Time and Work: The LCM Method and Pipes and Cisterns

Time and work questions often look tricky, but one idea solves most: treat the whole job as a fixed number of units.

The basic idea

If a person finishes a job in 10 days, they do 1/10 of it each day. Work done per day is the reciprocal of days taken.

Working together

If A takes 10 days and B takes 15 days, together they do 1/10 + 1/15 = 5/30 = 1/6 per day, so they finish in 6 days.

The LCM method

Choose a total amount of work equal to the LCM of the days. Here LCM(10, 15) = 30 units.

• A does 30 ÷ 10 = 3 units per day.
• B does 30 ÷ 15 = 2 units per day.
• Together: 5 units per day, so 30 ÷ 5 = 6 days.

No fractions needed.

When someone leaves or joins

Example: A and B start together; after 2 days A leaves. How long does B take to finish?

• Total work: 30 units. In 2 days, together they do 10 units.
• Remaining: 20 units. B does 2 per day, so 10 more days.

Efficiency

If A is twice as efficient as B, A takes half the time. If A and B together take 12 days, and A alone takes x days, B takes 2x days: 1/x + 1/(2x) = 1/12, so x = 18. A takes 18 days and B takes 36.

Pipes and cisterns

The same method applies:

• Inlet pipe: fills the tank (positive work).
• Outlet pipe or leak: empties it (negative work).

Example: Pipe A fills a tank in 12 hours; pipe B empties it in 18 hours. Both open: LCM = 36 units. A adds 3 per hour; B removes 2 per hour. Net +1 per hour, so 36 hours to fill.

Wages

If people share wages, divide in proportion to the work done (efficiency × days).

Common traps

1. Adding days instead of rates.
2. Forgetting negative work for outlets.
3. Ignoring people who leave partway.

Tips

• Pick the LCM as total work.
• Write units per day for each person.
• Check that answers make sense.

Practice

A few short problems a day will make this method automatic.

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