So, you know the ideal gas law is PV = nRT. In this problem, P and T are fixed. So, it is useful to move all of the constant terms to one side of the equation and move all of the non-constant terms to the other side of the equation. After some rearranging, you get:

V/n = RT/P = C

Since RT/P is constant (C), that means that:

V_{initial}/n_{initial} = C

and

V_{final}/n_{final} = C

or

V_{initial}/n_{initial} = V_{final}/n_{final}

Using this approach, you can also derive some to the other simplified ideal gas laws, like Charles' law (V_{1}/T_{1} = V_{2}/T_{2}) or Boyle's law (P_{1}V_{1} = P_{2}V_{2}).