Question

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the numbers of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ n\left(H\right)=25 \\ n\left(T\right)=26 \\ n\left(I\right)=26 \\ n\left(H\cap I\right)=9 \\ n\left(H\cap T\right)=11 \\ n\left(T\cap I\right)=8 \\ n\left(H\cap T\cap I\right)=3 \\ \left(\mathrm{i}\right)\mathrm{We}\mathrm{know}: \\ n\left(H\cup T\cup I\right)=n\left(H\right)+n\left(T\right)+n\left(I\right)-n\left(H\cap T\right)-n\left(T\cap I\right)-n\left(H\cap I\right)+n\left(H\cap T\cap I\right) \\ \Rightarrow n\left(H\cup T\cup I\right)=25+26+26-11-8-9+3=52 \\ \mathrm{Thus},52\mathrm{people}\mathrm{can}\mathrm{read}\mathrm{at}\mathrm{least}\mathrm{one}\mathrm{of}\mathrm{the}\mathrm{newspapers}. \\ \left(\mathrm{ii}\right)\mathrm{Now},\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{calculate}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{people}\mathrm{who}\mathrm{read}\mathrm{exactly}\mathrm{one}\mathrm{newspaper}. \\ \mathrm{We}\mathrm{have}: \\ n\left(H\right)+n\left(T\right)+n\left(I\right)-2n\left(H\cap T\right)-2n\left(T\cap I\right)-2n\left(H\cap I\right)+3n\left(H\cap T\cap I\right) \\ =25+26+26-22-16-18+9 \\ =30 \\ \mathrm{Thus},30\mathrm{people}\mathrm{can}\mathrm{read}\mathrm{exactly}\mathrm{one}\mathrm{newspaper}. \end{gathered}$$