Question

In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:
(i) how may drink tea and coffee both;
(ii) how many drink coffee but not tea.

Answer 1

Let A & B denote the sets of the persons who drink tea & coffee, respectively .
$$\begin{gathered} \mathrm{Given}: \\ n\left(A\cup B\right)=50 \\ n\left(A\right)=30 \\ n\left(A-B\right)=14 \\ \left(\mathrm{i}\right)n\left(A-B\right)=n\left(A\right)-n\left(A\cap B\right) \\ \Rightarrow 14=30-n\left(A\cap B\right) \\ \Rightarrow n\left(A\cap B\right)=16 \\ \mathrm{Thus},16\mathrm{persons}\mathrm{drink}\mathrm{tea}\mathrm{and}\mathrm{coffee}\mathrm{both}. \\ \left(\mathrm{ii}\right)n\left(A\cup B\right)=n\left(A\right)+n\left(B\right)-n\left(A\cap B\right) \\ \Rightarrow 50=30+n\left(B\right)-16 \\ \Rightarrow n\left(B\right)=36 \\ \mathrm{We}\mathrm{have}\mathrm{to}\mathrm{find}n\left(B-A\right) \\ \Rightarrow n\left(B-A\right)=n\left(B\right)-n\left(A\cap B\right) \\ \Rightarrow n\left(B-A\right)=36-16=20 \\ \mathrm{Thus},20\mathrm{persons}\mathrm{drink}\mathrm{coffee}\mathrm{but}\mathrm{not}\mathrm{tea}. \end{gathered}$$