In this guide
Time and work and speed and distance look like two chapters, but they are the same idea. Work = rate × time, and distance = speed × time. In both, rates can be added when two things act together and subtracted when they act against each other. If you set problems up as rates, pipes, trains and boats stop being separate formulas to memorise and become one method used in slightly different settings.
Time and work: the total-work method
Instead of working with fractions like 1/18 and 1/24, assume the total work is the LCM of the times. Each person's daily output is then a whole number.
If A takes 18 days and B takes 24 days, take the work as 72 units. A does 72 ÷ 18 = 4 units a day and B does 3. Together they do 7 units a day, so they finish in 72 ÷ 7 days.
Why it works: rates add. A's rate is 1/18 of the job a day and B's is 1/24. Scaling the job to 72 units just clears the denominators.
Efficiency and wages
"A is twice as efficient as B" means A's rate is twice B's. Assign rates 2 and 1 and proceed. When people are paid jointly for a job, wages are shared in the ratio of work done, which for the same number of days is the ratio of their rates.
Pipes and cisterns
A filling pipe adds work; an outlet or leak subtracts it. The method is the same, with a minus sign.
Speed and distance
Units
To convert km/h to m/s, multiply by 5/18. To go back, multiply by 18/5. The factor is 1,000 m ÷ 3,600 s = 5/18. Useful pairs: 18 km/h = 5 m/s, 36 = 10, 54 = 15, 72 = 20, 90 = 25.
Trains
| Situation | Distance to cover | Speed to use |
|---|---|---|
| Train passes a pole or a person standing still | Length of the train | Train's speed |
| Train passes a platform or a bridge | Train + platform | Train's speed |
| Two trains, opposite directions | Sum of lengths | Sum of speeds |
| Two trains, same direction | Sum of lengths | Difference of speeds |
| Train passes a person walking the same way | Length of the train | Train − walker |
The logic is relative speed. When two objects move towards each other, the gap closes at the sum of their speeds; when one chases the other, it closes at the difference.
Average speed
Average speed = total distance ÷ total time, not the average of the speeds. For equal distances at speeds x and y, it is 2xy ÷ (x + y). For a distance d each way, the total time is d/x + d/y, and 2d divided by that simplifies to 2xy ÷ (x + y).
Boats and streams
If the boat's speed in still water is b and the stream's speed is s:
- downstream speed d = b + s, and upstream speed u = b − s
- so b = (d + u) ÷ 2 and s = (d − u) ÷ 2
Worked examples
Example 1. A takes 18 days and B takes 24 days to do a job. How long do they take together?
Total work = 72 units. A = 4, B = 3 units a day.
72 ÷ 7 = 10 2/7 days, about 10.3 days.
Example 2. Two pipes fill a tank in 20 and 30 hours. An outlet empties it in 60 hours. With all three open, how long does the tank take to fill?
Take the tank as 60 units. Rates: +3, +2, −1. Net 4 units an hour.
60 ÷ 4 = 15 hours.
Example 3. A 360 m train at 54 km/h crosses a 240 m platform. How long does it take?
54 km/h = 15 m/s. Distance = 360 + 240 = 600 m.
600 ÷ 15 = 40 seconds.
Example 4. A boat's speed in still water is 16 km/h and the stream flows at 4 km/h. How long does a 60 km trip downstream and back take?
Downstream 20 km/h: 3 hours. Upstream 12 km/h: 5 hours.
Total 8 hours.
Example 5. A and B together finish a job in 12 days. A alone takes 20 days. How long does B take alone?
Total work = 60 units (LCM of 12 and 20). Together 5 a day; A does 3. So B does 2.
60 ÷ 2 = 30 days.
Example 6. A takes 10 days and B 15 days. They work together for 3 days, then A leaves. How long does the whole job take?
Total work = 30 units. A = 3, B = 2 a day. In 3 days they do 15 units.
B does the remaining 15 at 2 a day: 7.5 days. Total 10.5 days.
Example 7. Trains of 150 m and 100 m run towards each other at 60 km/h and 30 km/h. How long do they take to cross each other?
Relative speed = 90 km/h = 25 m/s. Distance = 250 m.
250 ÷ 25 = 10 seconds.
Example 8. A takes 10 days and B 15 days. Working together, they are paid ₹5,000. How should it be shared?
Rates are 3 : 2 (from 30 units), and they worked the same days.
A gets ₹3,000 and B gets ₹2,000.
Common mistakes
- Forgetting to add the platform or the other train's length.
- Mixing m/s and km/h in one calculation.
- Adding a leak's rate instead of subtracting it.
- Sharing wages by time taken instead of work done. The faster worker earns more, not less.
Practice set
- A takes 10 days and B 15 days. How long together? 6 days. 30 units; 3 + 2 = 5 a day.
- Equal distances at 36 km/h and 54 km/h. Average speed? 43.2 km/h. 2 × 36 × 54 ÷ 90.
- A 200 m train passes a pole in 8 seconds. Speed in km/h? 90 km/h. 25 m/s × 18/5.
- A boat goes 30 km downstream in 2 hours and back in 3 hours. Find the boat's and the stream's speed. 12.5 and 2.5 km/h. d = 15, u = 10.
- 12 people finish a job in 15 days. How many are needed to finish it in 10 days? 18. 12 × 15 = 180 person-days; 180 ÷ 10.
- A pipe fills a tank in 6 hours; a leak empties it in 12 hours. With both working, how long to fill? 12 hours. 12 units; +2 − 1 = 1 an hour.
- A 300 m train passes a person walking at 6 km/h in the same direction in 20 seconds. Find the train's speed. 60 km/h. Relative speed = 15 m/s = 54 km/h; 54 + 6.
- A is twice as efficient as B, and together they finish a job in 14 days. How long does A take alone? 21 days. Rates 2 and 1; work = 3 × 14 = 42; 42 ÷ 2.
What to do next
- Solve every work problem this week with the total-work method, even if you know the fraction formula.
- Learn the five km/h to m/s pairs above.
- Revise ratios and averages, since wages and average speed both use them.
- Test yourself on these topics in data sufficiency form.
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 Life Insurance Corporation of India website .
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