Find the mean of 8, 12, 16, 22, 10 and 4. Find the resulting mean, if each of the observations, given above, be :
decreased by `40%`.

Answer

To solve the problem step by step, we will first find the mean of the given observations and then calculate the new mean after decreasing each observation by 40%.

### Step 1: Find the Mean of the Given Observations

1. **List the observations**: The observations given 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** using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} = \frac{72}{6} = 12 \]

### Step 2: Calculate the New Mean After Decreasing Each Observation by 40%

1. **Determine the decrease for each observation**:
- To decrease by 40%, we can calculate 40% of each observation.
- 40% can be expressed as \( \frac{2}{5} \). Thus, we will subtract \( \frac{2}{5} \) of each observation from itself.

2. **Calculate the new values for each observation**:
- For 8:
\[ 8 - \frac{2}{5} \times 8 = 8 - 3.2 = 4.8 \]
- For 12:
\[ 12 - \frac{2}{5} \times 12 = 12 - 4.8 = 7.2 \]
- For 16:
\[ 16 - \frac{2}{5} \times 16 = 16 - 6.4 = 9.6 \]
- For 22:
\[ 22 - \frac{2}{5} \times 22 = 22 - 8.8 = 13.2 \]
- For 10:
\[ 10 - \frac{2}{5} \times 10 = 10 - 4 = 6 \]
- For 4:
\[ 4 - \frac{2}{5} \times 4 = 4 - 1.6 = 2.4 \]

3. **New observations**: The new observations after decreasing by 40% are 4.8, 7.2, 9.6, 13.2, 6, and 2.4.

4. **Sum the new observations**:
\[ 4.8 + 7.2 + 9.6 + 13.2 + 6 + 2.4 = 43.2 \]

5. **Calculate the new mean**:
\[ \text{New Mean} = \frac{\text{Sum of new observations}}{\text{Number of observations}} = \frac{43.2}{6} = 7.2 \]

### Final Answer
The mean of the original observations is 12, and the new mean after decreasing each observation by 40% is 7.2.