**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.** $8$ men complete a piece of work in $10$ days, working $9$ hours a day. In how many days will $6$ men complete the same work if they work $8$ hours a day?

Accepted Answer

B. $15$

The workforce-equivalence identity $M_1 D_1 T_1 = M_2 D_2 T_2$ holds when the work to be done is the same.
Plugging in: $8 	imes 10 	imes 9 = 6 	imes D_2 	imes 8 \;\Rightarrow\; 720 = 48 D_2 \;\Rightarrow\; D_2 = 15$ days.
Distractor check: option A (12) drops one factor; option D (20) inverts the ratio. The clean way to spot the right answer: fewer men working fewer hours should give *more* days, so anything less than 10 is impossible.

Answer

$12$

Answer

$18$

Answer

$20$