Distance, Speed & Time
1. Core Idea
One formula generates every question in this topic:
Distance = Speed x Time
The skill is not the formula — it is building the right table of who travels what, and spotting which quantity is shared between the two travellers (same distance? same time? total time?).
2. Must-Know Rules
| Quantity | Formula |
|---|---|
| Distance | S x T |
| Speed | D / T |
| Time | D / S |
| Average speed | Total distance / Total time |
| Same distance, two speeds | avg = 2ab/(a + b) |
| Same time, two speeds | avg = (a + b)/2 |
| Closing speed (towards each other) | s1 + s2 |
| Closing speed (same direction) | s1 - s2 |
Set up a table. For any two-traveller problem:
| Distance | Speed | Time | |
|---|---|---|---|
| A | |||
| B |
Fill in what you know, express the rest in one variable, and use the shared quantity to write the equation.
Relative motion.
- Approaching each other: the gap closes at
s1 + s2. Time to meet = gap / (s1 + s2). - Chasing: the gap closes at
s1 - s2. Time to catch = gap / (s1 - s2).
Unit consistency. Speed in km/h needs time in hours. 45 minutes is 0.75 hours, not 45.
3. Worked Examples
Example 1 — Same distance
A train covers a distance at 60 km/h and returns at 90 km/h. What is the average speed for the round trip?
Equal distances -> harmonic mean:
2(60)(90)/(60 + 90) = 10800/150 = 72 km/h.
Trap answer: 75 (the plain average). Correct answer must be below 75 because more time is spent at the slower speed.
Example 2 — Catching up
A cyclist leaves at 9:00 am travelling at 15 km/h. A car leaves the same point at 11:00 am at 60 km/h along the same road. At what time does the car catch the cyclist?
By 11:00 am, the cyclist has gone 2 x 15 = 30 km — that is the gap.
Closing speed = 60 - 15 = 45 km/h.
Time to close = 30/45 = 2/3 hour = 40 minutes.
The car catches up at 11:40 am.
Example 3 — Two legs
Sara drives 120 km at 40 km/h, then 180 km at 60 km/h. What is her average speed for the whole journey?
Time leg 1 = 120/40 = 3 h. Time leg 2 = 180/60 = 3 h.
Total distance 300 km, total time 6 h.
Average = 300/6 = 50 km/h.
(Here the times happen to be equal, so the plain average of 40 and 60 also gives 50. That is a coincidence of the numbers — do not rely on it.)
4. GRE Traps
- Averaging the two speeds. Only valid when the times are equal, not when the distances are.
- Unit mismatch. Minutes with km/h, or metres with km.
- Adding times when the travellers move simultaneously. They share the clock.
- Forgetting the head start in catch-up problems.
- "Speed increased by 25%" means x1.25, so time becomes 1/1.25 = 0.8 of the original — a 20% reduction, not 25%.
- Round trip distance. A round trip is twice the one-way distance; some questions give one and ask about the other.
- Stops and breaks. If a journey includes a 20-minute halt, average speed for the whole trip must include that time.
5. Speed Tricks
- Always draw the table. Three columns, one row per traveller. It converts the story into equations mechanically.
- Pick a convenient distance when none is given — often the LCM of the two speeds. For 60 and 90, take 180 km each way; times are 3 h and 2 h, total 5 h for 360 km -> 72 km/h. No formula needed.
- Convert km/h to m/s with x5/18 when a question mixes units.
- Estimate the direction first: the average speed always lands closer to the slower speed.
- For meeting problems, add speeds and divide the gap. One line.
6. Self-Check
Q1. A car travels 150 km in 2.5 hours. What is its speed?
Q2. Two trains 300 km apart travel towards each other at 70 km/h and 80 km/h. How long until they meet?
Q3. A jogger runs 4 km at 8 km/h and walks 4 km at 4 km/h. What is the average speed?
Answers
A1. 150/2.5 = 60 km/h.
A2. Closing speed 150 km/h; time = 300/150 = 2 hours.
A3. Equal distances: 2(8)(4)/12 = 64/12 = 16/3, about 5.33 km/h.
7. One-Line Summary for the Board
D = S x T, one row per traveller. Average speed = total distance / total time, never the average of speeds.