The mean of 4 numbers is 6 and the mean of other 6 numbers is 9. Find the mean of all the numbers.

Answer

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

### Step 1: Calculate the total sum of the first set of numbers.
The mean of the first 4 numbers is given as 6. The formula for the mean is:
\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]
Let the sum of the first 4 numbers be \( S_1 \). Therefore, we can write:
\[ 6 = \frac{S_1}{4} \]
Multiplying both sides by 4 gives:
\[ S_1 = 6 \times 4 = 24 \]

### Step 2: Calculate the total sum of the second set of numbers.
The mean of the second set of 6 numbers is given as 9. Let the sum of these 6 numbers be \( S_2 \). Using the mean formula:
\[ 9 = \frac{S_2}{6} \]
Multiplying both sides by 6 gives:
\[ S_2 = 9 \times 6 = 54 \]

### Step 3: Calculate the total sum of all numbers.
Now, we can find the total sum of all 10 numbers by adding \( S_1 \) and \( S_2 \):
\[ \text{Total Sum} = S_1 + S_2 = 24 + 54 = 78 \]

### Step 4: Calculate the mean of all numbers.
The total number of observations is \( 4 + 6 = 10 \). Therefore, the mean of all the numbers is:
\[ \text{Mean of all numbers} = \frac{\text{Total Sum}}{\text{Total Number of Observations}} = \frac{78}{10} = 7.8 \]

### Final Answer:
The mean of all the numbers is **7.8**.

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