If 5 men and 3 women can reap 18 acre of crop in 4 days; 3 men and 2 women can reap 22 acre of crop in 8 days, then how many men are required to join 21 women to reap 54 acre of crop in 6 days?

### Concept & Formula When different groups of people perform different amounts of work in a given time, we use the chain rule formula linking Men ($M$), Days ($D$), and Work ($W$): $$ \frac{M_1 \times D_1}{W_1} = \frac{M_2 \times D_2}{W_2} $$ We must first find the efficiency ratio between a man and a woman to convert all workers into a single unit. ### Step-by-Step Solution 1. **Find the Efficiency Ratio:** Let the daily work of 1 man be $m$ and 1 woman be $w$. Case 1: $(5m + 3w)$ reap 18 acres in 4 days. Case 2: $(3m + 2w)$ reap 22 acres in 8 days. Using the formula: $$ \frac{(5m + 3w) \times 4}{18} = \frac{(3m + 2w) \times 8}{22} $$ Simplify the denominators and numerators: $$ \frac{2(5m + 3w)}{9} = \frac{4(3m + 2w)}{11} $$ $$ \frac{5m + 3w}{9} = \frac{2(3m + 2w)}{11} $$ Cross-multiplying: $$ 11(5m + 3w) = 18(3m + 2w) $$ $$ 55m + 33w = 54m + 36w $$ $$ m = 3w $$ This means 1 man is as efficient as 3 women. 2. **Calculate the Required Men:** Let $x$ men join 21 women. Total workforce in terms of women = $x(3w) + 21w = (3x + 21)w$. Convert the first group to women: $5m + 3w = 5(3w) + 3w = 18w$. Now, 18 women reap 18 acres in 4 days. We need $(3x + 21)$ women to reap 54 acres in 6 days. $$ \frac{18 \times 4}{18} = \frac{(3x + 21) \times 6}{54} $$ $$ 4 = \frac{(3x + 21) \times 1}{9} $$ $$ 36 = 3x + 21 $$ $$ 3x = 15 \implies x = 5 $$ ### Exam Strategy & Shortcut Always convert variables ($m$, $w$) to a single unit as early as possible to minimize complex fractional calculations. Identifying that $m = 3w$ simplifies the equation dramatically. ### Common Pitfall Forgetting to include the work ($W$) in the denominator when setting up the chain rule equation. Work done (acres reaped) is not constant here. ### Final Answer Therefore, the correct answer is **5**.