1962 AHSME 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 \qquad \textbf{(B)}\ \frac{2 \pi + 1}{2} \qquad \textbf{(C)}\ \pi \qquad \textbf{(D)}\ \frac{2\pi-1}{2}\qquad \textbf{(E)}\ \pi-2

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.