The mean of six numbers is 21. If one number is excluded, then their mean is 19, the excluded number is ________.

Options

A. 31
B. 26
C. 28
D. 25

Answer

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

### Step 1: Calculate the total sum of the six numbers.
Given that the mean of six numbers is 21, we can use the formula for mean:

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

Let the sum of the six numbers be \( S \). Therefore, we have:

\[ 21 = \frac{S}{6} \]

To find \( S \), we can rearrange the equation:

\[ S = 21 \times 6 \]

Calculating this gives:

\[ S = 126 \]

### Step 2: Calculate the total sum of the five numbers after excluding one number.
When one number is excluded, the mean of the remaining five numbers is 19. Let the excluded number be \( x \). The sum of the remaining five numbers can be expressed as:

\[ \text{Sum of five numbers} = S - x \]

Using the mean formula again for the five numbers:

\[ 19 = \frac{S - x}{5} \]

Rearranging gives:

\[ S - x = 19 \times 5 \]

Calculating this gives:

\[ S - x = 95 \]

### Step 3: Substitute the value of \( S \) and solve for \( x \).
Now we can substitute \( S = 126 \) into the equation:

\[ 126 - x = 95 \]

To find \( x \), we rearrange this equation:

\[ x = 126 - 95 \]

Calculating this gives:

\[ x = 31 \]

### Conclusion
The excluded number is \( 31 \).

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