Question

A survey show that $63\%$ people watch news channel A whereas $76\%$ people watch news channel
B. if $x\%$ of people watch both news channels, then prove that $39$ $\le$ x $\le$ $63$

Answer 1

Given $n\left(A\right)=63$ and $n\left(B\right)=76$

$\Rightarrow n(A\cup B)=n\left(A\right)+n\left(B\right)-n(A\bigcap B)$

$\Rightarrow n(A\cup B)=63+76-n(A\bigcap B)$

Since $n(A\cup B)\le 100$

$\Rightarrow 63+76-n(A\bigcap B)\le 100$

$\Rightarrow n(A\bigcap B)\ge 39$

$\Rightarrow x\ge 39$

Generally $n(A\bigcap B)\le n\left(A\right)$ and $n(A\bigcap B)\le n\left(B\right)$

$\Rightarrow n(A\bigcap B)\le 63$ and $n(A\bigcap B)\le 76$

Combining we get ;

$\Rightarrow n(A\bigcap B)\le 63$

$\Rightarrow x\le 63$

$\Rightarrow 39\le x\le 63$