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The annual rent collected by a corporation from a certain building was \(\mathrm{x}\) percent more in 1998 than in 1997 and \(\mathrm{y}\) percent less in 1999 than in 1998. Was the annual rent collected by the corporation from the building more in 1999 than in 1997?
Let's understand what we're asked to determine: Was the annual rent collected in 1999 more than in 1997?
Here's where it gets interesting. We have a value that first increases by \(\mathrm{x\%}\), then decreases by \(\mathrm{y\%}\). The crucial insight is that when you decrease by \(\mathrm{y\%}\) after increasing by \(\mathrm{x\%}\), the \(\mathrm{y\%}\) applies to the already-increased amount, not the original base.
If we start with rent R in 1997:
So we need to know: Is \((1 + \mathrm{x}/100) \times (1 - \mathrm{y}/100) > 1\)?
For this question to be sufficient, we need to be able to definitively answer YES or NO to whether 1999 rent exceeds 1997 rent.
Statement 1 tells us: \(\mathrm{x} > \mathrm{y}\)
This means the percentage increase from 1997 to 1998 was greater than the percentage decrease from 1998 to 1999. But does this guarantee that 1999 rent > 1997 rent?
Let's test with concrete examples starting at $100 rent in 1997:
Scenario 1: Small percentages
Scenario 2: Large percentages
Notice how in the second scenario, even though \(\mathrm{x} > \mathrm{y}\), the final rent is actually less! This happens because the \(55\%\) decrease applies to the much larger $210 base, removing $115.50, which is more than the $110 we initially gained.
The problem isn't just about which percentage is bigger—it's about the interaction effect. When percentages are large, the fact that \(\mathrm{y\%}\) applies to an increased base can overwhelm the original gain.
Since we can get different answers (YES in scenario 1, NO in scenario 2), Statement 1 alone is NOT sufficient.
[STOP - Not Sufficient!]
This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: \(\mathrm{xy}/100 < \mathrm{x} - \mathrm{y}\)
This condition directly addresses the interaction between the increase and decrease percentages.
Let's decode what this inequality means:
Statement 2 essentially says: "The interaction penalty is smaller than the net gain."
When \(\mathrm{xy}/100 < \mathrm{x} - \mathrm{y}\), this ensures that even after accounting for the fact that \(\mathrm{y\%}\) applies to a larger base (the increased 1998 rent), we still have a net gain overall.
Let's visualize with our earlier example:
This is exactly the condition that ensures \((1 + \mathrm{x}/100) \times (1 - \mathrm{y}/100) > 1\), which means 1999 rent > 1997 rent.
Statement 2 alone is sufficient to answer YES to the question—the 1999 rent was definitely more than the 1997 rent.
[STOP - Sufficient!]
This eliminates choices C and E.
Statement 2 alone provides exactly the condition needed to guarantee that 1999 rent exceeds 1997 rent, while Statement 1 alone does not.
Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."