The mean of 7 observation is 10 and that of 3 observations is 5. What is the mean of all the 10 observations?

Options

A. 15
B. 10
C. 8.5
D. 7.5

Answer

To find the mean of all 10 observations, we can follow these steps:

### Step 1: Calculate the total sum of the first set of observations
We know that the mean of 7 observations is 10. The formula for the mean is:

\[ \text{Mean} = \frac{\text{Total Sum of Observations}}{\text{Number of Observations}} \]

Rearranging this gives us:

\[ \text{Total Sum of Observations} = \text{Mean} \times \text{Number of Observations} \]

For the first set of observations:

\[ \text{Total Sum of 7 Observations} = 10 \times 7 = 70 \]

### Step 2: Calculate the total sum of the second set of observations
We know that the mean of 3 observations is 5. Using the same formula:

\[ \text{Total Sum of 3 Observations} = 5 \times 3 = 15 \]

### Step 3: Calculate the total sum of all observations
Now, we can find the total sum of all 10 observations by adding the total sums from both sets:

\[ \text{Total Sum of All Observations} = \text{Total Sum of 7 Observations} + \text{Total Sum of 3 Observations} \]

\[ \text{Total Sum of All Observations} = 70 + 15 = 85 \]

### Step 4: Calculate the mean of all 10 observations
Now, we can find the mean of all 10 observations using the total sum we calculated:

\[ \text{Mean of All Observations} = \frac{\text{Total Sum of All Observations}}{\text{Total Number of Observations}} \]

\[ \text{Mean of All Observations} = \frac{85}{10} = 8.5 \]

### Final Answer
The mean of all 10 observations is **8.5**.

---