Time and Work by the LCM Method: Whole Numbers, No Fractions

The classic method for time and work uses fractions: if A finishes a job in 12 days, A does 1/12 of it each day. That is correct, but adding 1/12 and 1/18 under exam pressure is slow and error-prone. The LCM method removes the fractions by choosing a convenient size for the job.

Take the total work to be the LCM of the given numbers of days. Every worker then completes a whole number of units each day. Add the units of the people working together, divide the total work by that sum, and you have the time.

Pair questions give the time for A + B, B + C and C + A. Adding the three pair rates counts each person twice, so halve the sum to get everyone working together. Then subtract a pair from the total to isolate one person.

To find C alone, subtract the A + B rate from the total: 7.5 − 6 = 1.5 units a day, so C alone needs 60 ÷ 1.5 = 40 days.

Sometimes the question gives a ratio instead of times: "A is twice as efficient as B." Efficiency is units per day, so let A do 2 units a day and B do 1. If they finish together in 14 days, the job is 14 × 3 = 42 units, and each person's own time follows.

Efficiency and time are inversely proportional for the same job. If A is three times as efficient as B, A takes one-third of B's time.

Split the job into stages. Work out how many units were completed while everyone was present, subtract from the total, and divide what is left by the rate of those still working.

When a question talks about numbers of identical workers, use worker-days instead of the LCM. The job size is workers × days, and that product stays the same however the work is divided.

Payment is shared in proportion to the work done. When people work together for the whole job, that is the ratio of their daily units, which the LCM method gives you directly.

• ✘ Adding days instead of rates. Two workers are always faster than the faster one alone.
• ✘ Forgetting to halve the sum of the pair rates.
• ✘ Sharing wages by days worked instead of work done.
• ✘ Using the full-time rate for a worker who left partway through.

Pipes and tanks follow exactly the same method with outlets counted as negative work; see pipes and cisterns. More shortcuts are in the Quant cheat code.

Why does the LCM method work?

The size of the job is arbitrary, so you can pick any convenient number of units. Choosing the LCM of all the given days makes every worker's daily output a whole number, so you add and subtract integers instead of fractions such as 1/12 and 1/18.

How are wages shared when workers work for different numbers of days?

Wages follow the work actually done, not the time spent. Multiply each worker's daily units by the days they worked, then share the payment in the ratio of those totals.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

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