A man, a woman and a boy can do a piece of work in 6, 9 and 18 days respectively. How many boys must assist one man and one woman to do the work in 1 day? (N.M.A.T., 2006)

### Concept & Formula This problem requires converting the total time taken by individuals into their respective 1-day work rates, and summing them to achieve a target work rate of 1 (representing the full job completed in 1 day). $$ \text{Total 1-day work} = \frac{1}{\text{Days to complete}} $$ ### Step-by-Step Solution * **Individual 1-day Work Rates:** 1 man's 1-day work = $1/6$ 1 woman's 1-day work = $1/9$ 1 boy's 1-day work = $1/18$ * **Set up the Equation:** Let the number of boys required to assist be $x$. We need the total work to be completed in exactly 1 day. (1 man's work) + (1 woman's work) + ($x$ boys' work) = 1 $1/6 + 1/9 + x(1/18) = 1$ * **Solve for $x$:** Find a common denominator for the fractions on the left side, which is 18: $3/18 + 2/18 + x/18 = 1$ $(3 + 2 + x) / 18 = 1$ $(5 + x) / 18 = 1$ Multiply both sides by 18: $5 + x = 18$ $x = 18 - 5$ $x = 13$ ### Exam Strategy & Shortcut Use the Total Work (LCM) method. LCM of 6, 9, 18 is 18 units (Let Total Work = 18 units). Man's efficiency = $18 / 6 = 3$ units/day. Woman's efficiency = $18 / 9 = 2$ units/day. Boy's efficiency = $18 / 18 = 1$ unit/day. To finish the 18 units of work in 1 day, the required efficiency is 18 units/day. 1 Man + 1 Woman provide = $3 + 2 = 5$ units/day. Remaining efficiency needed = $18 - 5 = 13$ units/day. Since 1 boy provides 1 unit/day, exactly 13 boys are required. ### Common Pitfall A common error is to set up the equation equal to some unknown rather than recognizing that "completing the work in 1 day" means the sum of their 1-day fractional work must equal exactly 1. ### Final Answer Therefore, the correct answer is **13**.