2011 AMC 12B Problem 7

Problem 7 of 25EasierAlgebra

Let

x

and

y

be two-digit positive integers with mean

60.

What is the maximum value of the ratio

\dfrac{x}{y}?

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Solution

Since

\dfrac{x+y}{2}=60,

we have

x+y=120.

To maximize

\dfrac{x}{y}

we make

y

small. Because

x\le99,

it follows that

y=120-x\ge21.

Taking

x=99

and

y=21

gives the maximum

\dfrac{99}{21}=\dfrac{33}{7}.

Thus, the correct answer is B.

Concepts: optimization · bounding to limit cases

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.