8 men can complete a piece of work in 20 days. 8 women can complete the same work in 32 days. In how many days will 5 men and 8 women together complete the same work? (Bank P.O., 2010)

### Concept & Work Rates When different groups finish the same work in different times, we can calculate their individual $1$-day fractional work to find combined rates. $$ \text{1-day work} = \frac{1}{\text{Total man-days}} $$ ### Step-by-Step Solution * **Calculate 1-Day Work for Men:** $8$ men complete the work in $20$ days. Total man-days = $8 \times 20 = 160$. Therefore, $1$ man's $1$-day work = $1/160$. * **Calculate 1-Day Work for Women:** $8$ women complete the work in $32$ days. Total woman-days = $8 \times 32 = 256$. Therefore, $1$ woman's $1$-day work = $1/256$. * **Calculate Target Group's 1-Day Work:** The target group is $5$ men and $8$ women. Their combined $1$-day work = $5 \times (1/160) + 8 \times (1/256)$ $= 5/160 + 8/256$ Simplify the fractions: $= 1/32 + 1/32$ $= 2/32$ $= 1/16$ * **Calculate Total Time:** Since their combined $1$-day work is $1/16$, they will take $16$ days to complete the full work. ### Exam Strategy & Shortcut Use the equivalence method directly. $8$ men take $20$ days, so total work = $160 \text{ M-days}$. $8$ women take $32$ days, so total work = $256 \text{ W-days}$. Equating them: $160 \text{ M} = 256 \text{ W} \Rightarrow 5 \text{ M} = 8 \text{ W}$. The question asks for the time taken by "$5 \text{ men} + 8 \text{ women}$". Substitute $5 \text{ men}$ with $8 \text{ women}$: $(8 \text{ W}) + 8 \text{ W} = 16 \text{ W}$. We already know $8$ women take $32$ days. So, $16$ women (double the workforce) will take half the time: $32 / 2 = 16$ days. ### Common Pitfall A common trap is attempting to average the days or arbitrarily combining the numbers without establishing a fundamental equivalence between the work rate of a man and a woman. ### Final Answer Therefore, the correct answer is **16 days**.