In a container, there are 6 liters of a homogeneous mixture of two liquids (alpha and beta) in the ratio of $7: 2$, while in another container, there are 9 liters of another mixture with the same two liquids (alpha and beta), but in this case, the ratio is $4: 7$. By mixing the liquids from the two containers, what will be the new ratio?

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Solution

In the first container, the amount of liquid alpha is $\frac{7}{9} \cdot 6=\frac{14}{3}$ liters and of beta is $\frac{2}{9} \cdot 6=$ $\frac{4}{3}$ liters, while in the second container, the amounts of alpha and beta, respectively, are $\frac{4}{11} \cdot 9=\frac{36}{11}$ and $\frac{7}{11} \cdot 9=\frac{63}{11}$ liters. When we mix them, we obtain $\frac{14}{3}+\frac{36}{11}=\frac{262}{33}$ liters of alpha and $\frac{4}{3}+\frac{63}{11}=\frac{233}{33}$ liters of beta. Therefore, the final ratio of the mixture (alpha : beta) is $\frac{\frac{262}{33}}{\frac{233}{33}}=\frac{262}{233}$.