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On the number line shown, is zero halfway between \(\mathrm{r}\) and \(\mathrm{s}\)?
We need to determine whether zero is halfway between r and s on a number line.
For zero to be halfway between two points, those points must be equidistant from zero but on opposite sides. This means one point must be the negative of the other (like -3 and +3).
In mathematical terms, we're asking: Is \(\mathrm{r + s = 0}\)? Or equivalently: Is \(\mathrm{s = -r}\)?
This is a yes/no question. We need sufficient information to answer definitively YES (if \(\mathrm{r + s = 0}\)) or NO (if \(\mathrm{r + s ≠ 0}\)).
Important: The diagram shows us that \(\mathrm{r < s < t}\). While we cannot trust the scale or exact distances in GMAT diagrams, we can trust the relative ordering of points as labeled. This constraint must be respected in all our examples.
Statement 1: s is to the right of zero (\(\mathrm{s > 0}\)).
This tells us s is positive, but we know nothing about where r is located.
Let's test different scenarios that respect \(\mathrm{r < s < t}\):
Since we can get both YES and NO answers depending on where r is placed, Statement 1 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choices A and D.
Important: We now forget Statement 1 completely and analyze Statement 2 independently.
Statement 2: The distance between t and r equals the distance between t and -s.
Mathematically: \(\mathrm{|t - r| = |t - (-s)| = |t + s|}\)
From the diagram, we know \(\mathrm{r < s < t}\), which means \(\mathrm{t > r}\). Therefore:
So our equation becomes: \(\mathrm{t - r = |t + s|}\)
The absolute value \(\mathrm{|t + s|}\) creates two possible cases:
Case 1: If \(\mathrm{t + s ≥ 0}\), then \(\mathrm{|t + s| = t + s}\)
Case 2: If \(\mathrm{t + s < 0}\), then \(\mathrm{|t + s| = -(t + s)}\)
Let's verify with concrete examples that respect \(\mathrm{r < s < t}\):
Example 1: \(\mathrm{r = -4, s = 4, t = 4}\)
Example that works: \(\mathrm{r = -10, s = -5, t = -2.5}\)
Another example: \(\mathrm{r = -5, s = 5, t = 5.01}\)
Since we can construct valid examples giving both YES and NO answers, Statement 2 is NOT sufficient.
[STOP - Not Sufficient!] This eliminates choice B.
Now we use both statements together:
Key insight: Since \(\mathrm{s > 0}\) (Statement 1) and \(\mathrm{t > s}\) (from diagram), we know:
Since \(\mathrm{t + s > 0}\), we are in Case 1 from our Statement 2 analysis:
Let's verify we cannot construct a counterexample that satisfies both statements:
Test: Can we find values where \(\mathrm{r < s < t}\), \(\mathrm{s > 0}\), \(\mathrm{t - r = |t + s|}\), but \(\mathrm{r + s ≠ 0}\)?
From Statement 1, \(\mathrm{s > 0}\). From the diagram, \(\mathrm{t > s}\), so \(\mathrm{t > 0}\). This forces \(\mathrm{t + s > 0}\), which forces \(\mathrm{r = -s}\).
There is no way to satisfy both statements without having \(\mathrm{r + s = 0}\).
Therefore, the statements together ARE sufficient to answer the question with a definitive YES.
[SUFFICIENT!]
Statement 1 alone is not sufficient. Statement 2 alone is not sufficient. But both statements together ARE sufficient to determine that zero is halfway between r and s.
Answer Choice C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.