C = Charlie's age; G = Gwen's age.

Next year, C = (1/2)(G) ("of" means "multiply" in a word problem, so "half of Gwen's age" means multiply Gwen's age by 1/2).

Also next year, G = C+6.

Multiply both sides of the first equation by 2, to get rid of the 1/2. Then instead of C = (1/2)(G), we have 2C = G.

Now we have two equations in terms of G: G = C+6 and G =2C. Since both C+6 and 2C are equal to G, they must be equal to each other. C+6 = 2C. Subtract C from both sides: 6 = C.

So

**, Charlie will be 6 and Gwen will be 12 (6 years older than Charlie). This year, Gwen is 11.***next year*