One man, 3 women and 4 boys can do a piece of work in 96 hours, 2 men and 8 boys can do it in 80 hours, 2 men and 3 women can do it in 120 hours. 5 men and 12 boys can do it in :

### Concept & System of Work Equations Treat the 1-hour work of a man, woman, and boy as variables ($m$, $w$, $b$) in a system of linear equations. Solve this system systematically to find the individual 1-hour work rates, and then combine them for the target group. $$\text{Total 1-Hour Work} = \frac{1}{\text{Total Hours Taken}}$$ ### Step-by-Step Solution * **Set up the equations:** Let $m$, $w$, and $b$ be the 1-hour work of a man, woman, and boy. * (1) $1m + 3w + 4b = \frac{1}{96}$ * (2) $2m + 8b = \frac{1}{80} \Rightarrow 2(1m + 4b) = \frac{1}{80} \Rightarrow 1m + 4b = \frac{1}{160}$ * (3) $2m + 3w = \frac{1}{120}$ * **Solve for $w$:** Subtract Equation 2 from Equation 1. * $(1m + 3w + 4b) - (1m + 4b) = \frac{1}{96} - \frac{1}{160}$ * $3w = \frac{5 - 3}{480} = \frac{2}{480} = \frac{1}{240}$ * **Solve for $m$:** Substitute $3w = \frac{1}{240}$ into Equation 3. * $2m + \frac{1}{240} = \frac{1}{120}$ * $2m = \frac{1}{120} - \frac{1}{240} = \frac{2}{240} - \frac{1}{240} = \frac{1}{240}$ * $m = \frac{1}{480}$ * **Solve for $b$:** Substitute $2m = \frac{1}{240}$ into Equation 2. * $\frac{1}{240} + 8b = \frac{1}{80}$ * $8b = \frac{1}{80} - \frac{1}{240} = \frac{3}{240} - \frac{1}{240} = \frac{2}{240} = \frac{1}{120}$ * $b = \frac{1}{960}$ * **Calculate target group's work:** Find the 1-hour work of 5 men and 12 boys ($5m + 12b$). * $5(\frac{1}{480}) + 12(\frac{1}{960})$ * $= \frac{10}{960} + \frac{12}{960} = \frac{22}{960} = \frac{11}{480}$ * **Find total time:** Time is the reciprocal of the 1-hour work rate. * Time = $\frac{480}{11}$ hours. * $\frac{480}{11} = 43$ with a remainder of 7, which is $43 \frac{7}{11}$ hours. ### Exam Strategy & Shortcut Look for groupings. Notice that $1m+4b$ is exactly half of the second equation. This allows you to instantly isolate $3w$ from the first equation. Solving linear work systems is all about finding these overlapping blocks. ### Common Pitfall Getting lost in large denominators. Always look for the Least Common Multiple (LCM) for the denominators $96, 80, 120$ early on (which is $480$) to make fraction addition and subtraction seamless. ### Final Answer Therefore, the correct answer is **43 $\frac{7}{11}$ hours**.