Of the people who donated money to a certain local theater last year, 1/4 donated $20 or less and 2/3 donated more than $20 but less than $1,000. If the average (arithmetic mean) amount donated by the people who donated more than $20 but less than $1,000 was $180, what was the average amount donated by the people who donated $1,000 or more?
(1) The average amount donated by the people who donated less than $1,000 was $132.
(2) The average amount donated by the people who donated more than $20 was $360.
- A. Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.
- B. Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.
- C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- D. EACH statement ALONE is sufficient.
- E. Statements (1) and (2) TOGETHER are not sufficient.
Solution:
Understanding the Question
We need to find the average donation amount for people who donated $1,000 or more.
Given Information
- 1/4 of donors gave less than or equal to $20
- 2/3 of donors gave between $20 and $1000 (with average $180)
- The remaining fraction gave greater than or equal to $1000
- Since 1/4 plus 2/3 is equal to 11/12, this means 1/12 of donors gave greater than or equal to $1000
What We Need to Determine
To have sufficiency, we need to be able to calculate one specific value for the average donation of the group that gave greater than or equal to $1000.
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 $1000 was $132.
What Statement 1 Provides
This gives us the combined average of the first two groups:
- Group 1: Donors who gave less than or equal to $20
- Group 2: Donors who gave $20-$1000
Together, these groups represent 11/12 of all donors.
Calculation Analysis
Using the weighted average formula, we can find the average for the less than or equal to $20 group.
We know:
- Combined groups (1 and 2): 11/12 of donors with average $132
- Group 2 alone: 2/3 of donors with average $180
Setting up the weighted average:
(1/4 * A1 + 2/3 *180) divided by (11/12) = 132
Converting to common denominator (12) for easier calculation:
- (3/12 * A1 + 8/12*180) divided by (11/12) = 132
- (3A1 + 1440) divided by 11 = 132
- 3A1 + 1440 = 1452
- 3A1 = 12
- A1 = $4
Why This Isn’t Sufficient
We now know:
- Group 1 (less than or qraul to $20): average = $4
- Group 2 ($20 – $1000): average = $180
- Group 3 (greater than or equal to $1000): 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 – $1000
- Group 3: Donors who gave greater than or equal to $1000
Together, these groups represent 2/3 + 1/12 = 9/12 = 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: 2/3 of all donors (which is 8/9 of this combined group)
- Group 3: 1/12 of all donors (which is 1/9 of this combined group)
Setting up the weighted average:
(2/3 *180 + 1/12*A3) divided by (3/4) = 360
Converting fractions:
- (8/12*180 + 1/12*A3) divided by 9/12 = 360
- (1440 + A3) divided by 9 = 360
- 1440 + A3 = 3240
- A3 = $1800
Verification
We found exactly one value: the average donation for the greater than or equal to $1000 group is $1800.
[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 greater than or equal to $1000 group is $1800, while Statement 1 alone does not provide sufficient information.
Answer Choice B: Statement 2 alone is sufficient, but Statement 1 alone is not sufficient.
