Question

Given \( x<y<z \), compare the following quantities: \[ \frac{x + y + z}{3} \quad \text{and} \quad y \]

• A. Quantity A is greater
• B. Quantity B is greater
• C. The two quantities are equal
• D. The relationship cannot be determined from the information given.
• E.

Hint:

When comparing averages and individual values, remember that the middle value (median) can be larger or smaller than the average depending on the values of the other numbers.

Answer 1

The Correct Option is B

Step 1: the given condition.
We are given \( x<y<z \), meaning \( y \) is the middle value. We are asked to compare the average \( \frac{x + y + z}{3} \) with \( y \).
Step 2: Try an example.
For example, let \( x = 1, y = 2, z = 3 \). Then: \[ \frac{x + y + z}{3} = \frac{1 + 2 + 3}{3} = 2 \] Thus, the two quantities are equal in this case. However, if \( x = 1, y = 2, z = 10 \), then: \[ \frac{x + y + z}{3} = \frac{1 + 2 + 10}{3} = 4.33 \] Here, Quantity A is greater than Quantity B.
Step 3: Conclusion.
Based on the analysis and the specific examples, the relationship between the quantities can be that Quantity B is greater.
Final Answer: \[ \boxed{\text{The correct answer is (2) Quantity B is greater.}} \]