The mean of eight numbers is 40. If one number is excluded, their mean becomes 30. The excluded number is

Options

A. 30
B. 130
C. 110
D. 138

Answer

To find the excluded number, we can follow these steps:

### Step 1: Calculate the sum of the eight numbers.
The mean of eight numbers is given as 40. The formula for the mean is:

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

From this, we can express the sum of the eight numbers (let's call it \( S_8 \)) as follows:

\[ S_8 = \text{Mean} \times \text{Number of observations} = 40 \times 8 = 320 \]

### Step 2: Calculate the sum of the seven numbers.
When one number is excluded, the mean of the remaining seven numbers becomes 30. Using the same formula for the mean, we can express the sum of the seven numbers (let's call it \( S_7 \)):

\[ S_7 = \text{Mean} \times \text{Number of observations} = 30 \times 7 = 210 \]

### Step 3: Find the excluded number.
To find the excluded number (let's call it \( x \)), we can use the relationship between the sums of the eight and seven numbers:

\[ x = S_8 - S_7 = 320 - 210 = 110 \]

Thus, the excluded number is **110**.

### Summary of Steps:
1. Calculate the sum of the eight numbers using the mean.
2. Calculate the sum of the seven numbers using the new mean.
3. Subtract the sum of the seven numbers from the sum of the eight numbers to find the excluded number.