A sum of 5,600 is distributed among 12 people consisting of men, women and children. The ratio of the total amounts given to all men, all women and all children is 9:4:1. But the ratio of the amounts received by each man, woman and child is 3:2:1. Then the amount received by each women is

Given:

Total sum = ₹5,600

Total people = 12 (men, women, children)

Ratio of total amounts (men:women:children) = 9:4:1

Ratio of amounts per person (man:woman:child) = 3:2:1

Formula Used:

Total amount = (Number of people) × (Amount per person)

Calculations:

Let the total amounts given to men, women, and children be 9x, 4x, and 1x respectively.

⇒ 9x + 4x + 1x = 5600

⇒ 14x = 5600

⇒ x = \(\frac{5600}{14}\)

⇒ x = 400

Total amount for men = 9 × 400 = 3600

Total amount for women = 4 × 400 = 1600

Total amount for children = 1 × 400 = 400

Let the amount received by each man, woman, and child be 3y, 2y, and 1y respectively.

Let the number of men, women, and children be M, W, and C respectively.

Total amount for men = M × 3y = 3600

Total amount for women = W × 2y = 1600

Total amount for children = C × 1y = 400

Also, M + W + C = 12

From the equations above, we can express M, W, C in terms of y:

M = \(\frac{3600}{3y}\) = \(\frac{1200}{y}\)

W = \(\frac{1600}{2y}\) = \(\frac{800}{y}\)

C = \(\frac{400}{1y}\) = \(\frac{400}{y}\)

Substitute M, W, and C into the total number of people equation:

⇒ \(\frac{1200}{y}\) + \(\frac{800}{y}\) + \(\frac{400}{y}\) = 12

⇒ \(\frac{1200+800+400}{y}\) = 12

⇒ \(\frac{2400}{y}\) = 12

⇒ y = \(\frac{2400}{12}\)

⇒ y = 200

Amount received by each woman = 2y = 2 × 200 = 400

∴ The amount received by each woman is ₹400.