If it gives a contradiction then it isn't correct!

You are using your laws wrong. Here, let me restate them:

Associative Laws

(a + b) + c = a + (b + c)

(ab)c = a(bc)

Commutative Laws

a + b = b + a

ab = ba

Distributive Laws

a(b + c) = ab + ac

(a + b)c = ac + bc

What you are trying to use is the reverse of the Distributive Law, ac + bc = (a + b)c, but you are writing ac + bc = (ac + b)c. This is obviously not correct. You have to factor the c (in your case 1/3) from both terms.

That is to say

not

-Dan