If the mean of 10 observations is 20 and that of another 15 observations is 16 then the mean of 25 observations is ______

Options

A. 18
B. `18.2`
C. `17.6`
D. 17

Answer

To find the mean of 25 observations given the means of two separate groups of observations, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The mean of the first group of 10 observations is 20.
- The mean of the second group of 15 observations is 16.

2. **Calculate the Sum of the First Group**:
- The formula for mean is:
\[ \text{Mean} = \frac{\text{Sum of Observations}}{\text{Number of Observations}} \]
- For the first group:
\[ 20 = \frac{\text{Sum of 10 Observations}}{10} \]
- Rearranging gives:
\[ \text{Sum of 10 Observations} = 20 \times 10 = 200 \]

3. **Calculate the Sum of the Second Group**:
- For the second group:
\[ 16 = \frac{\text{Sum of 15 Observations}}{15} \]
- Rearranging gives:
\[ \text{Sum of 15 Observations} = 16 \times 15 = 240 \]

4. **Calculate the Total Sum of All Observations**:
- Now, add the sums of both groups:
\[ \text{Total Sum} = \text{Sum of 10 Observations} + \text{Sum of 15 Observations} = 200 + 240 = 440 \]

5. **Calculate the Mean of All 25 Observations**:
- The total number of observations is:
\[ 10 + 15 = 25 \]
- Now, using the total sum to find the mean:
\[ \text{Mean of 25 Observations} = \frac{\text{Total Sum}}{\text{Total Number of Observations}} = \frac{440}{25} \]
- Performing the division:
\[ \text{Mean of 25 Observations} = 17.6 \]

### Final Answer:
The mean of the 25 observations is **17.6**.