If the sum of five consecutive even numbers is 40 more than the average of those numbers, then find the middle number of the series?

Options

A. 30
B. 10
C. 20
D. 40

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the five consecutive even numbers
Let the first even number be \( x \). The five consecutive even numbers can be expressed as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \)
- Fourth number: \( x + 6 \)
- Fifth number: \( x + 8 \)

### Step 2: Calculate the sum of these numbers
The sum \( S \) of these five consecutive even numbers is:
\[ S = x + (x + 2) + (x + 4) + (x + 6) + (x + 8) \]
Combining like terms, we get:
\[ S = 5x + 20 \]

### Step 3: Calculate the average of these numbers
The average \( A \) of these five numbers is given by:
\[ A = \frac{S}{5} = \frac{5x + 20}{5} = x + 4 \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the sum of the five consecutive even numbers is 40 more than the average of those numbers:
\[ S = A + 40 \]
Substituting the expressions for \( S \) and \( A \):
\[ 5x + 20 = (x + 4) + 40 \]

### Step 5: Simplify the equation
Now, simplify the right side:
\[ 5x + 20 = x + 44 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 5x - x = 44 - 20 \]
\[ 4x = 24 \]

### Step 7: Solve for \( x \)
Dividing both sides by 4:
\[ x = 6 \]

### Step 8: Find the middle number
The middle number of the series (the third number) is:
\[ x + 4 = 6 + 4 = 10 \]

Thus, the middle number of the series is **10**.

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