56 men can complete a piece of work in 24 days. In how many days can 42 men complete the same piece of work? (Bank P.O., 2008)

### Concept & Man-Days Formula The total amount of work can be measured in "man-days" (number of men multiplied by the number of days). For the same piece of work, the total man-days remains constant. $$ M_1 \times D_1 = M_2 \times D_2 $$ ### Step-by-Step Solution * Given: $M_1 = 56$ men, $D_1 = 24$ days. * Given: $M_2 = 42$ men, $D_2 = ?$ days. * Total work = $56 \times 24$ man-days. * Equate the total work to the second scenario: $56 \times 24 = 42 \times D_2$. * Solve for $D_2$: $D_2 = \frac{56 \times 24}{42}$. * Simplify the fraction: Divide 56 and 42 by 14 to get $\frac{4 \times 24}{3}$. * Divide 24 by 3 to get 8. * $D_2 = 4 \times 8 = 32$ days. ### Exam Strategy & Shortcut Use the ratio method. The ratio of men is $56 : 42 = 4 : 3$. Since men and days are inversely proportional, the ratio of days will be $3 : 4$. If 3 parts = 24 days, then 1 part = 8 days. So, 4 parts = $4 \times 8 = 32$ days. ### Common Pitfall Setting up a direct proportion ($56/24 = 42/x$) instead of an inverse proportion, forgetting that fewer men will take *more* time, not less. ### Final Answer Therefore, the correct answer is **32**.