Question

In a group of $15$people, $7$ read French, $8$ read English while $3$ of them read none of them. How many read French and English both?

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is A

$3$

Find the number of people that read French and English both

Given: $\mathrm{n}\left(\mathrm{F}\right)=7,\mathrm{n}\left(\mathrm{E}\right)=8$ , $\mathrm{n}\left(\mathrm{N}\right)=3$ and the total number of people $=15$

Let $\mathrm{n}\left(\mathrm{F}\right)$ denote the number of people who read French, $\mathrm{n}\left(\mathrm{E}\right)$ denote the number of people who read English, $\mathrm{n}(\mathrm{E}\cap \mathrm{F})$ denote the number of people who read both English and French and $\mathrm{n}\left(\mathrm{N}\right)$ denotes the number of people who read none.

That means,

$$\begin{gathered} \mathrm{n}\left(\mathrm{F}\right)+\mathrm{n}\left(\mathrm{E}\right)–\mathrm{n}(\mathrm{E}\cap \mathrm{F})+\mathrm{n}\left(\mathrm{N}\right)=15 \\ \Rightarrow 7+8–\mathrm{n}(\mathrm{E}\cap \mathrm{F})+3=15 \\ \Rightarrow –\mathrm{n}(\mathrm{E}\cap \mathrm{F})+18=15 \\ \Rightarrow 18-15=\mathrm{n}(\mathrm{E}\cap \mathrm{F}) \\ \Rightarrow 3=\mathrm{n}(\mathrm{E}\cap \mathrm{F}) \\ \therefore \mathrm{n}(\mathrm{E}\cap \mathrm{F})=3 \end{gathered}$$

Hence, $3$ people read both French and English, so, the correct option is (A).