5 men and 2 boys working together can do four times as much work as a man and a boy. Working capacities of a man and a boy are in the ratio :

### Concept & Work Equivalence The total work done is directly proportional to the number of workers and their individual capacities. By translating the given condition into an algebraic equation, we can find the direct ratio of their working capacities. $$W \propto \text{Workers} \times \text{Capacity}$$ ### Step-by-Step Solution * **Set up the equation:** Let the working capacity of one man be $m$ and one boy be $b$. * Work done by 5 men and 2 boys = $(5m + 2b)$. * Work done by 1 man and 1 boy = $(1m + 1b)$. * The problem states that the first group does four times as much work as the second group. * $(5m + 2b) = 4 \times (1m + 1b)$. * **Solve the equation:** Expand the right side. * $5m + 2b = 4m + 4b$. * Group the variables by moving $4m$ to the left and $2b$ to the right. * $5m - 4m = 4b - 2b$. * $1m = 2b$. * **Find the ratio:** We need the ratio of the capacity of a man to a boy, which is $m/b$. * $m/b = 2/1$. * Therefore, the ratio is $2 : 1$. ### Exam Strategy & Shortcut Translate the word problem directly into an algebraic expression and solve for the ratio. No complex work formulas are needed here; it is a straightforward linear equation. ### Common Pitfall A common mistake is reading the ratio backwards and selecting $1 : 2$ instead of $2 : 1$. Always verify which quantity comes first in the question ("capacity of a man and a boy" means $m:b$). ### Final Answer Therefore, the correct answer is **2 : 1**.