If the arithmetic mean of 9, 11, 4, 7, p and 6 is 7, then the value of p

Options

A. 10
B. 5
C. 15
D. 20

Answer

To find the value of \( p \) such that the arithmetic mean of the numbers 9, 11, 4, 7, \( p \), and 6 is equal to 7, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Arithmetic Mean**:
The arithmetic mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the total count of numbers. The formula for the arithmetic mean \( \bar{x} \) is given by:
\[ \bar{x} = \frac{\text{Sum of all terms}}{n} \]
where \( n \) is the number of terms.

2. **Identifying the Terms**:
We have the numbers: 9, 11, 4, 7, \( p \), and 6. Thus, the total number of terms \( n \) is 6.

3. **Setting Up the Equation**:
According to the problem, the arithmetic mean is given as 7. Therefore, we can set up the equation:
\[ 7 = \frac{9 + 11 + 4 + 7 + p + 6}{6} \]

4. **Calculating the Sum of Known Terms**:
First, we calculate the sum of the known numbers:
\[ 9 + 11 + 4 + 7 + 6 = 37 \]
So, we can rewrite the equation as:
\[ 7 = \frac{37 + p}{6} \]

5. **Cross-Multiplying to Eliminate the Denominator**:
To eliminate the fraction, we cross-multiply:
\[ 7 \times 6 = 37 + p \]
This simplifies to:
\[ 42 = 37 + p \]

6. **Solving for \( p \)**:
Now, we isolate \( p \) by subtracting 37 from both sides:
\[ p = 42 - 37 \]
Thus, we find:
\[ p = 5 \]

### Final Answer:
The value of \( p \) is \( 5 \).

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