Time and work questions look different each time, but they follow one idea: work done is the product of rate and time. Once you learn to think in terms of one day's work, even long problems with several people, leaving workers or pipes become easy. This page covers the formulas, the efficiency shortcut and solved examples.
The basic idea
If a person completes a job in x days, their one day's work is 1/x of the job. The whole job is counted as 1. Work done = one day's work × number of days.
Core formulas
- If A takes x days and B takes y days, together they take xy/(x + y) days.
- If A and B together take t days and A alone takes x days, then B alone takes 1/(1/t − 1/x) days.
- If A is twice as efficient as B, A takes half the time B takes. Time is inversely proportional to efficiency.
- Men-days rule: (M1 × D1 × H1)/W1 = (M2 × D2 × H2)/W2, where M is men, D is days, H is hours per day and W is work.
Example 1. A can do a job in 12 days and B in 18 days. Together they take 12 × 18/30 = 7.2 days.
Example 2. A and B together finish a job in 10 days, and A alone in 15 days. B alone takes 1/(1/10 − 1/15) = 30 days.
The efficiency shortcut
Take the LCM of the days as the total work in units, and then find each person's units per day. For example, if A takes 12 days and B takes 18 days, take the total work as 36 units. A does 3 units a day, B does 2 units a day, and together they do 5 units a day, so 36/5 = 7.2 days. This avoids fractions and is much faster in exams.
Men, days and hours
Example 3. 12 men complete a job in 15 days. To find the days needed by 20 men: 12 × 15/20 = 9 days. More men means fewer days, so this is an inverse relationship.
When someone leaves or joins
Example 4. A and B can complete a job in 12 and 24 days. They work together for 4 days and then A leaves. Work done in 4 days = 4 × (1/12 + 1/24) = 4 × 3/24 = 1/2. B finishes the remaining half alone in 12 days.
Alternate days
Example 5. A takes 10 days and B takes 15 days, and they work on alternate days, starting with A. In two days, they do 1/10 + 1/15 = 1/6 of the job. So the job needs 6 such cycles, which is 12 days.
Sharing wages
Wages are shared in the ratio of work done, which is the ratio of one day's work if they worked together for the same time.
Example 6. A takes 6 days and B takes 12 days. Their work ratio is 1/6 : 1/12 = 2 : 1. On ₹900, A gets ₹600 and B gets ₹300.
Pipes and cisterns
An inlet pipe fills a tank, and an outlet pipe or a leak empties it. Treat inlet work as positive and outlet work as negative.
Example 7. An inlet fills a tank in 20 hours, and an outlet empties it in 30 hours. Together, they fill it in 1/(1/20 − 1/30) = 60 hours.
Three or more workers
Add all the one day's works. Example 8. A, B and C can complete a job in 10, 15 and 30 days. Together, one day's work = 1/10 + 1/15 + 1/30 = 3/30 + 2/30 + 1/30 = 6/30 = 1/5. So they finish in 5 days.
Efficiency in percentage
If A is 25% more efficient than B, the efficiency ratio A : B is 5 : 4, so the time ratio is 4 : 5. Example 9. If B takes 20 days, A takes 20 × 4/5 = 16 days.
A quick tip on choosing the method
Use the LCM method when days are given for each person, and use the men-days formula when the number of workers or hours per day changes. For pipes, always decide first which pipes fill and which empty, so you know which signs to use. Practise writing the total work as units, because it turns every problem into simple whole-number arithmetic.
Common mistakes to avoid
- Adding days instead of adding one day's work.
- Forgetting to subtract the outlet work in pipes problems.
- Mixing hours per day with days.
- Sharing wages by days worked instead of by work done.
Practice questions
- A takes 20 days and B takes 30 days. How long do they take together?
- 15 men finish a job in 20 days. How many days will 25 men need?
- A and B together take 12 days, and A alone takes 20 days. How long does B alone take?
- A pipe fills a tank in 15 hours and a leak empties it in 45 hours. How long to fill it with the leak?
- A takes 8 days and B takes 24 days. Share ₹1,600 between them according to work.
Answers: 1) 12 days, 2) 12 days, 3) 30 days, 4) 22.5 hours, 5) ₹1,200 and ₹400.
How to practise
Solve ten problems a day using the efficiency method. Revise ratio and proportion first if wages and efficiency feel hard, and test yourself in the mock tests.
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