Question

A club has 100 members, 30 of whom are lawyers. Within the club, 25 members are liars and 55 members are neither lawyers nor liars. What proportion of the lawyers are liars?

Answer 1

Let A = set of lawyers
B = set of liars and  
C = set of all members of the country club.

Let the number of members in any set Q be denoted by N(Q)
The proportions of the lawyers that are liars is equal to N (A intersection B)/N(A) . We are given that

N(S) = 100
N(A) = 30
N(B) = 25

N((A union B)') = 55

The last statement implies that N(A union B) = 100 - 55 = 45 .  
To use this information to calculate N(A intersection B) , we verify using a Venn diagram that  
N(A intersection B) = N(A) + N(B) - N(A union B)

Thus, N(A intersection B) = 30 + 25 - 45 = 10

the proportion of lawyers that are liars is = N(A intersection B)/N(A)
= 10/30 = 1/3