A sum of ₹ 6100 was divided among 8 men, 10 women and 12 children in such a way that each man received $25\%$ more than a woman and each woman received $25\%$ more than a child. How much did each woman receive?

### Concept & Strategy This problem tests weighted distribution. You must balance two things simultaneously: the ratio of the *individual* amounts received by a man, woman, and child, and the *total* number of people in each group. Establish a base unit for the smallest share (the child), build the other shares from it, and construct a master sum equation. ### Step-by-Step Solution * **Given:** Total Amount = ₹ $6100$. Group = $8$ Men, $10$ Women, $12$ Children. 1 Man ($M$) = 1 Woman ($W$) + $25\% = 1.25 W = \frac{5}{4} W$. 1 Woman ($W$) = 1 Child ($C$) + $25\% = 1.25 C = \frac{5}{4} C$. * **Calculation / Deduction:** Express all individual shares in terms of a child's share ($C$): Child's share = $C$. Woman's share = $\frac{5}{4} C$. Man's share = $\frac{5}{4} \times (\frac{5}{4} C) = \frac{25}{16} C$. Set up the total sum equation by multiplying the individual share by the number of people in that category: $$(8 \times M) + (10 \times W) + (12 \times C) = 6100$$ $$8 \left( \frac{25}{16} C \right) + 10 \left( \frac{5}{4} C \right) + 12C = 6100$$ $$\frac{200}{16} C + \frac{50}{4} C + 12C = 6100$$ Simplify the fractions: $$\frac{25}{2} C + \frac{25}{2} C + 12C = 6100$$ $$25C + 12C = 6100$$ $$37C = 6100$$ $$C = \frac{6100}{37}$$ We need the share of *each woman*: $$W = \frac{5}{4} \times C$$ $$W = \frac{5}{4} \times \frac{6100}{37}$$ $$W = 5 \times \frac{1525}{37} = \frac{7625}{37}$$ Perform the final division: $7625 \div 37 \approx 206.081$ So, each woman receives approximately ₹ $206.08$. ### Exam Strategy & Shortcut Instead of dealing with fractions, assume an integer value for the child's share that is easily divisible by $16$ (since $1.25 \times 1.25 = \frac{25}{16}$). Let Child = $16x$. Woman = $16x + 25\% = 20x$. Man = $20x + 25\% = 25x$. Total sum = $8(25x) + 10(20x) + 12(16x) = 200x + 200x + 192x = 592x$. $592x = 6100 \Rightarrow x = \frac{6100}{592}$. Woman's share = $20x = 20 \times \left( \frac{6100}{592} \right) = \frac{122000}{592} \approx 206.08$. This completely avoids fraction addition! ### Common Pitfall Students often confuse the total share of all women with the individual share of one woman. After finding $x$, they might multiply it by the whole group multiplier instead of just calculating the single unit value. Always double-check whether the question asks for "the share of women" (group) or "how much did each woman receive" (individual). ### Final Answer **Therefore, the correct answer is ₹ 206.08.**