The sum of seven consecutive even numbers of a set is 532. What is the average of first four consecutive even numbers of the same set ?

Options

A. 76
B. 75
C. 74
D. 73

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the first even number
Let the first even number be \( A \).

### Step 2: Write the seven consecutive even numbers
The seven consecutive even numbers can be expressed as:
- First number: \( A \)
- Second number: \( A + 2 \)
- Third number: \( A + 4 \)
- Fourth number: \( A + 6 \)
- Fifth number: \( A + 8 \)
- Sixth number: \( A + 10 \)
- Seventh number: \( A + 12 \)

### Step 3: Set up the equation for the sum
The sum of these seven numbers is given as 532. Therefore, we can write the equation:
\[ A + (A + 2) + (A + 4) + (A + 6) + (A + 8) + (A + 10) + (A + 12) = 532 \]

### Step 4: Simplify the equation
Combining like terms, we have:
\[ 7A + (2 + 4 + 6 + 8 + 10 + 12) = 532 \]
Calculating the sum of the constants:
\[ 2 + 4 + 6 + 8 + 10 + 12 = 42 \]
Thus, the equation simplifies to:
\[ 7A + 42 = 532 \]

### Step 5: Solve for \( A \)
Subtract 42 from both sides:
\[ 7A = 532 - 42 \]
\[ 7A = 490 \]
Now, divide by 7:
\[ A = \frac{490}{7} = 70 \]

### Step 6: Find the first four consecutive even numbers
Now that we have \( A = 70 \), the first four consecutive even numbers are:
- First: \( 70 \)
- Second: \( 72 \)
- Third: \( 74 \)
- Fourth: \( 76 \)

### Step 7: Calculate the average of the first four numbers
The average is calculated as:
\[ \text{Average} = \frac{\text{Sum of the numbers}}{\text{Number of numbers}} = \frac{70 + 72 + 74 + 76}{4} \]
Calculating the sum:
\[ 70 + 72 + 74 + 76 = 292 \]
Now, divide by 4:
\[ \text{Average} = \frac{292}{4} = 73 \]

### Final Answer
The average of the first four consecutive even numbers is **73**.

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