4 men and 10 women were put on a work. They completed $\frac{1}{3}$ of the work in 4 days. After this 2 men and 2 women were increased. They completed $\frac{2}{9}$ more of the work in 2 days. If the remaining work is to be completed in 3 days, then how many more women must be increased? (M.A.T., 2006)

### Concept & Fractional Work Progression When work is given in fractions over different time periods, calculate the 1-day work rate for each scenario to establish the efficiency relationship between workers. Then calculate the remaining work and determine the required workforce to finish it in the given time frame. $$\text{Work Rate} = \frac{\text{Work Done}}{\text{Days Taken}}$$ ### Step-by-Step Solution * **Analyze Scenario 1:** 4 men ($m$) and 10 women ($w$) do $\frac{1}{3}$ work in 4 days. * Their 1-day work = $\frac{1}{3} / 4 = \frac{1}{12}$. * So, $4m + 10w = \frac{1}{12}$. (Equation 1) * **Analyze Scenario 2:** 2 men and 2 women join. The team is now 6 men and 12 women. * They complete $\frac{2}{9}$ work in 2 days. * Their 1-day work = $\frac{2}{9} / 2 = \frac{1}{9}$. * So, $6m + 12w = \frac{1}{9}$. (Equation 2) * **Find efficiency relationship:** Multiply Equation 2 by $\frac{2}{3}$ to match the $m$ coefficients. * $\frac{2}{3}(6m + 12w) = \frac{2}{3}(\frac{1}{9}) \Rightarrow 4m + 8w = \frac{2}{27}$. (Equation 3) * Subtract Equation 3 from Equation 1: * $(4m + 10w) - (4m + 8w) = \frac{1}{12} - \frac{2}{27}$. * $2w = \frac{9}{108} - \frac{8}{108} = \frac{1}{108}$. * So, $w = \frac{1}{216}$ (1 woman does $\frac{1}{216}$ of the total work per day). * Substitute $w$ into Eq 1 to find $m$: * $4m + 10(\frac{1}{216}) = \frac{1}{12} \Rightarrow 4m + \frac{5}{108} = \frac{9}{108} \Rightarrow 4m = \frac{4}{108} \Rightarrow m = \frac{1}{108}$. * Therefore, 1 man is twice as efficient as 1 woman ($1m = 2w$). * **Calculate remaining work:** Total work done = $\frac{1}{3} + \frac{2}{9} = \frac{3}{9} + \frac{2}{9} = \frac{5}{9}$. * Remaining work = $1 - \frac{5}{9} = \frac{4}{9}$. * **Determine target workforce:** This $\frac{4}{9}$ work must be done in 3 days. * Required 1-day work = $\frac{4}{9} / 3 = \frac{4}{27}$. * Let the required number of women be $x$. Total workforce purely in women = $x \times w = x \times \frac{1}{216}$. * $x \times \frac{1}{216} = \frac{4}{27} \Rightarrow x = 216 \times \frac{4}{27} = 8 \times 4 = 32$ women total. * **Find additional women needed:** Current team is 6 men and 12 women. * Convert to women: $6(2w) + 12w = 12w + 12w = 24$ women. * We need 32 women equivalent, and we have 24. * Increase = $32 - 24 = 8$ women. ### Exam Strategy & Shortcut Instead of solving equations, notice the difference: Adding $2m+2w$ increases daily work from $\frac{1}{12}$ ($\frac{3}{36}$) to $\frac{1}{9}$ ($\frac{4}{36}$). Thus, the daily work of $2m+2w = \frac{1}{36}$. Since $6m+12w = \frac{1}{9}$ ($\frac{4}{36}$), subtract $3 \times (2m+2w)$ to find women's rate. This algebraic logic swiftly leads to $1m = 2w$. Remaining work is $\frac{4}{9}$ in 3 days $\rightarrow$ daily rate $\frac{4}{27}$. We currently have a team of 24 women. 24 women do $24/216 = \frac{1}{9}$ ($\frac{3}{27}$) per day. We need $\frac{4}{27}$ per day, so we are short by $\frac{1}{27}$ per day. Since 1 woman does $\frac{1}{216}$ per day, we need $\frac{1}{27} / \frac{1}{216} = 8$ more women. ### Common Pitfall A major pitfall is forgetting to subtract the current workforce (equivalent to 24 women) from the final required workforce (32 women) to find the *additional* women needed, leading to choosing a distractor option. ### Final Answer Therefore, the correct answer is **8**.