The mean of 16, 19, P, 21, 25, 28 is 22, find the value of P.

Answer

To find the value of \( P \) in the given set of numbers where the mean is 22, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mean Formula**:
The mean (average) of a set of numbers is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

2. **Identify the Given Values**:
We have the observations: 16, 19, \( P \), 21, 25, 28. The mean is given as 22, and the number of observations is 6.

3. **Set Up the Equation**:
Using the mean formula, we can set up the equation:
\[ 22 = \frac{16 + 19 + P + 21 + 25 + 28}{6} \]

4. **Calculate the Sum of Known Observations**:
First, calculate the sum of the known numbers:
\[ 16 + 19 + 21 + 25 + 28 = 109 \]
So, we can rewrite the equation as:
\[ 22 = \frac{109 + P}{6} \]

5. **Multiply Both Sides by 6**:
To eliminate the fraction, multiply both sides by 6:
\[ 22 \times 6 = 109 + P \]
This simplifies to:
\[ 132 = 109 + P \]

6. **Isolate \( P \)**:
Now, solve for \( P \) by subtracting 109 from both sides:
\[ P = 132 - 109 \]
This gives:
\[ P = 23 \]

7. **Conclusion**:
The value of \( P \) is 23.