If 12 men and 16 boys can do a piece of work in 5 days; 13 men and 24 boys can do it in 4 days, then the ratio of the daily work done by a man to that of a boy is

### Concept & Equating Total Work When two different groups complete the same piece of work in different numbers of days, the total work (Workforce $\times$ Days) for both groups is equal. Equating these allows us to find the efficiency relationship between the individual worker types. $$M_1 \times D_1 = M_2 \times D_2$$ ### Step-by-Step Solution * **Define variables:** Let the daily work of a man be $m$ and a boy be $b$. * **Calculate total work from the first group:** 12 men and 16 boys finish the work in 5 days. * Total Work = $5 \times (12m + 16b)$. * **Calculate total work from the second group:** 13 men and 24 boys finish the work in 4 days. * Total Work = $4 \times (13m + 24b)$. * **Equate and solve:** Since both groups are doing the same work, set the equations equal to each other. * $5(12m + 16b) = 4(13m + 24b)$. * $60m + 80b = 52m + 96b$. * Move $m$ terms to one side and $b$ terms to the other. * $60m - 52m = 96b - 80b$. * $8m = 16b$. * Simplify by dividing by 8. * $1m = 2b$. * **Determine ratio:** The ratio of a man's daily work to a boy's is $m/b = 2/1$. ### Exam Strategy & Shortcut Quickly cross-multiply the days and the workforce numbers mentally: $(12 \times 5)m = 60m$ vs $(13 \times 4)m = 52m$, difference is $8m$. For boys: $(16 \times 5)b = 80b$ vs $(24 \times 4)b = 96b$, difference is $16b$. Thus, $8m = 16b \Rightarrow m = 2b$. The ratio is $2:1$. ### Common Pitfall Miscalculating the distribution when multiplying the days across the workforce numbers (e.g., forgetting to multiply the boys by the days). ### Final Answer Therefore, the correct answer is **2 : 1**.