The wilson family invited 100 relatives to their christmas party. 90 guests had dutch heritage, 80 had german heritage and 75 had english heritage. what is the minimum number of people who claimed ancestry from all three heritages?

Let's put these people into the rooms to make it easier to understand.
Since we have 100 people, let it be 100 rooms for them.
 So, 90 Dutch people fill 90 rooms.
The task asks you to find the minimum common people so you have to make common rooms less in number. 
Let's empty the rooms: 
80 Germans fill 10 empty rooms and 70 Dutch people room.
There are 10 German rooms left, 20 Dutch rooms and 70 common rooms. Repeat this action and fill non common rooms with English and so forth.
What you have:
10 German to English and 20 Dutch to English. 

So, the answer is 45 (English, Dutch, German)