Time & Work

Never work with days. Work with one day's work, which is 1/days. Rates can be added; days cannot.

Four types:

1. Part of the work done. If A takes 12 days, in 5 days he does 5/12; the part left is 7/12.
2. Working together. Add the one-day works: 1/a + 1/b, then take the reciprocal. For two workers this is ab/(a+b).
3. One rate from the other. If A and B together take 8 days and A alone takes 12, then B's one-day work = 1/8 − 1/12 = 1/24, so B alone takes 24 days.
4. Efficiency. "A is twice as efficient as B" means A's rate is twice B's, so A takes half the time. Efficiency and time are inverse.

The single most common mistake is subtracting days instead of rates: from "together 8, A alone 12", writing 12 − 8 = 4 days for B. That is wrong — subtract the one-day works. A quick guard: two people working together must always finish faster than the faster one alone, so the joint answer must be smaller than both.

A fifth pattern combines these: some days worked jointly, then one leaves. Compute the fraction finished, then divide the remainder by the rate of whoever is left.

1 day's work = 1/n · together = 1/a + 1/b → time = ab/(a+b) · B alone = 1/(1/together − 1/a) · efficiency ∝ 1/time

A alone finishes a work in 12 days and B alone in 18 days. How long together? → 1/12 + 1/18 = 3/36 + 2/36 = 5/36 of the work per day, so the time is 36/5 = 7.2 days. Sanity check: 7.2 is less than 12, the faster worker's time, as it must be; and the shortcut ab/(a+b) = 216/30 = 7.2 agrees.