# 1962 AHSME Problems/Problem 5

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## Problem

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

## Solution (Intuitive)

The ratio of a circumference to a diameter always is the same so the answer is obviously C.

## Solution 2 (Full Proof)

Let us say that the radius of a circle is $r$. When the radius is increased by $1$, the new radius is $r+1$ so the diameter is $2r+2$. We know that the circumference of a circle is $2\pi r$ so $2 \cdot \pi \cdot (r+1) = \pi \cdot (2r+2)$. Finally, the problem asked for the ratio of the new circumference to the new diameter is $\frac{\pi \cdot (2r+2)}{2r+2}=\boxed{\pi}$.

~Mathfun1000

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