AHSME 1962 Problem 5

If the radius of a circle is increased by $1$ unit, the ratio of the new circumference to the new diameter is: $\textbf{(A)}\ \pi+2 \textbf{(B)}\ \frac{2\pi+1}{2} \textbf{(C)}\ \pi \textbf{(D)}\ \frac{2\pi-1}{2} \textbf{(E)}\ \pi-2$