed to use |
|---|---|---|
| Train crosses a pole, post or standing person | Length of the train | Speed of the train |
| Train crosses a platform, bridge or tunnel | Train length + platform length | Speed of the train |
| Two trains cross, opposite directions | Sum of both lengths | Sum of the speeds |
| Faster train overtakes slower, same direction | Sum of both lengths | Difference of the speeds |
| Train passes a man walking towards it | Length of the train | Sum of the speeds |
| Train passes a man walking away from it | Length of the train | Difference of the speeds |
A train 240 m long travelling at 72 km/h crosses a platform 360 m long. How long does it take?
- Convert the speed: 72 × 5/18 = 20 m/s.
- The train must clear its own length plus the platform: 240 + 360 = 600 m.
- 600 ÷ 20
Answer: 30 seconds.
Two trains 120 m and 180 m long run in opposite directions at 40 km/h and 50 km/h. How long do they take to cross each other?
- Opposite directions, so the relative speed is 40 + 50 = 90 km/h.
- 90 × 5/18 = 25 m/s.
- Distance = 120 + 180 = 300 m.
- 300 ÷ 25
Answer: 12 seconds.
A person has no length
When a train passes a pole or a person, the distance is just the train's length — a pole and a person are treated as points. Only platforms, bridges and other trains add length.
Relative speed again
Boats and streams
The current helps you one way and hinders you the other. That is all there is to it, and the two formulas for recovering the boat and stream speeds are worth memorising exactly.
The four relationships
- Downstream speed = boat speed + stream speedGoing with the current.
- Upstream speed = boat speed − stream speedGoing against it. If this is zero or negative the boat cannot make headway.
- Boat speed in still water = (downstream + upstream) ÷ 2The average of the two observed speeds.
- Stream speed = (downstream − upstream) ÷ 2Half the difference.
A boat covers 30 km downstream in 2 hours and returns in 3 hours. Find the speed of the boat in still water and the speed of the stream.
- Downstream speed = 30 ÷ 2 = 15 km/h.
- Upstream speed = 30 ÷ 3 = 10 km/h.
- Boat = (15 + 10) ÷ 2 = 12.5 km/h.
- Stream = (15 − 10) ÷ 2 = 2.5 km/h.
Answer: Boat 12.5 km/h, stream 2.5 km/h.
A round trip is never the plain average
Rowing a fixed distance down and back at 15 and 10 km/h gives an average of 2 × 15 × 10 / 25 = 12 km/h, not 12.5. The still-water speed and the average speed of the round trip are different numbers, and both appear in the options.
Question shapes
Races and head starts
Race questions are relative-speed questions with a specific vocabulary. Translate the phrase into "in the same time, A covers this and B covers that" and it becomes a ratio.
- "A beats B by x metres"
- In the time A finishes the race, B has covered (race length − x). Since the time is the same, the ratio of speeds equals the ratio of those distances. In a 200 m race won by 25 m, speeds are 200 : 175 = 8 : 7.
- "A beats B by t seconds"
- B needs t more seconds to finish the same distance. Use that to find B's speed, then compare.
- "A gives B a start of x metres"
- B runs (race length − x) while A runs the whole distance. A head start in metres shortens B's course.
- "A gives B a start of t seconds"
- B begins t seconds earlier, so A's running time is t less than B's for the same finish.
- Dead heat
- Both finish together, so their times are equal. Set the two times equal and solve for whatever the question left unknown.
In a 100 m race, A beats B by 20 m. If A runs at 5 m/s, what is B's speed?
- A covers 100 m at 5 m/s, taking 20 seconds.
- In those same 20 seconds B covers only 80 m.
- 80 ÷ 20
Answer: 4 m/s.
Read this twice
Traps that cost marks
Four recurring errors, none of them mathematical.
- Forgetting to convert units
- A train length in metres divided by a speed in km/h gives nonsense. Convert to m/s with 5/18 before dividing, every time.
- Averaging the two speeds
- Over equal distances the average speed is the harmonic mean 2uv/(u+v), not (u+v)/2. The plain average is right only when the TIMES are equal.
- Leaving out the train's own length
- Crossing a platform means covering the platform plus the train. Only a pole or a person can be treated as a point.
- Inverting the speed ratio the wrong way
- At 3/4 of the usual speed the time becomes 4/3 of usual, not 3/4. The slower you go, the longer it takes — check your answer against that before moving on.
Solved examples
Worked line by line
Read the steps rather than the answer. The method is what transfers to the next question.
Convert 72 km/h into metres per second.
- Multiply by 5/18.
- 72 × 5/18 = 360/18
Answer: 20 m/s.
A man travels 30 km at 30 km/h and another 30 km at 60 km/h. Find his average speed.
- The distances are equal, so use the harmonic mean.
- Average = 2 × 30 × 60 ÷ (30 + 60)
- 3600 ÷ 90
- Check the long way: 1 hour + 0.5 hours = 1.5 hours for 60 km.
Answer: 40 km/h — not 45.
A train 180 m long crosses a pole in 12 seconds. Find its speed in km/h.
- Crossing a pole means covering only the train's own length.
- 180 ÷ 12 = 15 m/s
- 15 × 18/5
Answer: 54 km/h.
A train 300 m long overtakes another train 200 m long moving in the same direction. Their speeds are 70 km/h and 50 km/h. How long does the overtaking take?
- Same direction, so relative speed = 70 − 50 = 20 km/h.
- 20 × 5/18 = 50/9 m/s.
- Distance = 300 + 200 = 500 m.
- 500 ÷ (50/9) = 500 × 9/50
Answer: 90 seconds.
A boat travels 24 km upstream in 3 hours. If the boat's speed in still water is 12 km/h, find the speed of the stream.
- Upstream speed = 24 ÷ 3 = 8 km/h.
- Upstream = boat − stream, so 8 = 12 − stream.
Answer: The stream flows at 4 km/h.
Two cars 300 km apart drive towards each other at 40 km/h and 60 km/h. After how long do they meet?
- Approaching, so relative speed = 40 + 60 = 100 km/h.
- 300 ÷ 100
Answer: 3 hours.
Walking at 5/6 of his usual speed, a man is 8 minutes late. Find his usual time.
- Speeds new : usual = 5 : 6.
- Times invert to new : usual = 6 : 5.
- The difference of 1 part is the 8 minutes of lateness.
- Usual time is 5 parts.
Answer: 40 minutes.
In a 200 m race A beats B by 25 m. Find the ratio of their speeds.
- In the time A runs 200 m, B runs 200 − 25 = 175 m.
- Equal times mean the speed ratio equals the distance ratio.
- 200 : 175, divided by 25.
Answer: 8 : 7.
A man walking at 6 km/h reaches his office 5 minutes early; at 4 km/h he is 5 minutes late. How far is the office?
- The two journeys differ by 10 minutes, which is 1/6 of an hour.
- Let the distance be d. Then d/4 − d/6 = 1/6.
- Common denominator: d(3 − 2)/12 = d/12 = 1/6.
- d = 12/6
Answer: 2 km.
A train travelling at 54 km/h passes a man walking at 6 km/h in the same direction in 20 seconds. Find the length of the train.
- Same direction, so relative speed = 54 − 6 = 48 km/h.
- 48 × 5/18 = 40/3 m/s.
- Passing a man covers only the train's length.
- Length = (40/3) × 20
Answer: 266.67 m, or 800/3 metres.
Practice
16 questions on Time, Speed & Distance
Work each one out before you reveal the answer — the explanation is where the marks are.
Q190 km/h expressed in metres per second is:
- A25 m/s
- B30 m/s
- C18 m/s
- D20 m/s
Q2A car travelling at 45 km/h covers, in 3 hours, a distance of:
- A120 km
- B135 km
- C90 km
- D150 km
Q3Travelling 60 km at 40 km/h and returning at 60 km/h, the average speed is:
- A45 km/h
- B50 km/h
- C48 km/h
- D52 km/h
Q4A train 150 m long crosses a pole in 10 seconds. Its speed is:
- A60 km/h
- B45 km/h
- C50 km/h
- D54 km/h
Q5A train 200 m long at 54 km/h crosses a bridge 250 m long in:
- A30 seconds
- B40 seconds
- C25 seconds
- D35 seconds
Q6Two trains 150 m and 100 m long run in opposite directions at 60 and 40 km/h. They cross in:
- A10 seconds
- B9 seconds
- C12 seconds
- D15 seconds
Q7A boat with a still-water speed of 12 km/h travels 30 km downstream in a 3 km/h current in:
- A2.5 hours
- B3 hours
- C2 hours
- D1.5 hours
Q8A boat covers 24 km upstream in 3 hours. If its still-water speed is 12 km/h, the stream flows at:
- A3 km/h
- B5 km/h
- C2 km/h
- D4 km/h
Q9Two runners cover the same distance with speeds in the ratio 4 : 5. Their times are in the ratio:
- A5 : 4
- B16 : 25
- C1 : 1
- D4 : 5
Q10Walking at 5/6 of his usual speed a man is 8 minutes late. His usual time is:
- A48 minutes
- B40 minutes
- C36 minutes
- D45 minutes
Q11In a 200 m race A beats B by 25 m. The ratio of their speeds is:
- A4 : 3
- B25 : 200
- C8 : 7
- D7 : 8
Q12Two cars 300 km apart approach each other at 40 and 60 km/h. They meet after:
- A2.5 hours
- B2 hours
- C4 hours
- D3 hours
Q13A cyclist rides 20 km at 10 km/h and 20 km at 20 km/h. His average speed is:
- A13.33 km/h
- B16 km/h
- C12 km/h
- D15 km/h
Q14A train 300 m long overtakes a 200 m train going the same way; speeds are 70 and 50 km/h. It takes:
- A45 seconds
- B90 seconds
- C60 seconds
- D75 seconds
Q15A man walking at 6 km/h is 5 minutes early; at 4 km/h he is 5 minutes late. The distance is:
- A3 km
- B1.5 km
- C2 km
- D2.5 km
Q16A train at 54 km/h passes a man walking at 6 km/h in the same direction in 20 seconds. The train is:
- A333.33 m long
- B400 m long
- C300 m long
- D266.67 m long
Questions
Time, Speed & Distance — FAQs
What is the basic formula for time, speed and distance?
Distance = Speed × Time, which rearranges to Speed = Distance ÷ Time and Time = Distance ÷ Speed. Every question on the topic is this equation with one quantity held constant; the difficulty is always in identifying which one.
How do I convert km/h to m/s?
Multiply by 5/18. So 72 km/h becomes 20 m/s. To go the other way, multiply by 18/5. Since 5/18 is less than 1, converting to m/s must always make the number smaller — a quick way to catch the mistake.
Why is average speed not the average of the two speeds?
Because you spend more time at the slower speed, so it carries more weight. Over equal distances the correct figure is the harmonic mean, 2uv/(u+v). At 30 and 60 km/h that gives 40, not 45. The plain average is right only when the two TIMES are equal, not the distances.
What is relative speed?
The rate at which the gap between two moving bodies changes. Moving in opposite directions the speeds add; moving in the same direction they subtract. This one idea covers trains crossing, boats in a current, and overtaking questions alike.
What distance does a train cover when crossing a platform?
Its own length plus the platform's length, because the rear of the train must clear the far end. Crossing a pole or a person covers only the train's length, since those are treated as points with no length of their own.
How do I find the speed of a boat and of the stream?
Boat speed in still water = (downstream + upstream) ÷ 2, and stream speed = (downstream − upstream) ÷ 2. If a boat does 15 km/h downstream and 10 km/h upstream, the boat is 12.5 km/h and the stream 2.5 km/h.
What does "A beats B by 20 metres" mean?
That in the time A finished the race, B had covered 20 m less. Since both ran for the same time, the ratio of their speeds equals the ratio of the distances they covered. In a 100 m race that is 100 : 80 = 5 : 4.
How do I solve "walking at 3/4 of his usual speed he is 10 minutes late"?
With the inverse ratio. Speeds are 3 : 4, so times are 4 : 3. The difference of one part is the 10 minutes of lateness, so the usual time is 3 parts — 30 minutes. Reversing the ratio gives 40 minutes, which is the trap answer.
Is the still-water speed the same as the average speed of a round trip?
No, and both are usually in the options. Rowing a fixed distance at 15 km/h down and 10 km/h up gives a still-water speed of 12.5 km/h but a round-trip average of 2 × 15 × 10 / 25 = 12 km/h.
How much of the exam is Time, Speed and Distance?
It carries the highest weight in railway exams — four to six questions in RRB NTPC, trains especially. Expect three to five in SSC CGL and CHSL and two to four in banking. Together with Time and Work it is one of the two biggest arithmetic blocks.
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