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 the average of all the 3 given odd numbers is 507 less than the square of the average of all the 3 given even numbers, then what is the smallest odd number?

Correct option is B

Given: 

​The smallest even number is 9 more than the largest odd number. ​

​Square of the average of all 3 given odd numbers is 507 less than the square of the average of all 3 given numbers. ​

Formula Used: 

Average: $$\frac{\text{Sum of all values}}{\text{Total number of values}}$$​ 

​$$\\(a + b)^2 = a^2 + b^2 + 2ab\\$$​​

Solution: 

Let the smallest odd number be x. 

Three consecutive odd number will be = x, x+2, x+4 

Let the smallest even number be y. 

Three consecutive even number will be = y, y+2, y+4 

As per the question: 

y = (x + 4) + 9

y = x + 13      (1)

Average of three consecutive odd numbers:

$$\frac{x + (x + 2) + (x + 4)}{3} = \frac{3x + 6}{3} = x + 2$$​   

Average of three consecutive even numbers: 

​$$\frac{y + (y + 2) + (y + 4)}{3} = \frac{3y + 6}{3} = y + 2$$

As per the condition given in the question: 

$$\\(x + 2)^2 = (y + 2)^2 - 507$$

​Substitute y from equation (1)

$$\\(x + 2)^2 = (x + 15)^2 - 507$$

Expand Both sides:

$$\\x^2 + 4 + 4x = x^2 + 225 + 30x - 507 \\\text{Simplify: } \\4x - 30x = 225 - 507 - 4 \\-26x = -286 \\x = \frac{286}{26} = 11$$​​​

​