**Time and Work — formulas used here.** - Work done in unit time. If a…

Question

**Time and Work — formulas used here.**

- Work done in unit time. If a worker finishes a job in $m$ days, the fraction of work done in one day is $	frac{1}{m}$.
- Combined work (two workers). A finishes in $X$ days, B in $Y$ days; together they finish in $	frac{XY}{X+Y}$ days.
- Combined work (three workers). A, B, C take $X$, $Y$, $Z$ days; together they take $	frac{XYZ}{XY+YZ+ZX}$ days.
- Efficiency–time inverse. If A is $k$ times as efficient as B, A takes $	frac{1}{k}$ as long.
- Workforce equivalence. $M_1 D_1 T_1 = M_2 D_2 T_2$ for the same work, where $M$ = workers, $D$ = days, $T$ = hours/day.

**Question.** A and B together can finish a job in $12$ days. They work together for $4$ days, after which B leaves. A then takes another $18$ days to finish the rest alone. How many days would A alone need to do the entire job?

Accepted Answer

B. $27$

Step 1 — work done in $4$ days together: $	frac{4}{12} = 	frac{1}{3}$ of the job.
Step 2 — A alone does the remaining $	frac{2}{3}$ in $18$ days. So A's daily rate is $\dfrac{2/3}{18} = \dfrac{1}{27}$.
Step 3 — at rate $	frac{1}{27}$ per day, A alone needs $27$ days to finish the whole job.
Distractor check: option A (24) comes from forgetting that the remaining is $	frac{2}{3}$ and assuming the rest is half. Option C (30) over-corrects.

Answer

$24$

Answer

$30$

Answer

$36$