Let m be the amount in mL you need of the 55% mixture. The amount of alcohol in the mixture is 0.55mL.
Let a be the amount in mL you need of the pure alcohol. We need 270mL altogether, so a+m=270.
The amount of alcohol in the final solution is 0.95×270=256.5mL.
So 0.55m+a=256.5. That is, 0.55m+270-m=256.5. Therefore, 0.45m=270-256.5=13.5. m=13.5/0.45=1350/45=30mL.
So we need 30mL of the mixture on hand and 270-30=240ml of pure alcohol.