In this guide
Time and work and speed and distance are staples of the arithmetic word problems in the SBI PO prelims. They also appear in the mains, as a statement in data sufficiency, as one side of a quantity comparison, or inside a caselet. The same few ideas carry across all of these formats, so it pays to learn them as methods rather than as a list of formulas.
The one idea behind both topics: rates add. Two workers together do the sum of their daily work. Two trains moving towards each other close the gap at the sum of their speeds. Once you think in rates, the formulas stop being things to memorise.
Time and work: the total-work method
Fractions such as 1/12 + 1/18 are slow to add under pressure. The total-work method (often called the LCM method) turns them into whole numbers.
- Take the total work as the LCM of the given days, in "units".
- Divide to get each worker's units per day.
- Add or subtract rates as the question needs.
- Divide the work left by the combined rate.
Why it works. The size of the job is arbitrary. Calling it 36 units instead of "1 job" just scales every rate by 36, and the answer in days does not change.
Example 1. Three workers. A, B and C can finish a job in 12, 18 and 36 days. How long will they take together?
- Total work = 36 units. A does 3 units a day, B 2 and C 1.
- Together: 6 units a day, so 36 ÷ 6 = 6 days.
Example 2. Pairs. A and B together take 10 days, B and C 15 days, and A and C 12 days. How long will all three take together?
- Total work = LCM of 10, 15 and 12 = 60 units.
- A + B = 6, B + C = 4, A + C = 5 units a day. Adding the three pairs counts each person twice: 2(A + B + C) = 15.
- A + B + C = 7.5 units a day, so 60 ÷ 7.5 = 8 days.
Example 3. A worker leaves. A can finish a job in 20 days and B in 30 days. They start together, and A leaves after 5 days. How many more days does B need?
- Total work = 60 units. A does 3 a day, B 2.
- In 5 days together: 5 × 5 = 25 units. Work left: 35 units.
- B alone: 35 ÷ 2 = 17.5 days more (22.5 days from the start).
Example 4. Efficiency. A is twice as efficient as B. Together they finish a job in 12 days. How long would A take alone?
- Rates are 2 : 1, so together 3 units a day, and the job is 3 × 12 = 36 units.
- A alone: 36 ÷ 2 = 18 days.
Example 5. Person-hours. 12 people working 8 hours a day finish a job in 15 days. How many days will 18 people working 10 hours a day take?
- Total work in person-hours is fixed: 12 × 15 × 8 = 1,440.
- 18 × D × 10 = 1,440, so D = 8 days.
Example 6. Wages. A alone takes 15 days and B alone 10 days. Together they are paid ₹4,500 for the job. How should it be shared?
- Total work = 30 units. A does 2 units a day, B 3.
- They work the same number of days, so pay is shared in the ratio of rates, 2 : 3. A gets ₹1,800 and B gets ₹2,700.
Pipes and cisterns
Pipes are time and work with one change: an outlet pipe does negative work.
Example 7. Pipe A fills a tank in 10 hours and pipe B in 15 hours; pipe C empties it in 30 hours. All three are opened together. How long to fill the tank?
- Total = 30 units. A fills 3 units an hour, B 2, and C empties 1.
- Net: 3 + 2 − 1 = 4 units an hour, so 30 ÷ 4 = 7.5 hours.
Speed and distance: the core tools
| Tool | Rule | Why |
|---|---|---|
| Unit change | km/h × 5/18 = m/s | 1 km = 1,000 m and 1 hour = 3,600 s; 1,000/3,600 = 5/18 |
| Average speed over equal distances | 2ab ÷ (a + b) | Average speed is total distance ÷ total time, not the average of speeds |
| Same direction | Relative speed = difference | One object gains on the other only by the extra speed |
| Opposite directions | Relative speed = sum | Both close the gap at once |
| Train passes a pole or a person | Distance = train's length | The front travels one train length until the back clears |
| Train passes a platform or another train | Distance = sum of lengths | The front must travel both lengths |
| Boats | Boat = (down + up) ÷ 2; stream = (down − up) ÷ 2 | Downstream = boat + stream; upstream = boat − stream |
Example 8. Average speed. A car covers the first half of a journey at 40 km/h and the second half at 60 km/h. Find the average speed.
- 2 × 40 × 60 ÷ 100 = 48 km/h, not 50.
- Check with a distance: take each half as 120 km. Time = 3 + 2 = 5 hours for 240 km, so 48 km/h.
Example 9. Trains in the same direction. Trains of 200 m and 250 m run in the same direction at 72 km/h and 54 km/h. How long does the faster train take to pass the slower one?
- Relative speed = 18 km/h = 5 m/s. Distance = 200 + 250 = 450 m.
- 450 ÷ 5 = 90 seconds.
Example 10. Trains in opposite directions. Trains of 150 m and 100 m run towards each other at 54 km/h and 36 km/h. How long do they take to cross?
- Relative speed = 90 km/h = 25 m/s. Distance = 250 m.
- 250 ÷ 25 = 10 seconds.
Example 11. Late and early. An aspirant walking to the exam centre at 4 km/h arrives 10 minutes late. At 5 km/h, they arrive 5 minutes early. How far is the centre?
- The two times differ by 15 minutes = 1/4 hour.
- d/4 − d/5 = 1/4, so d/20 = 1/4 and d = 5 km.
Example 12. Boats. A boat goes 30 km downstream in 2 hours and 18 km upstream in 2 hours. Find the boat's speed in still water and the stream's speed.
- Downstream 15 km/h, upstream 9 km/h.
- Boat = (15 + 9) ÷ 2 = 12 km/h; stream = (15 − 9) ÷ 2 = 3 km/h.
How these appear in the SBI PO paper
| Format | What it looks like | What to do |
|---|---|---|
| Word problem | A direct question with options | Use the method; check that the answer is in the right units |
| Quantity comparison | Quantity I from one problem, quantity II from another | Solve both fully; a wrong unit flips the answer |
| Data sufficiency | Two statements, each giving part of the data | Ask only whether the rate or the distance can be fixed; do not solve to the end |
| Caselet | Several workers or vehicles described in a paragraph | Put the rates in a small table first |
The format mix changes from year to year, so check recent papers. See quantity comparison and data sufficiency for those formats.
Common mistakes
- Averaging speeds. Equal distances need 2ab ÷ (a + b); equal times need the simple average (a + b) ÷ 2.
- Forgetting the length of the platform or of the second train.
- Mixing units. Convert km/h to m/s before dividing metres by speed.
- Treating days as additive. A takes 10 days and B takes 15 days does not mean 25 days together; rates add, days do not.
- In pairs questions, forgetting to halve. Adding three pair-rates gives twice the combined rate.
Practice set
- A takes 15 days and B 20 days to do a job. How long together?
- Pipe A fills a tank in 12 hours; pipe B empties it in 20 hours. Both are open. How long to fill?
- A 300 m train passes a 200 m platform in 25 seconds. Find its speed in km/h.
- A boat goes 24 km downstream in 2 hours and 24 km upstream in 3 hours. Find the stream's speed.
- A and B together finish a job in 8 days; A alone takes 12 days. How long does B take alone?
- 10 people working 6 hours a day finish a job in 16 days. How many people working 10 hours a day finish it in 8 days?
- A car goes from P to Q at 30 km/h and returns at 45 km/h. Find the average speed for the round trip.
- Trains of 120 m and 180 m run in opposite directions at 40 km/h and 50 km/h. How long do they take to cross each other?
Answers:
- 60/7 days, about 8.6. Total 60 units: A 4 a day, B 3; 60 ÷ 7.
- 30 hours. Total 60 units: A fills 5, B empties 3; net 2; 60 ÷ 2.
- 72 km/h. 500 m ÷ 25 s = 20 m/s; 20 × 18/5 = 72.
- 2 km/h. Down 12, up 8; (12 − 8) ÷ 2.
- 24 days. Total 24 units: together 3 a day, A 2; so B does 1.
- 12 people. 10 × 16 × 6 = 960 = x × 8 × 10, so x = 12.
- 36 km/h. 2 × 30 × 45 ÷ 75 = 2,700 ÷ 75.
- 12 seconds. Relative speed 90 km/h = 25 m/s; 300 m ÷ 25.
What to do next
- Redo examples 2, 3 and 11 without looking, using whole-number units only.
- Write the speed table from memory, including the "why" column.
- Solve ten mixed questions a day for a week, timed at about 45 seconds each.
- Revise interest and profit and loss, the other two arithmetic topics that feed DI caselets.
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 State Bank of India website .
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