The mean of `8` numbers is `7`. One more number is added and the mean is still `7`. Find the number that has been added.

Options

A. `7`
B. `8`
C. `9`
D. `6`

Answer

To solve the problem step by step, we will follow the logical reasoning based on the information given.

### Step 1: Understand the given information
We know that the mean of 8 numbers is 7.

### Step 2: Calculate the sum of the first 8 numbers
The formula for mean is:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]
Here, the mean is 7 and the number of observations is 8.

So, we can calculate the sum of the first 8 numbers:
\[ \text{Sum} = \text{Mean} \times \text{Number of observations} = 7 \times 8 = 56 \]

### Step 3: Add one more number and find the new mean
Now, when one more number is added, the total number of observations becomes:
\[ 8 + 1 = 9 \]
We are told that the mean remains 7 even after adding this new number.

### Step 4: Calculate the sum of the 9 numbers
Using the same mean formula for the new set of numbers:
\[ \text{Sum of 9 numbers} = \text{Mean} \times \text{Number of observations} = 7 \times 9 = 63 \]

### Step 5: Find the number that has been added
Let’s denote the number that has been added as \( x \). The sum of the first 8 numbers is 56, and the new sum with the added number is 63. Therefore, we can set up the equation:
\[ \text{Sum of 8 numbers} + x = \text{Sum of 9 numbers} \]
Substituting the known values:
\[ 56 + x = 63 \]

### Step 6: Solve for \( x \)
To find \( x \), we rearrange the equation:
\[ x = 63 - 56 \]
\[ x = 7 \]

### Conclusion
The number that has been added is **7**.

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