The average of four consecutive even numbers is one-fourth of the sum of these numbers. What is the difference between the first and last number?

Options

A. 4
B. 6
C. 2
D. 8

Answer

To solve the problem, we need to find the difference between the first and last of four consecutive even numbers, given that the average of these numbers is one-fourth of their sum.

### Step-by-Step Solution:

1. **Define the Even Numbers**:
Let's denote the first even number as \( 2x \). The four consecutive even numbers can then be expressed as:
- First number: \( 2x \)
- Second number: \( 2x + 2 \)
- Third number: \( 2x + 4 \)
- Fourth number: \( 2x + 6 \)

2. **Calculate the Sum of the Numbers**:
The sum of these four numbers is:
\[ (2x) + (2x + 2) + (2x + 4) + (2x + 6) = 8x + 12 \]

3. **Calculate the Average of the Numbers**:
The average of these four numbers is given by:
\[ \text{Average} = \frac{\text{Sum}}{4} = \frac{8x + 12}{4} = 2x + 3 \]

4. **Set Up the Equation**:
According to the problem, the average is one-fourth of the sum. Therefore, we can set up the equation:
\[ 2x + 3 = \frac{1}{4}(8x + 12) \]

5. **Simplify the Equation**:
To eliminate the fraction, multiply both sides by 4:
\[ 4(2x + 3) = 8x + 12 \]
This simplifies to:
\[ 8x + 12 = 8x + 12 \]
This equation is always true, indicating that our assumption about the numbers is correct.

6. **Find the Difference Between the First and Last Number**:
The first number is \( 2x \) and the last number is \( 2x + 6 \). The difference between the last and the first number is:
\[ (2x + 6) - (2x) = 6 \]

### Final Answer:
The difference between the first and last number is \( 6 \).

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