Introduction to Time and Work (LCM Method)

Core idea

If A finishes a job in n days, A does 1/n of the job per day.

Work = Rate × Time

LCM (total work) method

1. Take total work = LCM of the days.
2. Find each person's work per day.
3. Add rates for people working together.

Example: A takes 10 days, B takes 15 days. Total work = LCM(10, 15) = 30 units. A = 3 units/day, B = 2 units/day. Together = 5 units/day → 30 ÷ 5 = 6 days.

Formula for two people

Together time = (a × b)/(a + b) → (10 × 15)/25 = 6 days.

Efficiency

If A is twice as efficient as B, A takes half the time. Efficiency ∝ 1/Time.

Men–days rule

M₁ × D₁ × H₁ / W₁ = M₂ × D₂ × H₂ / W₂ (M = men, D = days, H = hours per day, W = work).

Example: 12 men finish a job in 10 days → 1 job = 120 man-days → 20 men need 6 days.

Analogy

Think of filling a 30-litre bucket. One tap pours 3 litres a minute, another 2 litres a minute. Together they pour 5 litres a minute, so the bucket is full in 6 minutes. In time-and-work, the "job" is the bucket and each worker is a tap.