In a group of 70 people, 37 like coffee, 52 like tea and each person likes at least one of the two drinks. How many people like both coffee and tea?

Let C denote the set of people who like coffee and T denote the set of people who like tea. Then C ∪ T is the set of people who like at least one of the two drinks and C ∩ T is the set of people who like both of the two drinks. 
Then n(C) = 37, n(T) = 52 and n(C ∪ T) = 70

Method 1:
We know that 
n(C ∪ T) = n(C) + n(T) - n(C ∩ T)
   .^..  70 = 37 + 52 - n(C ∩ T)
n(C ∩ T) = 37 + 52 - 70 
n(C ∩ T) = 89 - 70
n(C ∩ T) = 19
.^..  The number of people who like both coffee and tea are 19.

Method 2:
Let n(C ∩ T) = x
From the diagram we get
n(C ∪ T) = 37 - x + x + 52 – x
n(C ∪ T) = 89 - x
        70 = 89 - x
.^..       x = 19.
.^..  The number of people who like both coffee and tea are 19.