The initial ratio of water and milk in a mixture of 84 litres is 2 : 5. What quantity of water (in litres) should be added to the mixture so that the resulting mixture has 50% of water?

Given:

The initial ratio of water and milk in a mixture of 84 litres is 2 : 5.

We need to find the quantity of water to be added so that the resulting mixture has 50% water.

Formula used:

Let the initial amount of water be W and milk be M.

According to the ratio, W : M = 2 : 5

Total mixture = W + M = 84 litres

Calculations:

Let W be the quantity of water and M be the quantity of milk.

W = 2x, M = 5x

2x + 5x = 84

⇒ 7x = 84

⇒ x = 12

Initial quantity of water, W = 2x = 2 × 12 = 24 litres

Initial quantity of milk, M = 5x = 5 × 12 = 60 litres

Let y litres of water be added to make the water 50% of the mixture.

New quantity of water = 24 + y

New total mixture = 84 + y

According to the problem, 50% of the new mixture is water:

⇒ 24 + y = 0.5 × (84 + y)

⇒ 24 + y = 42 + 0.5y

⇒ 24 + y - 0.5y = 42

⇒ 24 + 0.5y = 42

⇒ 0.5y = 42 - 24

⇒ 0.5y = 18

⇒ y = 18 ÷ 0.5

⇒ y = 36

∴ The correct answer is option (3).