1959 AHSME Problems/Problem 12

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By adding the same constant to $20,50,100$ a geometric progression results. The common ratio is: $\textbf{(A)}\ \frac53 \qquad\textbf{(B)}\ \frac43\qquad\textbf{(C)}\ \frac32\qquad\textbf{(D)}\ \frac12\qquad\textbf{(E)}\ \frac{1}3$

Solution

Suppose that the constant is $x$. Then $20+x, 50+x, 100+x$ is a geometric progression, so $(20+x)(100+x) = (50+x)^2$. Expanding, we get $2000 + 120x + x^2 = 2500 + 100x + x^2$; therefore, $20x = 500$, so $x=25$.

Now we can calculate our geometric progression to be $20+25, 50+25, 100+25 = 45, 75, 125$. Therefore, the common ratio is $\frac{125}{75} = \frac{75}{45} = \boxed{\frac{5}{3}}$, and our answer is $\boxed{\textbf{A}}$.

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