Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3...

Understanding the Question

We need to find the average donation amount for people who donated \($1,000\) or more.

Given Information

• \(\frac{1}{4}\) of donors gave \(≤ $20\)
• \(\frac{2}{3}\) of donors gave between \($20\) and \($1,000\) (with average \($180\))
• The remaining fraction gave \(≥ $1,000\)
• Since \(\frac{1}{4} + \frac{2}{3} = \frac{11}{12}\), this means \(\frac{1}{12}\) of donors gave \(≥ $1,000\)

What We Need to Determine

To have sufficiency, we need to be able to calculate one specific value for the average donation of the \(≥ $1,000\) group.

Key Insight

This is a weighted average problem with three distinct groups. To find the average of one group, we typically need information about either:

• The overall average of all donors, OR
• The combined average of groups that include our target group

Analyzing Statement 1

Statement 1: The average amount donated by people who donated less than \($1,000\) was \($132\).

What Statement 1 Provides

This gives us the combined average of the first two groups:

• Group 1: Donors who gave \(≤ $20\)
• Group 2: Donors who gave \($20\)-\($1,000\)

Together, these groups represent \(\frac{11}{12}\) of all donors.

Calculation Analysis

Using the weighted average formula, we can find the average for the \(≤ $20\) group.

We know:

• Combined groups (1 and 2): \(\frac{11}{12}\) of donors with average \($132\)
• Group 2 alone: \(\frac{2}{3}\) of donors with average \($180\)

Setting up the weighted average:
\(\left(\frac{1}{4} \times \mathrm{A_1} + \frac{2}{3} \times 180\right) \div \left(\frac{11}{12}\right) = 132\)

Converting to common denominator (12) for easier calculation:

• \(\left(\frac{3}{12} \times \mathrm{A_1} + \frac{8}{12} \times 180\right) \div \left(\frac{11}{12}\right) = 132\)
• \((3\mathrm{A_1} + 1440) \div 11 = 132\)
• \(3\mathrm{A_1} + 1440 = 1452\)
• \(3\mathrm{A_1} = 12\)
• \(\mathrm{A_1} = $4\)

Why This Isn't Sufficient

We now know:

• Group 1 (\(≤ $20\)): average = \($4\)
• Group 2 (\($20\)-\($1,000\)): average = \($180\)
• Group 3 (\(≥ $1,000\)): average = ?

Without knowing the overall average of ALL donors, we cannot determine Group 3's average. The missing link prevents us from finding a unique value.

Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now we analyze Statement 2 independently, forgetting Statement 1 completely.

Statement 2: The average amount donated by people who donated more than \($20\) was \($360\).

What Statement 2 Provides

This gives us the combined average of Groups 2 and 3:

• Group 2: Donors who gave \($20\)-\($1,000\)
• Group 3: Donors who gave \(≥ $1,000\)

Together, these groups represent \(\frac{2}{3} + \frac{1}{12} = \frac{9}{12} = \) \(\frac{3}{4}\) of all donors.

Calculation Analysis

We can use the weighted average formula to find Group 3's average.

Within the "more than \($20\)" category:

• Group 2: \(\frac{2}{3}\) of all donors (which is \(\frac{8}{9}\) of this combined group)
• Group 3: \(\frac{1}{12}\) of all donors (which is \(\frac{1}{9}\) of this combined group)

Setting up the weighted average:
\(\left(\frac{2}{3} \times 180 + \frac{1}{12} \times \mathrm{A_3}\right) \div \left(\frac{3}{4}\right) = 360\)

Converting fractions:

• \(\left(\frac{8}{12} \times 180 + \frac{1}{12} \times \mathrm{A_3}\right) \div \left(\frac{9}{12}\right) = 360\)
• \((1440 + \mathrm{A_3}) \div 9 = 360\)
• \(1440 + \mathrm{A_3} = 3240\)
• \(\mathrm{A_3} = $1,800\)

Verification

We found exactly one value: the average donation for the \(≥ $1,000\) group is \($1,800\).

[STOP - Sufficient!]

Statement 2 is sufficient.

This eliminates choices C and E.

The Answer: B

Statement 2 alone provides enough information to determine that the average donation for the \(≥ $1,000\) group is \($1,800\), while Statement 1 alone does not provide sufficient information.

Answer Choice B: Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.