If the average of 7 consecutive even number is 44. Find the difference between first and last number

Answer

To solve the problem of finding the difference between the first and last of 7 consecutive even numbers given that their average is 44, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Concept of Consecutive Even Numbers**:
- Consecutive even numbers can be represented as:
- First number: \( x \)
- Second number: \( x + 2 \)
- Third number: \( x + 4 \)
- Fourth number: \( x + 6 \)
- Fifth number: \( x + 8 \)
- Sixth number: \( x + 10 \)
- Seventh number: \( x + 12 \)

2. **Calculate the Average**:
- The average of these 7 numbers is given as 44.
- The average can be calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of all numbers}}{\text{Total numbers}} \]
- Therefore, we can write:
\[ 44 = \frac{x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) + (x + 12)}{7} \]

3. **Simplify the Equation**:
- Combine the terms in the numerator:
\[ 44 = \frac{7x + (0 + 2 + 4 + 6 + 8 + 10 + 12)}{7} \]
- The sum of the constants (2, 4, 6, 8, 10, 12) is:
\[ 2 + 4 + 6 + 8 + 10 + 12 = 42 \]
- Thus, the equation becomes:
\[ 44 = \frac{7x + 42}{7} \]

4. **Multiply Both Sides by 7**:
- To eliminate the fraction, multiply both sides by 7:
\[ 44 \times 7 = 7x + 42 \]
- This simplifies to:
\[ 308 = 7x + 42 \]

5. **Isolate \( x \)**:
- Subtract 42 from both sides:
\[ 308 - 42 = 7x \]
- This gives:
\[ 266 = 7x \]
- Now, divide both sides by 7:
\[ x = \frac{266}{7} = 38 \]

6. **Identify the First and Last Numbers**:
- The first number is \( x = 38 \).
- The last number (7th number) is:
\[ x + 12 = 38 + 12 = 50 \]

7. **Calculate the Difference**:
- The difference between the last and first number is:
\[ 50 - 38 = 12 \]

### Final Answer:
The difference between the first and last number is **12**.