Question

The average (arithmetic mean) of 7, 9, and \(x\) is greater than 9.
Column A: \(x\)
Column B: 11

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

Hint:

A useful shortcut for average problems: If the average of a set of numbers is \(k\), the numbers "balance" around \(k\). Here, 7 is 2 below 9, and 9 is at 9. To make the average greater than 9, \(x\) must be more than 2 above 9 to compensate for the 7. So \(x\) must be greater than \(9+2=11\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves the concept of the arithmetic mean and requires solving an inequality to find the possible range of values for \(x\).
Step 2: Key Formula or Approach:
The formula for the arithmetic mean (average) of a set of numbers is:
\[ \text{Average} = \frac{\text{Sum of the numbers}}{\text{Count of the numbers}} \]
We are given an inequality involving the average, which we must solve for \(x\).
Step 3: Detailed Explanation:
The average of the three numbers 7, 9, and \(x\) is given by \(\frac{7 + 9 + x}{3}\).
We are told that this average is greater than 9. So, we can write the inequality:
\[ \frac{7 + 9 + x}{3}>9 \]
First, simplify the numerator:
\[ \frac{16 + x}{3}>9 \]
To solve for \(x\), multiply both sides of the inequality by 3:
\[ 16 + x>9 \times 3 \]
\[ 16 + x>27 \]
Subtract 16 from both sides of the inequality:
\[ x>27 - 16 \]
\[ x>11 \]
The result tells us that the value of \(x\) must be strictly greater than 11.
The quantity in Column A is \(x\).
The quantity in Column B is 11.
Step 4: Final Answer:
Since \(x\) must be greater than 11, the quantity in Column A is always greater than the quantity in Column B.