Time and Work: The LCM Method for SSC & RRB

Time and Work is one of the most frequently tested topics in SSC CGL, CHSL, MTS, GD Constable, and RRB NTPC and Group D quantitative aptitude sections. The standard textbook approach — treat the job as "1 unit" and express each person's rate as a fraction of a day — is correct, but adding and comparing fractions under a two-minute-per-question time limit is where most marks are lost to careless arithmetic. The LCM method solves the identical problems using whole numbers instead, which is faster to compute and much easier to sanity-check at a glance.

The core idea

Instead of treating the total work as "1 unit," assume the total work equals the LCM of the number of days each person takes. Because the LCM is divisible by every individual's day-count, each person's daily work (their "efficiency") comes out as a whole number instead of a fraction.

Once you have each person's efficiency in whole units per day, combined work, remaining work, and time-to-finish are all just arithmetic on whole numbers — no common denominators required.

Worked example 1: two workers together

A can complete a task in 12 days, B can complete it in 15 days. How long will they take working together?

LCM(12, 15) = 60. Assume total work = 60 units.

A's efficiency = 60 / 12 = 5 units/day. B's efficiency = 60 / 15 = 4 units/day. Combined efficiency = 5 + 4 = 9 units/day.

Time together = Total work / Combined efficiency = 60 / 9 = 6⅔ days.

Compare this to the fraction method: 1/12 + 1/15 = 5/60 + 4/60 = 9/60 = 3/20, so time = 20/3 = 6⅔ days — same answer, but the LCM method never required finding a common denominator by hand.

Worked example 2: someone leaves partway through

A and B can finish a task in 10 days and 15 days respectively. They work together for 3 days, after which B leaves. In how many more days will A finish the remaining work alone?

LCM(10, 15) = 30. Total work = 30 units.

A's efficiency = 30/10 = 3 units/day. B's efficiency = 30/15 = 2 units/day. Combined efficiency = 5 units/day.

Work done together in 3 days = 5 × 3 = 15 units. Remaining work = 30 − 15 = 15 units.

A alone finishes the remaining 15 units at 3 units/day = 15/3 = 5 more days.

Worked example 3: efficiency ratios

A is thrice as efficient as B. Together they finish a task in 15 days. How long would B alone take?

Let B's efficiency = 1 unit/day, so A's efficiency = 3 units/day. Combined efficiency = 4 units/day.

Total work = Combined efficiency × Time = 4 × 15 = 60 units.

B alone takes Total work / B's efficiency = 60 / 1 = 60 days.

Worked example 4: pipes and cisterns (the same method, negative efficiency)

Pipe A fills a tank in 10 hours. Pipe B, a leak, empties the full tank in 15 hours. If both are opened together, how long does it take to fill the tank?

LCM(10, 15) = 30. Total work (tank capacity) = 30 units.

Pipe A's efficiency = +30/10 = +3 units/hour (filling). Pipe B's efficiency = −30/15 = −2 units/hour (emptying, so negative).

Combined efficiency = 3 − 2 = 1 unit/hour.

Time to fill = 30 / 1 = 30 hours.

Pipes-and-cisterns questions are just Time and Work questions where an "emptying" pipe is a worker with negative efficiency — the LCM method handles both with the same steps.

Why this is faster under exam conditions

The fraction method requires finding a common denominator every time you combine two people's rates, which gets slower and more error-prone as the numbers get less friendly (try 1/13 + 1/17 by hand under time pressure). The LCM method front-loads that one calculation — the LCM itself — and every step after that is addition, subtraction, or simple division of whole numbers. For a 60-90 second target per question, that difference matters.

Practice questions

Try solving these using the LCM method before checking your working against the examples above:

1. A can do a piece of work in 8 days, B in 12 days. In how many days can they finish it together?
2. A, B and C can complete a task in 20, 30 and 60 days respectively. Working together, how long will they take?
3. A can do a job in 18 days. A and B together can do it in 8 days. In how many days can B alone do it?
4. A tank has two inlet pipes that fill it in 20 and 30 hours, and one outlet pipe that empties it in 15 hours. If all three are opened together, how long does it take to fill the tank?

Mastering the LCM method is less about memorizing a trick and more about reflexively assuming whole-number total work instead of fractional "1 unit" work — once that reflex is in place, most Time and Work variants (partial work, efficiency ratios, pipes and cisterns, wages) fall out of the same three or four lines of arithmetic.