$1 \cdot 113$ Two identical bottles are filled with alcohol solutions. In one bottle, the volume ratio of alcohol to water is $p: 1$, and in the other bottle, it is $q: 1$. If the two solutions are mixed together, the volume ratio of alcohol to water in the mixture is
(A) $\frac{p+q}{2}$.
(B) $\frac{p^{2}+q^{2}}{p+q}$.
(C) $\frac{2 p q}{p+q}$.
(D) $\frac{2\left(p^{2}+p q+q^{2}\right)}{3(p+q)}$.
(E) $\frac{p+q+2 p q}{p+q+2}$.
(30th American High School Mathematics Examination, 1979)
[Solution] If the total volume of the solution in each bottle is $x$ liters, then one bottle contains $\frac{p x}{p+1}$ liters of alcohol and $\frac{x}{p+1}$ liters of water; while the other bottle contains $\frac{q x}{q+1}$ liters of alcohol and $\frac{x}{q+1}$ liters of water.
In the mixed solution, the ratio of the volume of alcohol to water is
$$
\frac{\left(\frac{p}{p+1}+\frac{q}{q+1}\right) x}{\left(\frac{1}{p+1}+\frac{1}{q+1}\right) x}=\frac{p(q+1)+q(p+1)}{(q+1)+(p+1)}=\frac{p+q+2 p q}{p+q+2} \text {. }
$$

Therefore, the answer is $(E)$.