How to solve a mole fraction problem given the example

A sample of C2H2 (g) has a pressure of 8.2 kPa. After some time a portion of it reacts to form C6H6 (g). The total pressure of the mixture of gases is then 4.4 kPa. Assume the volume and the temperature do not change. Find the mole fraction of C2H2 (g) that has undergone reaction.
If you may please explain your steps in getting to the solution, that would help me understand the problem so I can solve similar ones.

Start with the ideal gas law PV=nRT

Initially P1 = 8.2 kPa and n1 = moles of C2H2 (no C6H6)
After the reaction P2 = 4.4 kPa and n2 = moles of C2H2 + moles of C6H6

Since T1 = T2 and V1 = V2 and R is R, dividing the 2 equations results in...

P2 / P1 = n2 / n1

now we know that 3 moles of C2H2 will react to form 1 mole of C6H6

if we let x = the extent of reaction or the amount of C6H6 produced then we know that the final moles of C2H2 = n1 - 3*x so...

the total moles of gas mixture in the final gas (n2) = x + n1 - 3x or simplified to n2 = n1 - 2x

substitute back into P2 / P1 = n2 / n1 and assume a basis of n1 = 1 mole (its arbitrary because you are just looking for mole fractions), then simplify the equation and solve for x

the resulting fraction of C2H2 that underwent the reaction is (initial C2H2 - final C2H2) / intial C2H2 or (n1 - (n1 - 3x))/n1 or 3x / n1