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.