The ratio of two numbers is 5 : 9. If 6 is added to the greater number and 6 is subtracted from the smaller number, then greater number becomes 6 times the smaller number. What are the numbers?

10 and 18

The numbers be x : y 

 x  :  y 

 5  :  9

-6    +6

________

 1  :  6

Using cross - multiplication method, 

$\frac{5R - 6}{9R+6} = \frac{1}{6}$

30R - 9R = 6 + 36

      21R  = 42

         R = 2

Therefore, the numbers are 10, 18