Eight consecutive numbers are given. If the average of the two numbers that appear in the middle is 6, then the sum of the eight given numbers is:

Options

A. 54
B. 664
C. 36
D. 48

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Understand the Problem
We are given that there are eight consecutive numbers, and the average of the two middle numbers is 6. We need to find the sum of all eight numbers.

**Hint:** Remember that consecutive numbers can be represented in terms of a starting number.

### Step 2: Represent the Consecutive Numbers
Let the first number be \( x \). Then, the eight consecutive numbers can be represented as:
- \( x, x+1, x+2, x+3, x+4, x+5, x+6, x+7 \)

**Hint:** The middle numbers in a set of eight numbers are the 4th and 5th numbers.

### Step 3: Identify the Middle Numbers
The two middle numbers in this case are:
- 4th number: \( x + 3 \)
- 5th number: \( x + 4 \)

**Hint:** The average of two numbers is found by adding them together and dividing by 2.

### Step 4: Set Up the Equation for the Average
The average of the two middle numbers is given as 6. Therefore, we can set up the equation:
\[ \frac{(x + 3) + (x + 4)}{2} = 6 \]

**Hint:** Simplify the equation to find the value of \( x \).

### Step 5: Solve the Equation
Now, simplify the equation:
\[ \frac{2x + 7}{2} = 6 \]
Multiply both sides by 2:
\[ 2x + 7 = 12 \]
Subtract 7 from both sides:
\[ 2x = 5 \]
Divide by 2:
\[ x = 2.5 \]

**Hint:** Now that you have \( x \), you can find all eight consecutive numbers.

### Step 6: Find the Eight Consecutive Numbers
Using \( x = 2.5 \), the eight consecutive numbers are:
- \( 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5 \)

**Hint:** To find the sum, you can either add them directly or use a formula.

### Step 7: Calculate the Sum of the Eight Numbers
To find the sum:
\[ 2.5 + 3.5 + 4.5 + 5.5 + 6.5 + 7.5 + 8.5 + 9.5 \]
This can be simplified as:
\[ (2.5 + 9.5) + (3.5 + 8.5) + (4.5 + 7.5) + (5.5 + 6.5) = 12 + 12 + 12 + 12 = 48 \]

**Hint:** Alternatively, you can use the formula for the sum of an arithmetic series.

### Final Answer
The sum of the eight consecutive numbers is \( 48 \).