If N & S are two numbers on the number line then what is

On the GMAT we can often see such statement: \(k\) is halfway between \(m\) and \(n\) on the number line. Remember this statement can ALWAYS be expressed as: \(\frac{m+n}{2}=k\).

If N & S are two numbers on the number line then what is the value of S?

(1) Distance between S & 0 is half the distance between N & 0 --> the distance between two numbers \(a\) and \(b\) on the number line is simply: \(|a-b|\), so we are told that \(|s-0|=\frac{|n-0|}{2}\) --> so we have that \(2|s|=|n|\) --> either \(2s=n\) or \(2s=-n\). In any case we can not get the single numerical value of \(s\). Not sufficient.

(2) -4 is halfway between N & S --> \(\frac{n+s}{2}=-4\) --> \(n+s=-8\). Not sufficient.

(1)+(2) If \(2s=n\) then \(n+s=3s=-8\) --> \(s=-\frac{8}{3}\) but if \(2s=-n\) then \(n+s=-2s+s=-s=-8\), so \(s=8\). Two different values, hence not sufficient.

Answer: E.