jmc

The number of ways to choose one white ball and one black ball is $5k$, since there are 5 choices for the white ball and $k$ choices for the black ball. The number of ways to pick any 2 balls out of $(k+5)$ balls is $\left(\genfrac{}{}{0ex}{}{k+5}{2}\right)=\frac{(k+5)(k+4)}{2}$. So we have to solve for $k$ in the equation $$\frac{5k}{\frac{(k+5)(k+4)}{2}}=\frac{10}{21}.$$After clearing the denominators, we can simplify to $210k=10(k+5)(k+4)$, giving the quadratic $10k^2-120k+200=0$. This is the same as $k^2-12k+20=0$, which factors as $(k-2)(k-10)=0$, so its solutions are $k=2$ or $k=10$. Since the problem is asking for the smallest value, $\boxed{2}$ is the correct answer.