Question

What is the average of the terms in set J?
1. The sum of any three terms in Set J is 21
2. Set J consists of 12 total terms.

• A. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed
• B. Both statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient
• C. EACH statement ALONE is sufficient to answer the question asked
• D. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked
• E. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked

Hint:

In problems about sets, a condition that applies to "any" subset (like "any three terms") is extremely restrictive. It often implies that all elements of the set are identical, which simplifies the problem dramatically.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question asks for the average (arithmetic mean) of the terms in a set. To find the average, we need two pieces of information: the sum of the terms and the number of terms.

Step 2: Key Formula or Approach:
\[ \text{Average} = \frac{\text{Sum of terms}}{\text{Number of terms}} \] We need to evaluate if the statements, alone or together, provide the necessary information to calculate this value uniquely.

Step 3: Detailed Explanation:
Analyzing Statement (1): The sum of any three terms in Set J is 21
Let the terms in Set J be \(j_1, j_2, j_3, j_4, \dots\).
The statement says that for any three terms we choose, their sum is 21. Let's assume the set has at least four terms to test this condition.
Pick three terms: \(j_1, j_2, j_3\). We have: \[ j_1 + j_2 + j_3 = 21 \] Now, pick another set of three terms, replacing one term, for example \(j_1, j_2, j_4\): \[ j_1 + j_2 + j_4 = 21 \] Comparing these two equations: \[ j_1 + j_2 + j_3 = j_1 + j_2 + j_4 \] \[ j_3 = j_4 \] This logic applies to any pair of terms in the set. This means all terms in Set J must be equal to each other. Let's call the value of each term \(j\).
So, for any three terms, we have: \[ j + j + j = 21 \] \[ 3j = 21 \] \[ j = 7 \] This means every term in Set J is 7.
The average of a set of numbers that are all identical is simply that number itself. So, the average of the terms in Set J is 7.
This statement provides a unique value for the average.
Therefore, Statement (1) ALONE is sufficient.
Analyzing Statement (2): Set J consists of 12 total terms.
This statement tells us the number of terms in the set. However, it gives no information about the values of these terms or their sum.
The average could be anything. For example, if all terms are 1, the average is 1. If all terms are 10, the average is 10.
Therefore, Statement (2) ALONE is not sufficient.

Step 4: Final Answer:
Statement (1) is sufficient to find the average, while statement (2) is not. The correct option is (D).