Is r negative?

\(r\,\,\mathop < \limits^? \,\,0\)

\(\left( 1 \right)\,\,\,\,r < s\,\,\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,\left( {r,s} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\\\
\,{\text{Take}}\,\,\left( {r,s} \right) = \left( { - 1,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \\
\end{gathered} \right.\)

\(\left( 2 \right)\,\,\,\,r = - s\,\,\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,\left( {r,s} \right) = \left( {1, - 1} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\\\
\,{\text{Take}}\,\,\left( {r,s} \right) = \left( { - 1,1} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\ \\
\end{gathered} \right.\)

\(\left( {1 + 2} \right)\,\,\,\,\left\{ \begin{gathered}\\
\,\,r = 0\,\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)} \,\,\,\,s = 0\,\,\,\, \Rightarrow \,\,\,\,\,\left( 1 \right)\,\,\,{\text{contradicted}} \hfill \\\\
\,\,r > 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)} \,\,\,\,s < 0\,\,\,\, \Rightarrow \,\,\,\,\,\left( 1 \right)\,\,\,{\text{contradicted}} \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,r \geqslant 0\,\,\,{\text{impossible}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

(1) r can be positive or negative. Insufficient

(2) When r is positive, s is negative and vice versa. Insufficient

(1)+(2), r is smaller than s; r is negative and s is positive. C is correct.

Since the question stem does not give us any information, let us go straight to the statements.

From statement I alone, we can only gather that r is smaller than s. But, this does not tell us anything about the sign of r. r may be positive, negative or ZERO.
Statement I alone is insufficient. Answer options A and D can be eliminated. Possible answer options at this stage are B, C or E.

From statement II alone, r = -s. From this, we see that the sign of r depends on the sign of s. If s is positive, r is negative and vice versa. Additionally, if s = 0, then r = 0. Clearly insufficient to answer whether r is negative.

Answer option B can be eliminated. The possible answer options are C or E.

Combining the data from statements I and II, we have the following:
From statement II, if s is positive, r is negative. This satisfies the data given in the first statement that r<s.

But if s is negative, r becomes positive. But, according to statement I, r has to be less than s. Therefore, we can conclude that s cannot be negative and hence r cannot be positive if it has to satisfy the combination of statements.

Similarly, s cannot be ZERO as well since that would make both r and s equal, which is not allowed.

So, from the combination of statements, we can answer that r is definitely negative. The combination of statements is sufficient. Answer option E can be eliminated.
The correct answer option is C.

Hope that helps!

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