In this guide
Every question in this topic rests on one line: distance = speed × time. What makes CDS questions tricky is not the formula but the setting: mixed units, trains with length, two objects moving at once, or a current helping one way and hindering the other. Each setting has a small rule, and each rule is the same formula applied carefully.
The good news is that these questions are very learnable. Once you know which distance to use and which speed to use, the arithmetic is usually one division.
Units first
- km/h to m/s: multiply by 5/18. Why: 1 km = 1,000 m and 1 hour = 3,600 s, and 1,000/3,600 = 5/18.
- m/s to km/h: multiply by 18/5.
- Common pairs: 18 km/h = 5 m/s, 36 km/h = 10 m/s, 54 km/h = 15 m/s, 72 km/h = 20 m/s, 90 km/h = 25 m/s.
Average speed
Average speed = total distance ÷ total time, never the plain average of the speeds.
For two equal distances at speeds a and b, the average speed is 2ab ÷ (a + b). Why: if each distance is d, the total time is d/a + d/b, and dividing 2d by that gives 2ab/(a + b). This is always less than (a + b)/2, because you spend longer at the slower speed.
Speed and time are inversely proportional
For a fixed distance, if the speed becomes k times, the time becomes 1/k times. So walking at 3/4 of the usual speed takes 4/3 of the usual time. Many "late by so many minutes" questions are solved with this one fact.
Relative speed
| Two bodies move | Relative speed |
|---|---|
| Towards each other or away from each other | Sum of the speeds |
| In the same direction | Difference of the speeds |
When one body chases another, the gap closes at the difference of their speeds.
Trains
| A train crosses | Distance covered |
|---|---|
| A pole, a person standing still or a signal | Its own length |
| A platform, a bridge or a tunnel | Its own length + that length |
| Another train | The sum of both lengths, at the relative speed |
| A person walking | Its own length, at the relative speed |
Boats and streams
- Downstream speed = boat speed + stream speed
- Upstream speed = boat speed − stream speed
- Boat speed in still water = (downstream + upstream) ÷ 2
- Stream speed = (downstream − upstream) ÷ 2
Races
"A beats B by 10 m in a 100 m race" means that when A finishes 100 m, B has run 90 m. So their speeds are in the ratio 100 : 90. Chain ratios the same way when three runners are involved.
Worked questions
Question 1: A car covers the first 120 km at 60 km/h and the next 150 km at 50 km/h. Find its average speed for the whole journey.
- Time = 120 ÷ 60 + 150 ÷ 50 = 2 + 3 = 5 hours.
- Average speed = 270 ÷ 5 = 54 km/h. The plain average, 55, is a trap option.
Question 2: Walking at 3/4 of the usual speed, a person reaches the office 20 minutes late. Find the usual time.
- At 3/4 of the speed, the time is 4/3 of the usual time, which is 1/3 of the usual time extra.
- 1/3 of the usual time = 20 minutes, so the usual time = 60 minutes.
Question 3: Two trains, 120 m and 180 m long, run at 50 km/h and 40 km/h. How long do they take to cross each other when moving in opposite directions, and in the same direction?
- Distance = 120 + 180 = 300 m in both cases.
- Opposite: 90 km/h = 25 m/s, so time = 300 ÷ 25 = 12 seconds.
- Same direction: 10 km/h = 25/9 m/s, so time = 300 × 9 ÷ 25 = 108 seconds.
Question 4: A train passes a pole in 15 seconds and a 100 m platform in 25 seconds. Find its length and speed.
- The extra 10 seconds are spent covering the platform's 100 m, so the speed = 10 m/s = 36 km/h.
- Length = 10 × 15 = 150 m.
Question 5: A boat goes 36 km downstream in 3 hours and 24 km upstream in 4 hours. Find the speed of the boat in still water and of the stream.
- Downstream = 12 km/h, upstream = 6 km/h.
- Boat = (12 + 6) ÷ 2 = 9 km/h. Stream = (12 − 6) ÷ 2 = 3 km/h.
Question 6: A thief runs at 8 km/h. A police officer starts 10 minutes later from the same spot and runs at 10 km/h. How long after starting does the officer catch the thief?
- In 10 minutes, the thief covers 8 × 1/6 = 4/3 km.
- The gap closes at 10 − 8 = 2 km/h, so time = (4/3) ÷ 2 = 2/3 hour = 40 minutes.
Practice set
- Convert 36 km/h to m/s.
- A 240 m train running at 72 km/h passes a pole. How long does it take?
- Two people start from the same point and walk in the same direction at 5 km/h and 3 km/h. How far apart are they after 3 hours?
- Convert 18 m/s to km/h.
- A person travels to a place at 40 km/h and returns along the same route at 60 km/h. Find the average speed.
- In a 100 m race, A beats B by 10 m and B beats C by 10 m. By how many metres does A beat C?
- A boat's speed in still water is 10 km/h and the stream flows at 2 km/h. How long does it take to go 48 km downstream and come back?
- A 200 m train crosses a 250 m bridge in 30 seconds. Find its speed in km/h.
Answers
- 10 m/s. 36 × 5/18.
- 12 seconds. 72 km/h = 20 m/s, and 240 ÷ 20 = 12.
- 6 km. The gap grows at 2 km/h for 3 hours.
- 64.8 km/h. 18 × 18/5.
- 48 km/h. 2 × 40 × 60 ÷ 100.
- 19 m. When A runs 100, B runs 90. C runs 90 for every 100 of B's, so C runs 81.
- 10 hours. 48 ÷ 12 + 48 ÷ 8 = 4 + 6.
- 54 km/h. Distance 450 m in 30 s is 15 m/s, and 15 × 18/5 = 54.
What to do next
- Memorise the km/h and m/s pairs from 18 to 90
- For every train question, write "distance = ?" and "speed = ?" before solving
- Revise time and work, since both topics are rate problems
- Solve the speed and distance questions from the last five CDS papers under time
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Union Public Service Commission website .
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