Find the mean of 8, 12, 16, 22, 10 and 4. Find the resulting mean, if each of the observations, given above, be :
increased by `25%`.

Answer

To find the mean of the given set of observations and the resulting mean after increasing each observation by 25%, we can follow these steps:

### Step 1: Calculate the Mean of the Original Observations
1. **List the Observations**: The observations are 8, 12, 16, 22, 10, and 4.
2. **Sum the Observations**:
\[ 8 + 12 + 16 + 22 + 10 + 4 = 72 \]
3. **Count the Number of Observations**: There are 6 observations.
4. **Calculate the Mean**:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} = \frac{72}{6} = 12 \]

### Step 2: Increase Each Observation by 25%
1. **Calculate 25% of Each Observation**:
- For 8: \( 25\% \text{ of } 8 = \frac{25}{100} \times 8 = 2 \)
- For 12: \( 25\% \text{ of } 12 = \frac{25}{100} \times 12 = 3 \)
- For 16: \( 25\% \text{ of } 16 = \frac{25}{100} \times 16 = 4 \)
- For 22: \( 25\% \text{ of } 22 = \frac{25}{100} \times 22 = 5.5 \)
- For 10: \( 25\% \text{ of } 10 = \frac{25}{100} \times 10 = 2.5 \)
- For 4: \( 25\% \text{ of } 4 = \frac{25}{100} \times 4 = 1 \)

2. **Add the 25% Increases to Each Observation**:
- New observations:
- \( 8 + 2 = 10 \)
- \( 12 + 3 = 15 \)
- \( 16 + 4 = 20 \)
- \( 22 + 5.5 = 27.5 \)
- \( 10 + 2.5 = 12.5 \)
- \( 4 + 1 = 5 \)

3. **List the New Observations**: The new observations are 10, 15, 20, 27.5, 12.5, and 5.

### Step 3: Calculate the Mean of the New Observations
1. **Sum the New Observations**:
\[ 10 + 15 + 20 + 27.5 + 12.5 + 5 = 90 \]
2. **Count the Number of New Observations**: There are still 6 observations.
3. **Calculate the New Mean**:
\[ \text{New Mean} = \frac{\text{Sum of New Observations}}{\text{Number of Observations}} = \frac{90}{6} = 15 \]

### Final Answers
- The mean of the original observations is **12**.
- The resulting mean after increasing each observation by 25% is **15**.