1965 AHSME Problems/Problem 15

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Problem

The symbol $25_b$ represents a two-digit number in the base $b$. If the number $52_b$ is double the number $25_b$, then $b$ is:

$\textbf{(A)}\ 7 \qquad  \textbf{(B) }\ 8 \qquad  \textbf{(C) }\ 9 \qquad  \textbf{(D) }\ 11 \qquad  \textbf{(E) }\ 12$

Solution

We begin by converting both $25_b$ and $52_b$ to base $10$. $25_b = 2b+5$ in base $10$ and $52_b = 5b+2$ base $10$. The problem tells us that $5b+2 = 4b+10$, yielding $\boxed{8}$ as our final answer.