Given that $p$ is a positive number, $n$ is a negative number, and $|p|\gt |n|$, which of the following expressions has the greatest value?

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We can substitute values, such as $p=2$ and $n=-1$. The second choice will result in the largest value.

To maximize the value of a fraction, we want the numerator to be as large as possible and the denominator as small as possible.

Among the choices for the numerator $p-n$ vs $p+n$, $p-n$ will result in the greater value since $p$ is positive and $n$ is negative.

To minimize the denominator, among the choices, $n$ is the smallest.