The average of 8 consecutive 8 integers is `23/2` . What is the average of first three integers?

Options

A. 9
B. `19/2`
C. 8
D. 10

Answer

To solve the problem step by step, we need to find the average of the first three integers from a set of 8 consecutive integers whose average is \( \frac{23}{2} \).

### Step 1: Understand the average of consecutive integers
The average of a set of numbers is calculated by dividing the sum of those numbers by the count of numbers. For 8 consecutive integers, if we denote the first integer as \( x \), the integers can be represented as:
- \( x, x+1, x+2, x+3, x+4, x+5, x+6, x+7 \)

### Step 2: Calculate the sum of the integers
The sum of these integers can be calculated as:
\[ \text{Sum} = x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6) + (x+7) \]
This simplifies to:
\[ \text{Sum} = 8x + (0 + 1 + 2 + 3 + 4 + 5 + 6 + 7) = 8x + 28 \]

### Step 3: Set up the equation for the average
Given that the average of these integers is \( \frac{23}{2} \), we can set up the equation:
\[ \frac{8x + 28}{8} = \frac{23}{2} \]

### Step 4: Solve for \( x \)
To eliminate the fraction, multiply both sides by 8:
\[ 8x + 28 = 8 \cdot \frac{23}{2} \]
Calculating the right side:
\[ 8 \cdot \frac{23}{2} = 4 \cdot 23 = 92 \]
Now, our equation becomes:
\[ 8x + 28 = 92 \]
Subtract 28 from both sides:
\[ 8x = 92 - 28 = 64 \]
Now divide by 8:
\[ x = \frac{64}{8} = 8 \]

### Step 5: Identify the first three integers
The first three integers are:
- \( x = 8 \)
- \( x + 1 = 9 \)
- \( x + 2 = 10 \)

### Step 6: Calculate the average of the first three integers
The average of the first three integers is:
\[ \text{Average} = \frac{8 + 9 + 10}{3} = \frac{27}{3} = 9 \]

### Final Answer
The average of the first three integers is \( 9 \).

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