Working $8$ hours a day, $12$ men can do a work in $30$ days. Working $4$ hours a day, $18$ men can do the work in

### Concept & Work Equivalence Formula When the total amount of work is constant, the product of workers ($M$), days ($D$), and hours per day ($H$) remains constant. $$ M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2 $$ ### Step-by-Step Solution * **Given**: - $M_1 = 12$, $D_1 = 30$, $H_1 = 8$ - $M_2 = 18$, $H_2 = 4$, $D_2 = x$ * **Calculation**: $$ 12 \times 30 \times 8 = 18 \times x \times 4 $$ $$ 2880 = 72x $$ $$ x = \frac{2880}{72} = 40 $$ ### Exam Strategy & Shortcut Compare ratios. Men change by a factor of $\frac{18}{12} = \frac{3}{2}$. Hours change by a factor of $\frac{4}{8} = \frac{1}{2}$. The total product on the left changes by $\frac{3}{2} \times \frac{1}{2} = \frac{3}{4}$. To keep the work constant, days must change by the reciprocal factor $\frac{4}{3}$. $30 \times \frac{4}{3} = 40$ days. ### Common Pitfall Setting up direct proportions instead of recognizing that increasing men or hours decreases the total days required (inverse proportionality). ### Final Answer Therefore, the correct answer is **40 days**.