The gases X & Y react with each other to produce gas Z according to the equation: 2X + Y ->2Z. 100cm3 of gas X was mixed with 10cm3 of gas Y, in a freely moving sealed gas syringe The reaction went to completion. What was the final volume of the syringe?

What's the key information we need to extract from the question:
reaction went to completionmolar ratios of X, Y and ZGas Y is the limiting factor & Gas X is in excess
If gas Y is the limiting factor, and the reaction went to completion, then we know all of gas Y was used up in the reaction. Let's write this under the equation. Now using the molar ratios, we know that is 10cm3 of Y reacts with 20cm3 of X to make 20cm3 of Z (which we work out from the equation).
Because the reaction went to completion, we know that 20cm3 of X and 10cm3 of Y have been used up, making 20cm3 of Z. Remember though, that gas X was in excess, so only 20cm3 of the 100cm3 we started with has reacted. Therefore, the final volume of gas in the syringe is going to be 80cm3 of X, 0cm3 of Y and 20cm3 of Z, leaving a total of 100cm3