Vessel A contains milk and water in the ratio 4 : 5. Vessel B contains milk and water in the ratio 2 : 1. If x litres mixture of A is mixed with y litres mixture of B, then the ratio of milk to water in the mixture becomes 8 : 5. Find the ratio x : y.

Correct option is A

Given:

Vessel A contains milk and water in the ratio 4:5.

Vessel B contains milk and water in the ratio 2:1.

x litres of mixture from Vessel A is mixed with y litres of mixture from Vessel B.

After mixing, the ratio of milk to water in the resulting mixture is 8:5.

Solution:

Milk and water in Vessel A:

In Vessel A, the ratio of milk to water is 4:5.

Therefore, in x litres of mixture from Vessel A:

Milk =$$\frac{4}{9}x$$ litres

Water $$= \frac{5}{9}x$$​ litres

Milk and water in Vessel B:

In Vessel B, the ratio of milk to water is 2:1.

Therefore, in y litres of mixture from Vessel B:

Milk =$$\frac{2}{3}y$$​ litres

Water = $$\frac{1}{3}y$$​ litres

Total milk in the mixture = Milk from Vessel A + Milk from Vessel B

$$\text{Total milk} = \frac{4}{9}x + \frac{2}{3}y$$​

​$$\text{Total water} = \frac{5}{9}x + \frac{1}{3}y$$​​

The ratio of milk to water is given as 8:5, so:

$$\frac{\frac{4}{9}x + \frac{2}{3}y}{\frac{5}{9}x + \frac{1}{3}y} = \frac{8}{5}$$​

​$$5 \times \frac{4}{9}x + 5 \times \frac{2}{3}y = 8 \times \frac{5}{9}x + 8 \times \frac{1}{3}y$$​​

$$\frac{20}{9}x + \frac{10}{3}y = \frac{40}{9}x + \frac{8}{3}y$$

20x + 30y = 40x + 24y

6y = 20x

​$$\frac{x}{y} = \frac{6}{20} = \frac{3}{10}$$

Alternate Method:

Milk in Vessel A=$$\frac 49 \times 13 \times 9 =52$$

Milk in Vessel B =$$\frac 23\times 13 \times 9 =78$$ 

The ratio of milk to mixture = $$\frac 8{13} \times 13 \times 9$$ = 72 

Thus, x and y is added in the 3 : 10