There are 3 consecutive odd numbers and 3 consecutive even numbers. The smallest even number is 9 more than largest odd number . If the square of average of all the 3 given odd number is 507 less than the square of the average of all the 3 given number , what is the smallest odd number.

Options

A. 11
B. 13
C. 17
D. 19

Explanation

To solve the problem step by step, let's denote the three consecutive odd numbers and the three consecutive even numbers mathematically.

1. **Define the Odd Numbers:**
Let the smallest odd number be \( x \). Then the three consecutive odd numbers can be represented as:
- First odd number: \( x \)
- Second odd number: \( x + 2 \)
- Third odd number: \( x + 4 \)

2. **Define the Even Numbers:**
Let the smallest even number be \( y \). Then the three consecutive even numbers can be represented as:
- First even number: \( y \)
- Second even number: \( y + 2 \)
- Third even number: \( y + 4 \)

3. **Set Up the Relationship:**
According to the problem, the smallest even number is 9 more than the largest odd number. The largest odd number is \( x + 4 \). Therefore, we can write the equation:
\[ y = (x + 4) + 9 \]
Simplifying this gives:
\[ y = x + 13 \quad \text{(Equation 1)} \]

4. **Calculate the Averages:**
The average of the three odd numbers is:
\[ \text{Average of odd numbers} = \frac{x + (x + 2) + (x + 4)}{3} = \frac{3x + 6}{3} = x + 2 \]
The average of the three even numbers is:
\[ \text{Average of even numbers} = \frac{y + (y + 2) + (y + 4)}{3} = \frac{3y + 6}{3} = y + 2 \]

5. **Set Up the Square Relationship:**
The problem states that the square of the average of the odd numbers is 507 less than the square of the average of the even numbers:
\[ (x + 2)^2 = (y + 2)^2 - 507 \]

6. **Expand the Squares:**
Expanding both sides gives:
\[ x^2 + 4x + 4 = y^2 + 4y + 4 - 507 \]
Simplifying this leads to:
\[ x^2 + 4x + 4 = y^2 + 4y - 503 \quad \text{(Equation 2)} \]

7. **Substitute Equation 1 into Equation 2:**
Substitute \( y = x + 13 \) into Equation 2:
\[ x^2 + 4x + 4 = (x + 13)^2 + 4(x + 13) - 503 \]
Expanding the right side:
\[ x^2 + 4x + 4 = (x^2 + 26x + 169) + (4x + 52) - 503 \]
This simplifies to:
\[ x^2 + 4x + 4 = x^2 + 30x - 282 \]

8. **Rearranging the Equation:**
Subtract \( x^2 \) from both sides:
\[ 4x + 4 = 30x - 282 \]
Rearranging gives:
\[ 282 + 4 = 30x - 4x \]
Simplifying:
\[ 286 = 26x \]
Thus,
\[ x = \frac{286}{26} = 11 \]

9. **Conclusion:**
The smallest odd number is \( x = 11 \).