Each of the 20 people working in a certain office contributed either $9, $10, or $11 toward an office party. What was the average (arithmetic mean) amount contributed per person in the office? The number of people who contributed $9 was the same as the number of people who contributed $11. The number of people who contributed $9 was more than the number of people who contributed $10.

Each of the 20 people working in a certain office contributed either $9, $10, or $11 toward an office party....

GMAT Data Sufficiency : (DS) Questions

Source: Mock

Data Sufficiency

DS - Statistics

HARD

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Notes

Post a Query

Each of the 20 people working in a certain office contributed either $9, $10, or $11 toward an office party. What was the average (arithmetic mean) amount contributed per person in the office?

The number of people who contributed $9 was the same as the number of people who contributed $11.
The number of people who contributed $9 was more than the number of people who contributed $10.

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

Solution

Understanding the Question

We need to find the average contribution per person when 20 people each contribute $\$9$, $\$10$, or $\$11$.

Given Information

20 people total
Each person contributes exactly one of: $\$9$, $\$10$, or $\$11$
We need the arithmetic mean (average) contribution

What We Need to Determine

Can we find one specific value for the average contribution? To answer this question definitively, we'd need to know exactly how many people contributed each amount.

Key Insight

Notice that $\$9$ and $\$11$ are symmetric around $\$10$ – they're each $\$1$ away from the middle value. This symmetry will be crucial in our analysis.

Analyzing Statement 1

Statement 1 tells us: The number of $\$9$ contributors equals the number of $\$11$ contributors.

This creates perfect symmetry! Here's why this matters:

Each person who contributes $\$9$ is $\$1$ below $\$10$
Each person who contributes $\$11$ is $\$1$ above $\$10$
With equal numbers of each, they balance out perfectly

Think of it like a seesaw: if you have equal weights at equal distances from the center, they balance. The remaining people all contributed $\$10$, so the average must be exactly $\$10$.

Let's visualize this: If 5 people gave $\$9$ and 5 people gave $\$11$, the 5 people below $\$10$ by $\$1$ perfectly cancel out the 5 people above $\$10$ by $\$1$. The other 10 people gave $\$10$. Result: average = $\$10$.

No calculations needed – this is a conceptual certainty based on the symmetry principle.

[STOP - Statement 1 is SUFFICIENT!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: More people contributed $\$9$ than contributed $\$10$.

This gives us a relationship between $\$9$ and $\$10$ contributors, but tells us nothing about the $\$11$ contributors. Let's test some scenarios:

Scenario 1: Many $\$11$ contributors

Suppose 8 people gave $\$9$, 7 gave $\$10$, and 5 gave $\$11$
The average would be pulled both down (by the $\$9$s) and up (by the $\$11$s)

Scenario 2: Few $\$11$ contributors

Suppose 10 people gave $\$9$, 8 gave $\$10$, and 2 gave $\$11$
Now the many $\$9$ contributors would strongly pull the average down

Since we don't know how many contributed $\$11$, we can't determine a unique average. The average could be anywhere depending on the number of $\$11$ contributors.

Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Statement 1 alone gives us a unique average ($\$10$) due to the perfect symmetry it creates, while Statement 2 leaves too much uncertainty about the $\$11$ contributors.

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

Answer Choices Explained

Statement (1) ALONE is sufficient but statement (2) ALONE is not sufficient.

Statement (2) ALONE is sufficient but statement (1) ALONE is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

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