Correlation of two sums sharing a common term

Let $X$, $Y$, $Z$ be pairwise uncorrelated with variances $64$, $36$ and $225$ respectively. Put $U=X+Y$ and $V=X+Z$. What is the correlation between $U$ and $V$?

$\tfrac{32}{85}$. Only the shared term contributes covariance: $\text{Cov}(U,V)=\text{Var}(X)=64$. The variances are $64+36=100$ and $64+225=289$, so $\rho=\tfrac{64}{10\cdot17}=\tfrac{32}{85}$.