Two vessels of equal capacity, and both completely filled, contain mixtures of milk and water in the ratio of 5 : 1 and 6 : 5, respectively. Both the mixtures are now mixed thoroughly. What is the ratio of milk to water in the new mixture so obtained?

The first ratio of mixtures of milk and water = 5:1

quantity of milk in the first mixture = $$\frac{5}{5+1}$$ = $$\frac{5}{6}$$    Eq.(i)

quantity of water in the first mixture = $$\frac{1}{5+1}$$ = $$\frac{1}{6}$$     Eq.(ii)

The second ratio of mixtures of milk and water = 6:5

quantity of milk in the second mixture = $$\frac{6}{6+5}$$ = $$\frac{6}{11}$$     Eq.(iii)

quantity of water in the second mixture = $$\frac{5}{6+5}$$ = $$\frac{5}{11}$$    Eq.(iv)

Ratio of milk to water in the new mixture = Eq.(i)+Eq.(iii) : Eq.(ii)+Eq.(iv)

= $$\frac{5}{6}+\frac{6}{11}$$ : $$\frac{1}{6}+\frac{5}{11}$$

= $$\frac{55+36}{66}$$ : $$\frac{11+30}{66}$$

= $$\frac{91}{66}$$ : $$\frac{41}{66}$$

= 91 : 41