Question

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 like both coffee and tea?

Answer 1

Let A denote the set of the people who like tea & B denote the set of the people who like coffee.

$$\begin{gathered} \mathrm{Given}: \\ n(A\cup B)=70 \\ n\left(A\right)=52 \\ n\left(B\right)=37 \\ \mathrm{To}\mathrm{find}: \\ n(A\cap B) \\ \mathrm{We}\mathrm{know}: \\ n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\cap B) \\ \Rightarrow 70=52+37-n(A\cap B) \\ \Rightarrow n(A\cap B)=19 \\ \mathrm{Therefore},19\mathrm{people}\mathrm{like}\mathrm{both}\mathrm{tea}\&\mathrm{coffee}. \end{gathered}$$