2011 AMC 12B Problems/Problem 7

Problem

Let $x$ and $y$ be two-digit positive integers with mean $60$ . What is the maximum value of the ratio $\frac{x}{y}$ ?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ \frac{33}{7} \qquad \textbf{(C)}\ \frac{39}{7} \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ \frac{99}{10}$

Solution

If $x$ and $y$ have a mean of $60$ , then $\frac{x+y}{2}=60$ and $x+y=120$ . To maximize $\frac{x}{y}$ , we need to maximize $x$ and minimize $y$ . Since they are both two-digit positive integers, the maximum of $x$ is $99$ which gives $y=21$ . $y$ cannot be decreased because doing so would increase $x$ , so this gives the maximum value of $\frac{x}{y}$ , which is $\frac{99}{21}=\boxed{\frac{33}{7}\ \textbf{(B)}}$

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2011 AMC 12B Problems/Problem 7

Problem

Solution

See also