The mean of five numbers is `30`. If one number is excluded, their mean becomes `28`. The excluded number is

Options

A. 28
B. 30
C. 35
D. 38

Explanation

To find the excluded number from the set of five numbers, we can follow these steps:

### Step 1: Understand the Mean Formula
The mean (average) of a set of numbers is calculated using the formula:
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

### Step 2: Calculate the Sum of the Five Numbers
We know that the mean of five numbers is 30. Therefore, we can express this as:
\[ \frac{x_1 + x_2 + x_3 + x_4 + x_5}{5} = 30 \]
Multiplying both sides by 5 gives:
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 30 \times 5 = 150 \]
Let’s denote this as Equation (1):
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 150 \]

### Step 3: Calculate the Sum of the Remaining Four Numbers
When one number (let's assume \(x_5\)) is excluded, the mean of the remaining four numbers becomes 28:
\[ \frac{x_1 + x_2 + x_3 + x_4}{4} = 28 \]
Multiplying both sides by 4 gives:
\[ x_1 + x_2 + x_3 + x_4 = 28 \times 4 = 112 \]
Let’s denote this as Equation (2):
\[ x_1 + x_2 + x_3 + x_4 = 112 \]

### Step 4: Find the Excluded Number
Now, we can find the excluded number \(x_5\) by substituting the value from Equation (2) into Equation (1):
\[ x_1 + x_2 + x_3 + x_4 + x_5 = 150 \]
Substituting Equation (2) into this gives:
\[ 112 + x_5 = 150 \]
To find \(x_5\), we can rearrange this equation:
\[ x_5 = 150 - 112 = 38 \]

### Conclusion
The excluded number is \(38\).