The total number of men, women and children working in a factory is 18. They earn Rs 4000 in a day. If the sum of the wages of all men, all women and all children is in the ratio of 18 : 10 : 12 and if the wages of an individual man, woman and child is in the ratio 6 : 5 : 3, then how much a child earns in a day?

Given:

Total number of men, women, and children = 18

Total wages in a day = Rs. 4000

Sum of wages ratio (men : women : children) = 18 : 10 : 12

Individual wage ratio (man : woman : child) = 6 : 5 : 3

Formula Used:

Total wages = (Number of men × Wage of one man) + (Number of women × Wage of one woman) + (Number of children × Wage of one child)

Calculation:

Let the number of men, women, and children be m, w, and c respectively.

m + w + c = 18     ...(1)

Let the wages of one man, woman, and child be 6x, 5x, and 3x respectively.

The total wages are distributed in the ratio 18 : 10 : 12.

Total wages = Rs. 4000

Wages of all men = (18 / (18 + 10 + 12)) × 4000 = (18 / 40) × 4000 = Rs. 1800

Wages of all women = (10 / 40) × 4000 = Rs. 1000

Wages of all children = (12 / 40) × 4000 = Rs. 1200

For men: Total wages of all men = m × 6x = 1800     ...(2)

For women: Total wages of all women = w × 5x = 1000     ...(3)

For children: Total wages of all children = c × 3x = 1200     ...(4)

From equation (2):

⇒ m × 6x = 1800

⇒ m × x = 300     ...(5)

From equation (3):

⇒ w × 5x = 1000

⇒ w × x = 200     ...(6)

From equation (4):

⇒ c × 3x = 1200

⇒ c × x = 400     ...(7)

From equations (5), (6), and (7):

m = 300 / x, w = 200 / x, c = 400 / x

Substitute these values into equation (1):

⇒ (300 / x) + (200 / x) + (400 / x) = 18

⇒ (300 + 200 + 400) / x = 18

⇒ 900 / x = 18

⇒ x = 900 / 18

⇒ x = 50

Wage of one child = 3x

⇒ Wage of one child = 3 × 50

⇒ Wage of one child = Rs. 150

The wage of one child in a day is Rs. 150.