The annual rent collected by a corporation from a certain building was x percent more in 1998 than in 1997...

Understanding the Question

Let's understand what we're asked to determine: Was the annual rent collected in 1999 more than in 1997?

Given Information

• 1998 rent was \(\mathrm{x\%}\) more than 1997 rent
• 1999 rent was \(\mathrm{y\%}\) less than 1998 rent
• We need a yes/no answer about whether 1999 rent > 1997 rent

What We Need to Determine

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:

• 1998 rent = \(\mathrm{R} \times (1 + \mathrm{x}/100)\)
• 1999 rent = \(\mathrm{R} \times (1 + \mathrm{x}/100) \times (1 - \mathrm{y}/100)\)

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.

Analyzing Statement 1

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?

Testing Different Scenarios

Let's test with concrete examples starting at $100 rent in 1997:

Scenario 1: Small percentages

• \(\mathrm{x} = 10\%, \mathrm{y} = 5\%\)
• 1998 rent: $100 → $110 (increased by \(10\%\))
• 1999 rent: $110 → $104.50 (decreased by \(5\%\) of $110, which is $5.50)
• Result: $104.50 > $100 ✓

Scenario 2: Large percentages

• \(\mathrm{x} = 110\%, \mathrm{y} = 55\%\)
• 1998 rent: $100 → $210 (more than doubled)
• 1999 rent: $210 → $94.50 (lost \(55\%\) of $210, which is $115.50!)
• Result: $94.50 < $100 ✗

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 Key Insight

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.

Conclusion

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.

Analyzing Statement 2

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.

Understanding What Statement 2 Provides

Let's decode what this inequality means:

• The left side \((\mathrm{xy}/100)\) represents the "interaction effect"—how much extra loss occurs because \(\mathrm{y\%}\) applies to an increased base
• The right side \((\mathrm{x} - \mathrm{y})\) represents the "net percentage difference"—the simple arithmetic difference between the gain and loss

Statement 2 essentially says: "The interaction penalty is smaller than the net gain."

Why This Guarantees a Net Increase

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:

• With \(\mathrm{x} = 110\%\) and \(\mathrm{y} = 55\%\):
  • Interaction effect: \(\mathrm{xy}/100 = 110 \times 55/100 = 60.5\)
  • Net difference: \(\mathrm{x} - \mathrm{y} = 110 - 55 = 55\)
  • Since \(60.5 > 55\), Statement 2 is FALSE here
  • And indeed, we saw the rent decreased from $100 to $94.50!

This is exactly the condition that ensures \((1 + \mathrm{x}/100) \times (1 - \mathrm{y}/100) > 1\), which means 1999 rent > 1997 rent.

Conclusion

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.

The Answer: B

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."