Problem: One evening two performances was given. There was 600 persons who could see them both, but only 80 saw the first one, and 125 saw the other one. 450 persons missed both of the performances. How many persons saw both?
My solution: We get the following information: 80 people see the first performance and 125 sees the second one. 205 persons sees the performance totally. We know that 150 persons sees one performance. Therefore, 205 - 150 = 55 persons sees both performances.
Am I thinking correctly? I have no access to any solutions manual.
Yes, this is correct.
$$ |A\cap B|=|A|+|B|-|A\cup B|=80+125-(600-450)=55\;. $$