The average of 6 consecutive integers is `15/2`. What is the average of last three integers?

Options

A. 8
B. 9
C. `17/2`
D. 7

Answer

To solve the problem step by step, we will find the average of the last three integers after determining the six consecutive integers based on the given average.

### Step-by-Step Solution:

1. **Understanding the Problem**: We know that the average of 6 consecutive integers is \( \frac{15}{2} \). We need to find the average of the last three integers.

2. **Setting Up the Integers**: Let the first integer be \( x \). Therefore, the six consecutive integers can be represented as:
- First integer: \( x \)
- Second integer: \( x + 1 \)
- Third integer: \( x + 2 \)
- Fourth integer: \( x + 3 \)
- Fifth integer: \( x + 4 \)
- Sixth integer: \( x + 5 \)

3. **Calculating the Average**: The average of these integers is given by the formula:
\[ \text{Average} = \frac{\text{Sum of integers}}{\text{Number of integers}} \]
The sum of the six integers is:
\[ x + (x + 1) + (x + 2) + (x + 3) + (x + 4) + (x + 5) = 6x + 15 \]
Therefore, the average can be expressed as:
\[ \frac{6x + 15}{6} = \frac{15}{2} \]

4. **Setting Up the Equation**: Now we set the average equal to \( \frac{15}{2} \):
\[ \frac{6x + 15}{6} = \frac{15}{2} \]

5. **Cross-Multiplying to Solve for \( x \)**:
\[ 2(6x + 15) = 15 \cdot 6 \]
Simplifying this gives:
\[ 12x + 30 = 90 \]

6. **Isolating \( x \)**:
\[ 12x = 90 - 30 \]
\[ 12x = 60 \]
\[ x = 5 \]

7. **Finding the Last Three Integers**: Now that we have \( x = 5 \), we can find the last three integers:
- Fourth integer: \( 5 + 3 = 8 \)
- Fifth integer: \( 5 + 4 = 9 \)
- Sixth integer: \( 5 + 5 = 10 \)

8. **Calculating the Average of the Last Three Integers**:
\[ \text{Average} = \frac{8 + 9 + 10}{3} = \frac{27}{3} = 9 \]

### Final Answer:
The average of the last three integers is **9**.