16 men can finish a work in 24 days and 48 boys can finish the same work in 16 days. 12 men started the work and after 4 days 12 boys joined them. In how many days can they finish the remaining work? (R.R.B., 2008)

### Concept & Work Equivalence To solve problems involving different groups (like men and boys) working together, we first need to establish a relationship between their work efficiencies. We equate the total work done by each group to find the ratio of their efficiencies. $$M_1 \times D_1 = M_2 \times D_2$$ ### Step-by-Step Solution * **Find efficiency ratio:** 16 men finish the work in 24 days, and 48 boys finish it in 16 days. * Total work in terms of men = $16 \times 24 = 384$ man-days. * Total work in terms of boys = $48 \times 16 = 768$ boy-days. * Equating the total work: $384 \text{ men} = 768 \text{ boys} \Rightarrow 1 \text{ man} = 2 \text{ boys}$. * **Calculate work done:** 12 men started the work and worked for 4 days. * Work done so far = $12 \times 4 = 48$ man-days. * **Calculate remaining work:** Remaining work = Total work - Work done = $384 - 48 = 336$ man-days. * **Determine new workforce:** After 4 days, 12 boys joined the 12 men. * Total workforce = 12 men + 12 boys. * Convert boys to men using our ratio ($2 \text{ boys} = 1 \text{ man}$): $12 \text{ boys} = 6 \text{ men}$. * Effective workforce = $12 \text{ men} + 6 \text{ men} = 18 \text{ men}$. * **Find remaining time:** Time = Remaining Work / Effective Workforce = $336 / 18$. * $336 / 18 = 112 / 6 = 56 / 3 = 18 \frac{2}{3}$ days. * Since $18 \frac{2}{3}$ is not among the specific numeric options, the answer is "None of these". ### Exam Strategy & Shortcut Always convert everything into a single unit of work (either all men or all boys) as early as possible. Treating "1 man-day" as a standard unit of work allows you to simply track the total pool of work and subtract from it as days pass. ### Common Pitfall A frequent mistake is adding men and boys directly ($12 + 12 = 24$) without converting them to a common efficiency base. Always convert first! ### Final Answer Therefore, the correct answer is **None of these**.