The mean of a set of 7 numbers is 81. If a number is excluded, the mean of the remaining numbers becomes 78. What is the value of the excluded number?

Options

A. 98
B. 100
C. 101
D. 99

Answer

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

### Step 1: Calculate the total sum of the 7 numbers
The mean of a set of numbers is calculated by dividing the total sum of the numbers by the count of numbers.

Given that the mean of the 7 numbers is 81, we can find the total sum (S) of these numbers using the formula:

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

So,

\[ 81 = \frac{S}{7} \]

Multiplying both sides by 7 gives:

\[ S = 81 \times 7 = 567 \]

### Step 2: Calculate the total sum of the remaining 6 numbers
When one number is excluded, the mean of the remaining 6 numbers becomes 78. We can find the total sum of these remaining numbers (S') using the same formula:

\[ 78 = \frac{S'}{6} \]

Multiplying both sides by 6 gives:

\[ S' = 78 \times 6 = 468 \]

### Step 3: Find the excluded number
The excluded number (let's call it E) can be found by subtracting the total sum of the remaining numbers from the total sum of the original 7 numbers:

\[ E = S - S' \]

Substituting the values we found:

\[ E = 567 - 468 = 99 \]

Thus, the value of the excluded number is **99**.

### Summary of Steps:
1. Calculate the total sum of the 7 numbers using the mean.
2. Calculate the total sum of the remaining 6 numbers using the new mean.
3. Subtract the total sum of the remaining numbers from the total sum of the original numbers to find the excluded number.

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