Time and Work Notes for Competitive Exams

Time and work is a favourite chapter in every competitive examination. Questions are about workers, machines, pipes and tanks completing a job in a certain time, and they test your logical thinking and speed. The chapter looks lengthy because of fractions, but once you learn the LCM method and the idea of efficiency, you can solve almost all questions in under a minute. This note explains the basic principles, the shortcuts, wages, pipes and cisterns, and solves many types of problems step by step in simple language.

The Core Idea: Work, Rate and Time

Work is what has to be done; rate (also called efficiency) is the amount of work done in one unit of time; and time is how long it takes. The basic relation is Work = Rate × Time. If a person can finish a job in n days, the part of work done in one day is 1/n. If two people work together, their rates add up. The total time is the whole work divided by the combined rate. For two persons who can do a job alone in a and b days, the time together is ab/(a + b) days.

The LCM Method (Unit Method)

Instead of using fractions, assume the total work to be the LCM of the days given. Then the daily work of each person is a whole number. Take this example: A can do a work in 12 days and B in 18 days. The LCM of 12 and 18 is 36 units. A does 36/12 = 3 units per day and B does 36/18 = 2 units per day. Together they do 5 units per day, so they need 36/5 = 7.2 days (7 1/5 days). This method also works when a person joins or leaves, because you can subtract the work done so far from the total.

Efficiency and Time Are Inversely Proportional

If A takes 10 days and B takes 15 days, their efficiencies are in the ratio 1/10:1/15 = 3:2. A more efficient person takes less time. If A is twice as efficient as B and together they finish a job in 12 days, then A's efficiency is 2e and B's is e. The total work is 3e × 12 = 36e. A alone would take 36e/2e = 18 days and B alone would take 36 days. Remember the connection: if the efficiency ratio is p:q, the ratio of days taken is q:p.

Different Types of Problems

Worked Examples

Example 1 (leaving before completion). A can do a job in 10 days and B in 15 days. They start together, but A leaves 5 days before the work is completed. In how many days is the work finished? Take the work as 30 units: A does 3 and B does 2 per day. Let the total time be x days; B works for x days, A works for x − 5 days. So 3(x − 5) + 2x = 30, which gives 5x = 45 and x = 9 days. Check: A works 4 days (12 units) and B works 9 days (18 units), total 30 units.

Example 2 (joining later). A can complete a work in 20 days and B in 30 days. A starts the work and B joins after 5 days. Total work is 60 units; A does 3 per day and B does 2 per day. In 5 days, A does 15 units, leaving 45 units. Together they do 5 units per day, so they need 9 more days. The total time is 14 days.

Example 3 (alternate days). A can do a job in 12 days and B in 18 days. They work on alternate days, starting with A. Total work is 36 units; A does 3 and B does 2. In a two-day cycle they do 5 units. After 14 days (7 cycles) they have completed 35 units. One unit remains for the 15th day, which is A's turn; A completes it in 1/3 of a day. The total time is 14 1/3 days.

Example 4 (wages). A and B complete a job in 10 and 15 days, and the total wage is 6,000 rupees. The efficiency ratio is 3:2, so the wages are shared in the ratio 3:2: A gets 3,600 rupees and B gets 2,400 rupees. When they work for different numbers of days, wages depend on the total work done by each, not just the efficiency.

Example 5 (man-days-hours). 8 men working 6 hours a day take 12 days to finish a job. How many days will 6 men working 8 hours a day take? The total man-hours is 8 × 12 × 6 = 576. The daily man-hours for the new group is 6 × 8 = 48. So the number of days is 576/48 = 12 days.

Pipes and Cisterns

Pipes and cisterns follow the same logic. An inlet pipe does positive work by filling the tank, and an outlet pipe does negative work by emptying it. Suppose pipe A fills a tank in 20 minutes, pipe B fills it in 30 minutes, and pipe C empties it in 40 minutes. Take the capacity as 120 units: A gives 6 units per minute, B gives 4 units, and C removes 3 units. The net rate is 6 + 4 − 3 = 7 units per minute, so the time to fill the tank is 120/7 = 17 1/7 minutes. If the tank is already full and only the outlet is open, the time to empty is based on its own rate. Questions with a leak can be solved by finding the rate of the leak from the difference in time.

Tips for Speed

Always choose a convenient total work (the LCM). Write the daily rates of every person at the start. For three workers A, B and C, if you know the rate of A + B, B + C and A + C, add all three and divide by two to get the combined rate of A + B + C. Convert "hours per day" and "days" to a total man-hours. When a worker is "x times as efficient", assign efficiency numbers directly. For work done by machines of different kinds, treat the machines like workers. Check that the answer is smaller than the time of the fastest worker alone when they work together.

Common Mistakes

Students often add the times instead of the rates. They forget that a person who leaves works fewer days than the person who stays. They treat an outlet as an inlet in cistern questions. They calculate wages in the ratio of days instead of the ratio of work done. They also make arithmetic mistakes in fractions, which the LCM method easily avoids. Write the steps clearly and verify the answer by substituting the numbers.

Practice Questions to Try Yourself

Test your understanding with these quick questions and check your answers. First, A can finish a task in 8 days and B in 24 days; how long will they take together? With 24 units of work, A does 3 and B does 1 per day, so they do 4 per day and finish in 6 days. Second, A and B together can do a job in 6 days, and A alone takes 10 days; how long will B alone take? With 30 units, together they do 5 per day and A does 3, so B does 2 per day and takes 15 days. Third, 15 men can build a wall in 20 days; how many men are needed to build it in 12 days? The man-days are 300, so 300/12 = 25 men are needed. Fourth, a tap fills a tank in 6 hours but a leak empties it in 12 hours; with the leak, how long does it take? The net rate is 1/6 − 1/12 = 1/12, so it takes 12 hours. Solve similar questions with different numbers every day, and you will quickly gain speed and accuracy in this chapter.

Quick Revision

• Work = rate × time; together time = ab/(a + b) for two workers.
• Take total work as the LCM of the days; efficiency is inversely proportional to time.
• Wages are in the ratio of work done; outlet pipes have negative rates.
• Man-days-hours: (M1 D1 H1)/W1 = (M2 D2 H2)/W2.

Time and work becomes easy when you think in units of work and daily rates. Practise each of the types shown above, and you will find that this chapter turns from a worry into a source of quick marks.

Related Notes to Read Next

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

• Profit, Loss and Interest Notes for Competitive Exams
• Number System Notes for Competitive Exams
• Average, Mixture and Alligation Notes
• All Quantitative Aptitude notes

Frequently Asked Questions (FAQ)

What is the basic formula for time and work?

Work equals rate multiplied by time. If a person completes a job in n days, the work done in one day is 1/n of the job.

How do I find the time taken by two workers working together?

If one worker takes a days and the other takes b days, the time together is ab/(a + b) days.

What is the LCM method in time and work?

Assume the total work as the LCM of the days taken by individual workers, then find the units of work done per day by each worker to avoid fractions.

What is the relation between efficiency and time?

Efficiency and time are inversely proportional. A worker who is twice as efficient takes half the time.

How are wages divided among workers?

Wages are divided in the ratio of the work done by each worker, which is the product of efficiency and days worked.

How do I solve alternate day work problems?

Find the work done in one cycle of two days, multiply by the number of complete cycles, and add the time for the remaining work.

How do pipes and cisterns questions work?

Treat inlet pipes as positive work and outlet pipes as negative work. Add the rates to get the net rate and divide the capacity by it.

What is the man-days-hours formula?

The formula is (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 A is thrice as efficient as B, what is the ratio of the time taken?

The ratio of the time taken by A and B is 1:3, because efficiency and time are inversely proportional.

What if a worker leaves before the work is completed?

Take the total time as x, let the worker who left work for fewer days, and form an equation of total work done equal to the entire work.

How can I avoid mistakes in time and work questions?

Use the LCM method, write the rate of each worker clearly, treat outlets as negative, and check the answer by substitution.

Is time and work important for exams like CSAT and SSC?

Yes. It is a regular topic in CSAT, SSC, banking and railway examinations, and it also helps in learning other chapters such as pipes and speed-distance.