Time and work

1. Meaning, assumptions and CSAT relevance

Time and work belongs to basic numeracy and can also appear within a data interpretation or reasoning problem. The wording may concern labourers building a wall, machines producing components, typists processing records or pipes filling a tank. Despite these different settings, the mathematical structure remains the same: a quantity of work is produced at a specified rate over a period of time.

The governing equation is Work = Rate × Time. If the whole job is represented by 1 and A finishes it in 12 days, A's rate is 1/12 job per day. If B finishes the same job in 18 days, B's rate is 1/18 job per day. Working together, they complete 5/36 of the job per day and therefore require 36/5, or 7.2 days.

Standard aptitude questions normally assume a constant work rate, identical work quality, uninterrupted availability and no loss of efficiency from working together. These are modelling assumptions, not universal facts about workplaces. If a question specifies fatigue, machine breakdowns, learning effects or interference between workers, those conditions replace the usual assumptions.

• Identify whether the question asks for elapsed time, actual working time, workforce size, efficiency or completed work.
• Use a consistent time unit throughout: days, hours or minutes.
• Do not add completion times to calculate joint completion time; add rates.

2. Unit-work method, LCM method and efficiency

The unit-work method takes total work as 1. The least common multiple, or LCM, method instead selects a convenient total that makes individual rates integers. For completion times of 12 and 18 days, choose 36 work units. A produces 3 units daily and B produces 2. Their combined rate is 5 units daily, so completion takes 36/5 days. The LCM is a computational convenience, not a separately given physical quantity.

For two workers completing the same task independently in x and y days, their joint time is xy/(x + y). This follows from adding 1/x and 1/y and taking the reciprocal. For three or more workers, adding rates is safer than memorising additional formulae. Also, if A and B together take t days while A alone takes a days, B's rate is 1/t − 1/a.

Efficiency means output per unit time. If A:B efficiency is 3:2, their times for the same job are in the ratio 2:3. If A is 25% more efficient than B, their rate ratio is 5:4 and their time ratio is 4:5. Thus, A takes 20% less time, not 25% less time. This distinction is a frequent source of errors.

• Use fractional rates when denominators are simple or total work is already normalised.
• Use LCM units when several workers have different integer completion times.
• Given rates for A+B, B+C and C+A, add them and divide by 2 to obtain the rate of A+B+C.

3. Changing teams, alternate days and partial completion

When workers join or leave, split the job into stages. Calculate work completed in each stage, subtract it from total work, and divide the remainder by the next stage's rate. Suppose A takes 10 days and B takes 15 days. With total work set to 30 units, their rates are 3 and 2 units daily. If A works alone for 2 days, 24 units remain. Together they finish these in 24/5 days, making total elapsed time 6.8 days.

Alternate-day problems require attention to the starting worker and the order of work. Suppose A completes a job in 6 days and B in 12 days, working on alternate days with A starting. Choose 12 units: A does 2 daily and B does 1. Each two-day cycle completes 3 units. After three cycles, 9 units are complete; A adds 2 on day 7 and B finishes the last unit on day 8.

Do not mechanically divide total work by average daily output in an alternating schedule. Whole cycles may be grouped, but the last incomplete cycle must be checked day by day. For example, a worker beginning the final day may complete the remaining work before the next worker's turn. Fractional final days are normally allowed unless the question restricts work to full shifts or indivisible tasks.

• Record the work remaining after every team change.
• Distinguish completion during the nth day from completion after n full days.
• Check whether a percentage refers to total work or only to the work remaining.

4. Worker-days, working hours and wages

For equally efficient workers, output is proportional to the number of workers and their working time. If daily hours also vary, use Work = k × Workers × Days × Hours per day, where k is output per worker-hour. For different groups, an efficiency multiplier may be included. The general comparison is W1/W2 = (M1 × D1 × H1 × E1)/(M2 × D2 × H2 × E2), provided the work is comparable.

For example, 12 equally efficient workers working 6 hours daily for 10 days supply 720 worker-hours. To complete the same work in 5 days at 8 hours daily, the required workforce is 720/(5 × 8) = 18. If the second team must complete 50% more work under the same conditions, it needs 27 workers. A worker-day alone is insufficient when working hours change.

When payment is explicitly proportional to work contributed, divide wages according to rate multiplied by actual time worked. If A and B have efficiencies 3:2 and work for 4 and 6 days respectively, their contributions are 12:12, so payment is shared equally. However, an hourly wage contract may produce a different division. Never assume work-linked payment when the question specifies a different rule.

• For fixed work and unchanged hours and efficiency, workers × days is constant.
• For fixed workforce and efficiency, more work requires proportionately more time.
• Round workforce requirements upward when a whole number of workers is necessary to meet a deadline.

5. Pipes, cisterns and a reliable solving strategy

Pipes and cisterns are time-and-work problems in which tank capacity replaces total work. An inlet filling a tank in x hours has rate +1/x tank per hour; an outlet emptying a full tank in y hours has rate −1/y. A pipe filling in 6 hours and an outlet emptying in 9 hours yield a net filling rate of 1/18 tank per hour, so an initially empty tank fills in 18 hours.

Initial conditions matter. If that tank is already one-third full, only two-thirds remains, requiring 12 hours at the same net rate. If the outlet rate equals the inlet rate, the water level does not rise under the constant-rate model. If outflow exceeds inflow, the tank cannot fill from empty. These conclusions assume the stated rates apply while the pipes operate; real discharge rates can vary with water pressure.

A dependable examination approach is to label total work, individual rates and elapsed time before calculating. Choose convenient work units, organise any schedule into stages, and use the options for a final plausibility check. Two positive-rate workers together must finish sooner than either alone. Adding a leak must increase filling time. Increasing workforce while holding work, hours and efficiency fixed must reduce completion time. Such checks detect many arithmetic and interpretation errors.

• Avoid premature decimal conversion; fractions often preserve accuracy.
• Apply signed rates only while the relevant inlet, outlet or worker is active.
• Treat indivisible outputs and sequential operations separately when simultaneous work is not possible.

Previous year questions

No UPSC question has been asked directly on this micro-topic yet. Use the practice questions below.

Practice questions