Question

In a beauty contest, half the number of judges voted for Miss A, 2/3 of them voted for
Miss B,10 voted for both and 6 did not vote for either of them. Find how many
judges in all, were present?

Answer 1

Let the number of judges be $x$, then from the given information we have $n(A) = \dfrac{x}{2}$,  $n(B) = \dfrac{2x}{3}$,  $n(A \cap B) = 10$ and  $n(A' \cap B') = 6$ 

 $x = n(A) + n(B) - n(A\cap B) +n(A'\cap B')$ 
 $x = \dfrac{x}{2}+ \dfrac{2x}{3}- 10+6$ 

It is easy to solve $x$.