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Choice A is correct. Let $x$ represent the percentage of $300$ that is $75$. This can be written as $\frac{x}{100}\left(300\right)=75$, or $3x=75$. Dividing both sides of this equation by $3$ yields $x=25$. Therefore, $25\%$ of $300$ is $75$.

Choice B is incorrect. $50\%$ of $300$ is $150$, not $75$.

Choice C is incorrect. $75\%$ of $300$ is $225$, not $75$.

Choice D is incorrect. $225\%$ of $300$ is $675$, not $75$.