Time, Speed, Distance and Work for CSAT

Time, speed, distance and work are among the most frequently tested topics of the UPSC CSAT paper and every other aptitude test. They look different, but they follow the same logic: a relation between three quantities. If you understand the idea of "rate" (how much happens in one unit of time), you can solve almost every question in this chapter. This note gives you the basic formulas, the shortcuts for trains, boats and races, the methods for work and pipes, and some solved examples with the working shown, all in simple language.

Basic Relation: Speed, Distance and Time

The base formula is Distance = Speed × Time, which also gives Speed = Distance/Time and Time = Distance/Speed. To convert a speed from km/h to m/s, multiply by 5/18; to convert from m/s to km/h, multiply by 18/5. For example, 54 km/h = 54 × 5/18 = 15 m/s. Keep units consistent: if distance is in km and time is in minutes, convert first. A useful rule is the proportionality of the three quantities: for the same distance, speed and time are inversely proportional; for the same speed, distance and time are directly proportional; and for the same time, distance and speed are directly proportional.

So if the speeds of two persons are in the ratio 3:4, the time taken for the same distance is in the ratio 4:3. A common application: a person walking at 3/4 of his usual speed reaches 20 minutes late. The time taken is 4/3 of the usual time, so the extra 1/3 of the usual time equals 20 minutes. Therefore the usual time is 60 minutes.

Average Speed

Average speed is the total distance divided by the total time, not the average of the speeds. When the distance is the same at two speeds x and y, the average speed is 2xy/(x + y). For 40 km/h going and 60 km/h returning, the average is 2 × 40 × 60/(100) = 48 km/h. When the time is the same at two speeds, the average speed is the simple average (x + y)/2. Remember this difference, since many students use the wrong formula.

Relative Speed

When two objects move in opposite directions, their relative speed is the sum of their speeds. When they move in the same direction, the relative speed is the difference. The time to meet or overtake = the distance between them divided by the relative speed. For example, two persons start from points 120 km apart and move towards each other at 40 km/h and 20 km/h; they meet after 120/60 = 2 hours. If they run on a circular track, the time to meet for the first time at the starting point is based on the LCM of their individual times, and the time to meet anywhere is the length of the track divided by the relative speed.

Problems on Trains

Boats and Streams

Let the speed of the boat in still water be b and the speed of the stream be s. Then downstream speed = b + s and upstream speed = b − s. Also, b = (downstream + upstream)/2 and s = (downstream − upstream)/2. If a boat has a speed of 10 km/h in still water and the stream flows at 2 km/h, the downstream speed is 12 km/h and the upstream speed is 8 km/h. To travel 24 km downstream takes 2 hours, and the same distance upstream takes 3 hours. The same method is used for airplanes flying with or against the wind, and for a person walking on a moving escalator.

Races and Circular Tracks

In a race of 100 m, if A beats B by 20 m, then when A finishes 100 m, B has run 80 m, so their speeds are in the ratio 100:80 = 5:4. If A beats B by 10 seconds, it means that B reaches the end 10 seconds after A. "Start" is the distance by which the slower runner gets a head start. These problems are solved by finding the speed ratio and then applying the proportion.

Time and Work

The key idea is that work = rate × time, where the rate is the part of work done in one day. If A can complete a job in 10 days, A's one-day work is 1/10. If B can do it in 15 days, B's one-day work is 1/15. Working together, they do 1/10 + 1/15 = 1/6 in one day, so they take 6 days. A neat method is to assume the total work as the LCM of the days: with 30 units of work, A does 3 units a day and B does 2 units a day, so together 5 units a day and 30/5 = 6 days. For persons joining or leaving the work, calculate the work done in each period step by step.

Alternate days: if A and B work on alternate days starting with A, then in two days they do 1/10 + 1/15 = 1/6, so six such cycles, which is 12 days, will finish the whole work. Efficiency: a person who is twice as efficient takes half the time. Men-days formula: (M1 × D1 × H1)/W1 = (M2 × D2 × H2)/W2, where M is men, D is days, H is hours per day and W is work. If 12 men can finish a work in 20 days, then 15 men will finish it in 12 × 20/15 = 16 days.

Pipes and Cisterns

Pipes work like workers. An inlet pipe fills a tank (positive work), and an outlet pipe empties it (negative work). If one pipe fills the tank in 6 hours and another empties it in 12 hours, then in one hour the net filling is 1/6 − 1/12 = 1/12 of the tank, so the tank will be filled in 12 hours when both are open. If the leak and the inlet are not known, form equations based on the given times.

Problem-solving Tips

Always draw a small diagram for movement questions. Write the given units. Convert km/h and m/s carefully. For work questions, assume the total work as the LCM of the days to avoid fractions. For boat questions, write the two equations for downstream and upstream. If the question gives the ratio of speeds, directly use the inverse ratio for time. Check if the answer is sensible: a train cannot cross a platform faster than it crosses a pole.

Common Mistakes

Students often use the average of speeds when the distances are equal; forget to add the platform length for the train; mix up upstream and downstream; or ignore a person who joins the work late. Another error is to use a total of 100 percent work but treat negative pipes like positive ones. Slow down on the first reading and write the data clearly before calculating.

Two More Solved Examples

Example 1. A can finish a work in 12 days and B in 24 days. They work together for 4 days, and then A leaves. How many more days does B need? Take the total work as 24 units. A does 2 units a day and B does 1 unit a day, so together they do 3 units a day. In 4 days they finish 12 units, leaving 12 units. B alone does 1 unit a day, so B needs 12 more days. Example 2. A man rows 15 km downstream in 3 hours and returns upstream in 5 hours. The downstream speed is 5 km/h and the upstream speed is 3 km/h. The speed of the boat in still water is (5 + 3)/2 = 4 km/h, and the speed of the stream is (5 − 3)/2 = 1 km/h. Practise writing each step like this, because a neat solution helps you find mistakes quickly and builds the habit of accuracy.

Quick Revision

• Distance = speed × time; km/h to m/s: multiply by 5/18.
• Average speed for equal distances: 2xy/(x + y).
• Downstream = b + s; upstream = b − s.
• Together work: 1/a + 1/b; use the LCM method.

Time, speed, distance and work are about careful thinking, not heavy maths. Learn the few formulas, practise each type with five questions, and you will be able to solve these questions quickly and correctly in the examination.

Related Notes to Read Next

Continue your preparation with these related notes from the same subject:

• Logical Reasoning for CSAT Notes for UPSC
• Data Interpretation for CSAT Notes for UPSC
• Permutation, Combination and Probability for CSAT
• All CSAT notes

Frequently Asked Questions (FAQ)

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

Distance equals speed multiplied by time. From this, speed equals distance divided by time and time equals distance divided by speed.

How do I convert km/h to m/s?

Multiply the speed in km/h by 5/18 to get m/s. For example, 54 km/h equals 15 m/s.

What is the formula for average speed over equal distances?

It is 2xy/(x + y), where x and y are the two speeds. It is not the simple average of the speeds.

What is relative speed?

Relative speed is the speed of one object as seen from another. It is the sum of speeds in opposite directions and the difference of speeds in the same direction.

How long does a train take to cross a platform?

The time equals the length of the train plus the length of the platform divided by the speed of the train.

What is downstream and upstream speed?

Downstream speed is the speed of the boat plus the speed of the stream, and upstream speed is the speed of the boat minus the speed of the stream.

How do I find the speed of the stream?

The speed of the stream equals half the difference of the downstream and upstream speeds.

How do I solve time and work problems quickly?

Take the total work as the LCM of the days taken by the workers, find the work done per day by each and add or subtract as required.

If A does a job in 10 days and B in 15 days, how long will they take together?

They will take 6 days, because the work done per day is 1/10 plus 1/15, which equals 1/6.

What is the men-days formula?

The formula is (M1 × D1 × H1)/W1 = (M2 × D2 × H2)/W2, where M is the number of men, D is days, H is hours per day and W is work.

How do pipes and cisterns questions work?

Treat an inlet pipe as positive work and an outlet pipe as negative work, and then add the rates to find the net time to fill or empty the tank.

What is the relation between speed and time for the same distance?

For a fixed distance, speed and time are inversely proportional. If speed becomes 3/4, time becomes 4/3 of the original.