There are six consecutive positive even numbers. The sum of the average of the first three consecutive numbers and that of the next three consecutive numbers is 62. What is the sum of the first three consecutive numbers?

Options

A. 84
B. 94
C. 86
D. 92

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the six consecutive positive even numbers
Let the first even number be \( X \). Therefore, the six consecutive positive 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: \( X + 10 \)

### Step 2: Calculate the average of the first three numbers
The first three numbers are \( X \), \( X + 2 \), and \( X + 4 \). To find the average, we add these three numbers and divide by 3:
\[ \text{Average of first three} = \frac{X + (X + 2) + (X + 4)}{3} = \frac{3X + 6}{3} = X + 2 \]

### Step 3: Calculate the average of the next three numbers
The next three numbers are \( X + 6 \), \( X + 8 \), and \( X + 10 \). Similarly, we calculate their average:
\[ \text{Average of next three} = \frac{(X + 6) + (X + 8) + (X + 10)}{3} = \frac{3X + 24}{3} = X + 8 \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the sum of the averages of the first three and the next three numbers is 62:
\[ (X + 2) + (X + 8) = 62 \]
This simplifies to:
\[ 2X + 10 = 62 \]

### Step 5: Solve for \( X \)
Subtract 10 from both sides:
\[ 2X = 62 - 10 \]
\[ 2X = 52 \]
Now, divide by 2:
\[ X = 26 \]

### Step 6: Find the sum of the first three consecutive numbers
The first three consecutive even numbers are:
- \( X = 26 \)
- \( X + 2 = 28 \)
- \( X + 4 = 30 \)

Now, we calculate their sum:
\[ \text{Sum of first three} = 26 + 28 + 30 = 84 \]

### Final Answer
The sum of the first three consecutive positive even numbers is **84**.

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