Question

A survey shows that $61\%,46\%$, and $29\%$ of the people watched “$3$ idiots”, “Rajneeti” and “Avatar” respectively. $25\%$ of people watched exactly two of the three movies and $3\%$ watched none. What percentage of people watched all three movies ?

• A. $39\%$
• B. $11\%$
• C. $14$
• D. $7\%$

Answer 1

The correct option is D

$7\%$

Explanation for correct answer:

Step-1: Express given data.

Let $A$represent the people who watched ‘3 idiots’, $B$ represents the people who watched ‘Rajneeti’ and $C$ represent the people who watched ‘Avatar’.

So given,

$$\begin{gathered} n\left(A\right)=61 \\ n\left(B\right)=46 \\ n\left(C\right)=29 \end{gathered}$$

$n\left(A\cap B\right)+n\left(B\cap C\right)+n\left(A\cap C\right)-3\left(A\cap B\cap C\right)=25$

$n\left(A\cup B\cup C\right)=100-3=97$

$n\left(A\cap B\cap C\right)=?$

Step-2: Apply formula $\mathit{n}\left(A\cup B\cup C\right)\mathbf{=}\mathit{n}\left(A\right)\mathbf{+}\mathit{n}\left(B\right)\mathbf{+}\mathit{n}\left(C\right)\mathbf{-}\mathit{n}\left(A\cap B\right)\mathbf{-}\mathit{n}\left(B\cap C\right)\mathbf{-}\mathit{n}\left(A\cap C\right)\mathbf{+}\mathit{n}\left(A\cap B\cap C\right)$

$\Rightarrow$$n\left(A\cup B\cup C\right)=n\left(A\right)+n\left(B\right)+n\left(C\right)-\left(n\left(A\cap B\right)+n\left(B\cap C\right)+n\left(A\cap C\right)-3n\left(A\cap B\cap C\right)\right)-2n\left(A\cap B\cap C\right)$

$\Rightarrow$ $97=61+46+29-25-2n\left(A\cap B\cap C\right)$

$\Rightarrow$ $14=2n\left(A\cap B\cap C\right)$

$\Rightarrow$$n\left(A\cap B\cap C\right)=7$

Therefore, $7\%$ percentage of people watched all three movies.

Hence, correct answer is option $\left(D\right)$