David, when submitting a problem for CMIMC, wrote his answer as $100\frac{x}{y}$, where $x$ and $y$ are two positive integers with $x < y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's. What is the smallest possible value of $x + y$?

Let's think step by step and output the final answer within \boxed{}.
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