Ratio and Proportion

Concept:Find the weighted average of the fractions of water and milk from the two mixtures when mixed in a given ratio.Explanation:Vessel 1 has water and milk in the ratio $1:2$. So the fraction of water is $\frac{1}{3}$ and milk is $\frac{2}{3}$.Vessel 2 has water and milk in the ratio $2:5$. So the fraction of water is $\frac{2}{7}$ and milk is $\frac{5}{7}$.The mixing ratio (V1 : V2) is $1:4$, so V1 contributes $\frac{1}{5}$ of the total volume and V2 contributes $\frac{4}{5}$.Final water fraction: $\frac{1}{5} \times \frac{1}{3} + \frac{4}{5} \times \frac{2}{7} = \frac{1}{15} + \frac{8}{35} = \frac{7}{105} + \frac{24}{105} = \frac{31}{105}$.Final milk fraction: $\frac{1}{5} \times \frac{2}{3} + \frac{4}{5} \times \frac{5}{7} = \frac{2}{15} + \frac{20}{35} = \frac{14}{105} + \frac{60}{105} = \frac{74}{105}$.Thus the ratio of water to milk is $31:74$.Answer:$31:74$