Time, Speed and Distance Notes for Competitive Exams

Time, Speed and Distance (TSD) is one of the most scoring chapters of Quantitative Aptitude. It appears every year in UPSC CSAT, SSC CGL/CHSL, bank exams and state PCS papers. The whole chapter rests on one simple relation: Distance = Speed × Time. Once you understand this and learn a few shortcuts for average speed, relative speed, trains and boats, most questions can be solved in under a minute.

Quick Facts: Time, Speed and Distance at a Glance

• Basic formula: Speed = Distance ÷ Time; Distance = Speed × Time; Time = Distance ÷ Speed.
• Unit conversion: 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h.
• Average speed = Total distance ÷ Total time (never the simple average of speeds, unless times are equal).
• Relative speed: Add speeds when bodies move in opposite directions; subtract when they move in the same direction.
• Boats: Downstream speed = Boat + Stream; Upstream speed = Boat − Stream.

1. The Basic Relationship

Speed tells us how much distance is covered in one unit of time. If a cyclist covers 60 km in 3 hours, the speed is 60 ÷ 3 = 20 km/h. The three quantities are linked like this:

Proportionality Rules (Very Useful for Shortcuts)

• Distance constant: Speed and time are inversely proportional. If speed ratio is 3 : 4, time ratio is 4 : 3.
• Time constant: Distance is directly proportional to speed.
• Speed constant: Distance is directly proportional to time.

Example: A man walking at 3/4 of his usual speed reaches office 20 minutes late. Find his usual time.
Speed ratio new : usual = 3 : 4, so time ratio = 4 : 3. The difference of 1 part = 20 minutes, so usual time = 3 parts = 60 minutes.

2. Unit Conversion

Many mistakes happen because units are mixed. Remember that 1 km = 1000 m and 1 hour = 3600 seconds, so:

• km/h to m/s: multiply by 5/18. Example: 72 km/h = 72 × 5/18 = 20 m/s.
• m/s to km/h: multiply by 18/5. Example: 15 m/s = 15 × 18/5 = 54 km/h.

Tip: Speeds like 18, 36, 54, 72, 90 and 108 km/h convert neatly to 5, 10, 15, 20, 25 and 30 m/s. Examiners use these numbers often.

3. Average Speed

Average speed is always total distance divided by total time. Students lose marks by simply adding two speeds and dividing by 2.

• Equal distances at speeds x and y: Average speed = 2xy ÷ (x + y) (harmonic mean).
• Equal times at speeds x and y: Average speed = (x + y) ÷ 2.
• Three equal distances at x, y, z: Average speed = 3xyz ÷ (xy + yz + zx).

Example: A car goes from A to B at 40 km/h and returns at 60 km/h. Average speed = 2 × 40 × 60 ÷ 100 = 48 km/h, not 50 km/h.

4. Relative Speed

When two objects move together, we look at how fast the gap between them changes. This is relative speed.

Example: Two persons start from points 90 km apart and walk towards each other at 4 km/h and 5 km/h. They meet after 90 ÷ (4 + 5) = 10 hours.

Chase example: A thief runs at 8 km/h. A policeman starts 30 minutes later at 10 km/h. The thief's lead is 4 km. Relative speed = 2 km/h, so time to catch = 4 ÷ 2 = 2 hours.

5. Problems on Trains

Train questions are TSD questions with one twist: the length of the train must be included in the distance.

Solved Example 1: A 150 m long train runs at 54 km/h. How long will it take to cross a 250 m platform?
Speed = 54 × 5/18 = 15 m/s. Distance = 150 + 250 = 400 m. Time = 400 ÷ 15 = 26.67 seconds.

Solved Example 2: Two trains 120 m and 180 m long run at 50 km/h and 40 km/h in opposite directions. Time to cross each other?
Relative speed = 90 km/h = 25 m/s. Distance = 300 m. Time = 12 seconds.

Shortcut: If two trains start at the same time from A and B towards each other and take a and b hours to reach their destinations after meeting, then S1 : S2 = √b : √a.

6. Boats and Streams

A boat moving with the current (downstream) gets extra help, while a boat moving against the current (upstream) is slowed down.

• Let speed of boat in still water = B and speed of stream = S.
• Downstream speed (D) = B + S
• Upstream speed (U) = B − S
• Speed of boat in still water = (D + U) ÷ 2
• Speed of stream = (D − U) ÷ 2

Solved Example: A boat covers 24 km downstream in 2 hours and the same distance upstream in 3 hours. D = 12 km/h, U = 8 km/h. Boat speed = (12 + 8) ÷ 2 = 10 km/h; stream speed = (12 − 8) ÷ 2 = 2 km/h.

Round-trip shortcut: If a boat with still-water speed B goes a distance and returns in total time T in a stream of speed S, the one-way distance = T × (B² − S²) ÷ (2B).

7. Races and Circular Tracks (Short Note)

• "A beats B by x metres" means when A finishes, B is x metres behind.
• On a circular track of length L, two runners moving in the same direction meet first after L ÷ (S1 − S2); in opposite directions after L ÷ (S1 + S2).
• They meet again at the starting point after the LCM of their individual times to complete one round.

Common Mistakes to Avoid

1. Mixing km/h with metres and seconds. Convert first.
2. Taking the simple average of speeds for a round trip.
3. Forgetting to add the platform or bridge length in train questions.
4. Adding speeds when trains move in the same direction.
5. Confusing boat speed in still water with downstream speed.

How to Prepare This Topic for CSAT and SSC

• Day 1: Learn the basic formula, unit conversion and proportionality rules. Solve 20 basic questions.
• Day 2: Practise average speed and relative speed, including chase and meeting problems.
• Day 3: Solve 25 train and 15 boat questions from previous year papers.
• Revision: Keep a one-page formula sheet and revise it weekly. Track the time you take per question; the target is 60 to 75 seconds.

UPSC CSAT usually frames TSD questions with a reasoning angle, such as comparing two travellers or finding who reaches first. Draw a quick line diagram, mark speeds and times, and the answer often becomes obvious without long calculation.

Practice Questions with Answers

1. A bus covers 180 km in 4 hours. What is its speed in m/s? Answer: 45 km/h = 12.5 m/s.
2. A person travels half the distance at 30 km/h and the other half at 20 km/h. Find the average speed. Answer: 2 × 30 × 20 ÷ 50 = 24 km/h.
3. A 200 m long train crosses a pole in 10 seconds. Find its speed in km/h. Answer: 20 m/s = 72 km/h.
4. Two trains of 100 m each move in the same direction at 46 km/h and 36 km/h. How long will the faster train take to pass the slower one? Answer: Relative speed 10 km/h = 25/9 m/s; time = 200 ÷ (25/9) = 72 seconds.
5. A swimmer's speed in still water is 6 km/h and the river flows at 2 km/h. How long will the swimmer take to go 16 km upstream? Answer: 16 ÷ 4 = 4 hours.

Try these questions without looking at the answers first, and time yourself. If you can solve all five within six minutes, your basics are ready for mixed-level CSAT and SSC questions.

Related Notes

• Time and Work Notes: LCM Method, Pipes and Wages
• Ratio and Proportion Notes with Solved Examples
• Average, Mixture and Alligation Notes
• Time, Speed, Distance and Work for CSAT

Frequently Asked Questions

What is the basic formula of time, speed and distance?

The basic formula is Distance = Speed × Time. From it, Speed = Distance ÷ Time and Time = Distance ÷ Speed.

How do you convert km/h into m/s?

Multiply the speed in km/h by 5/18. For example, 72 km/h × 5/18 = 20 m/s.

What is the formula for average speed when distances are equal?

When equal distances are covered at speeds x and y, average speed = 2xy ÷ (x + y).

What is relative speed?

Relative speed is the rate at which the distance between two moving objects changes. It is the sum of speeds in opposite directions and the difference in the same direction.

How do you find the time a train takes to cross a platform?

Add the length of the train and the platform, then divide by the speed of the train in the same units.

What distance is covered when a train crosses a pole?

Only the length of the train, because a pole has negligible length.

What is downstream and upstream speed?

Downstream speed is boat speed plus stream speed. Upstream speed is boat speed minus stream speed.

How do you find the speed of the stream?

Speed of stream = (Downstream speed − Upstream speed) ÷ 2.

Is time, speed and distance asked in UPSC CSAT?

Yes. CSAT Paper II regularly includes questions on speed, relative motion and travel time, often mixed with logical reasoning.

Why is average speed not the simple average of two speeds?

Because the body spends more time at the lower speed. Average speed must be calculated as total distance divided by total time.

How many questions should I practise on this topic?

Around 100 to 150 mixed questions, including previous year papers, are usually enough to become confident.

Which is the fastest method for train problems?

Convert speed to m/s, decide the total length to be covered, choose the correct relative speed and divide. A table of cases helps you do this quickly.