For an ideal solution with Raoult's law, the partial pressure of A in vapour is PA=PA0yA where yA=0.8 is the mole fraction in vapour phase. Thus PA=55×0.8=44 kNm−2.
Using the equilibrium condition yA=PtotalPA0χA where χA is mole fraction in liquid and Ptotal=PA0χA+PB0(1−χA):
0.8=55χA+15(1−χA)55χA
0.8(55χA+15−15χA)=55χA
44χA+12−12χA=55χA
12=23χA
χA=0.5217