Given that $$\frac{x}{y} \lt 1$$, and both $$x$$ and $$y$$ are positive integers, which one of the following expressions must be greater than 1?

Correct.
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__Plug In__ $$x = 1$$ and $$y = 5$$:
> $$\frac{y}{x} = \frac{5}{1}=5$$.
Since $$5 \gt 1$$, this answer choice cannot be eliminated for this Plug-In. All the other answer choices give a result that is smaller than 1 for the same Plug-In and thus are eliminated since it is not required that they are greater than 1. Therefore, $$\frac{y}{x}$$ is the correct answer.
__Alternative explanation__:
Since both $$x$$ and $$y$$ are positive, it is possible to multiply the original inequality by $$y$$ without flipping the sign. Thus, the inequality $$\frac{x}{y} \lt 1$$ becomes $$x \lt y$$, which can then be further divided by $$x$$ to get $$1 \lt \frac{y}{x}$$.

Incorrect.
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__Plug In__ $$x = 1$$ and $$y = 5$$:
> $$\frac{x}{y^2} = \frac{1}{5^2} = \frac{1}{25}$$.
Since $$\frac{1}{25}$$ is **not** greater than 1, this expression does not need to be greater than 1. Eliminate this answer choice.

Incorrect.
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__Plug In__ $$x = 1$$ and $$y = 5$$:
> $$\frac{x^2}{y} = \frac{1^2}{5} = \frac{1}{5}$$.
Since $$\frac{1}{5}$$ is **not** greater than 1, this expression does not need to be greater than 1. Eliminate this answer choice.

Incorrect.
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__Plug In__ $$x = 1$$ and $$y = 5$$:
> $$\frac{x^2}{y^2} = \frac{1^2}{5^2} = \frac{1}{25}$$.
Since $$\frac{1}{25}$$ is **not** greater than 1, this expression does not need to be greater than 1. Eliminate this answer choice.

Incorrect.
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__Plug In__ $$x = 1$$ and $$y = 5$$:
> $$\sqrt{\frac{x}{y}} = \sqrt{\frac{1}{5}}=\frac{1}{\sqrt{5}}$$.
Since $$\frac{1}{\sqrt{5}}$$ is **not** greater than 1, this expression does not need to be greater than 1. Eliminate this answer choice.

$$\frac{x}{y^2}$$

$$\frac{x^2}{y}$$

$$\frac{x^2}{y^2}$$

$$\frac{y}{x}$$

$$\sqrt{\frac{x}{y}}$$