The sum of three numbers is 68. If the ratio between first and second be 2 : 3 and that between second and third be 5 : 3, then the second number?

Let the 3 numbers are a, b, c.
Then, a/b = 2/3, b/c = 5/3
⇒ a/b = (2 x 5) / (3 x 5) = 10/15
and b/c = (5 x 3)/(3 x 3) = 15/9
⇒ a : b : c = 10 : 15 : 9
Let the number be 10y, 15y and 9y.
Then, 10y + 15y + 9y = 68
⇒ 34y = 68
⇒ y = 2
∴ Second number = 15y
= 15 x 2
= 30.