The ratio of two numbers is 3 : 5. If eight is added to the first, and seven to the second, then the ratio becomes 2 : 3. What will be the ratio become if six is added to each?

9 : 14

  3  :  5

  2  :  3

________

+8     +7

⇒ $\frac{3R+8}{5R+7} = \frac{2}{3}$

Using the cross - multiplication method, 

10R - 9R = (8 x 3) - (7 x 2)

⇒     R    = 10

The numbers = 3 x 10, 5 x 10 = 30, 50

When 6 is added = 36 : 56 = 9 : 14