A man, a woman and a boy can complete a job in 3, 4 and 12 days respectively. How many boys must assist 1 man and 1 woman to complete the job in $\frac{1}{4}$ of a day ?

### Concept & Formula To solve problems where individuals work together, we calculate their combined 1-day work. The total work is assumed to be 1 unit. $$ \text{Total 1-day work} = \frac{1}{\text{Total days to complete work}} $$ We can set up an equation adding the fractional work of the man, the woman, and the unknown number of boys to equal the required total 1-day work rate. ### Step-by-Step Solution * **Individual 1-day Work:** 1 man's 1-day work = $1/3$ 1 woman's 1-day work = $1/4$ 1 boy's 1-day work = $1/12$ * **Required Rate:** The job needs to be completed in $1/4$ of a day. Therefore, the required 1-day work of the combined group is: $1 / (1/4) = 4$ units of work per day. * **Set up the Equation:** Let the number of boys required be $x$. (1 man's work) + (1 woman's work) + ($x$ boys' work) = 4 $1/3 + 1/4 + x(1/12) = 4$ * **Solve for $x$:** Find a common denominator for the fractions (which is 12): $4/12 + 3/12 + x/12 = 4$ $(7 + x)/12 = 4$ $7 + x = 48$ $x = 48 - 7 = 41$ ### Exam Strategy & Shortcut Use the LCM method for total work. LCM of 3, 4, 12 is 12 (Total Work = 12 units). Efficiency of 1 Man = $12 / 3 = 4$ units/day. Efficiency of 1 Woman = $12 / 4 = 3$ units/day. Efficiency of 1 Boy = $12 / 12 = 1$ unit/day. They need to finish the work in $1/4$ day. So, required efficiency per day = $12 / (1/4) = 48$ units/day. Efficiency of 1 Man + 1 Woman = $4 + 3 = 7$ units/day. Remaining efficiency needed = $48 - 7 = 41$ units/day. Since 1 boy does 1 unit/day, 41 boys are needed. ### Common Pitfall A common mistake is misinterpreting "$\frac{1}{4}$ of a day" as 4 days, or forgetting to invert the fraction to find the 1-day work rate of the combined group (which is 4, not $1/4$). ### Final Answer Therefore, the correct answer is **41**.