Vessels A and B contain mixtures of milk and water in the ratio 4 : 5

1.	Vessels A and B contain mixtures of milk and water in the ratio 4 : 5 and 5 : 1 respectively. In what ratio should quantities of mixtures be taken from A to B to form a mixture in which milk to water is in the ratio 5 : 4 ? (a) 2 : 5 (b) 2 : 3 (c) 4 : 3 (d) 5 : 2
Answer» (d) 5 : 2 Let the required ratio be x : y. Milk in x litres of 1st mixture = \(\big(x\times\frac49\big)\) litres = \(\frac{4x}{9}\) litres Water in x litres of 1st mixture = \(\big(x-\frac{4x}9\big)\) llitres = \(\frac{5x}{9}\) litres Milk in y litres of 2nd mixture = \(\big(y\times\frac56\big)\) llitres = \(\frac{5y}{6}\) litres Water in y litres of 2nd mixture = \(\big(y-\frac{5y}6\big)\) llitres = \(\frac{y}{6}\) litres ∴ Milk : Water = \(\big(\frac{4x}{9}+\frac{5y}{6}\big)\) : \(\big(\frac{5x}{9}+\frac{y}{6}\big)\) = \(\big(\frac{8x+15y}{18}\big):\big(\frac{10x+3y}{18}\big)\) = (8x + 15y) : (10x +3y) Given, \(\frac{8x+15y}{10x+3y}=\frac54\) ⇒ 32x + 60y = 50x + 15y ⇒ 18x = 45y ⇒ \(\frac{x}{y} = \frac{45}{18}=\frac52\) ∴ Required ratio = 5 : 2

Vessels A and B contain mixtures of milk and water in the ratio 4 : 5 and 5 : 1 respectively. In what ratio should quantities of mixtures be taken from A to B to form a mixture in which milk to water is in the ratio 5 : 4 ? (a) 2 : 5 (b) 2 : 3 (c) 4 : 3 (d) 5 : 2

Answer»

(d) 5 : 2

Let the required ratio be x : y.

Milk in x litres of 1st mixture = \(\big(x\times\frac49\big)\) litres = \(\frac{4x}{9}\) litres

Water in x litres of 1st mixture = \(\big(x-\frac{4x}9\big)\) llitres = \(\frac{5x}{9}\) litres

Milk in y litres of 2nd mixture = \(\big(y\times\frac56\big)\) llitres = \(\frac{5y}{6}\) litres

Water in y litres of 2nd mixture = \(\big(y-\frac{5y}6\big)\) llitres = \(\frac{y}{6}\) litres

∴ Milk : Water = \(\big(\frac{4x}{9}+\frac{5y}{6}\big)\) : \(\big(\frac{5x}{9}+\frac{y}{6}\big)\)

= \(\big(\frac{8x+15y}{18}\big):\big(\frac{10x+3y}{18}\big)\)

= (8x + 15y) : (10x +3y)

Given, \(\frac{8x+15y}{10x+3y}=\frac54\) ⇒ 32x + 60y = 50x + 15y

⇒ 18x = 45y ⇒ \(\frac{x}{y} = \frac{45}{18}=\frac52\)

∴ Required ratio = 5 : 2