Question

A survey shows that in a city, $45$% citizens like tea, whereas $65$% citizens like coffee. If $x$% like both tea and coffee, then

• A. $x=45$
• B. $x=10$
• C. $10\le x\le 45$
• D. $45\le x\le 55$

Answer 1

The correct option is C $10\le x\le 45$

Let the population of the city be $100.$
$T$ and $C$ represent the people who like tea and coffee respectively.
$n\left(T\right)=45,n\left(C\right)=65$

$n(T\cup C)=n\left(T\right)+n\left(C\right)-n(T\cap C)$
$\Rightarrow n(T\cup C)=110-n(T\cap C)\Rightarrow n(T\cap C)=110-n(T\cup C)$
We know that,
$n(T\cup C)\le 100\Rightarrow -n(T\cup C)\ge -100\Rightarrow 110-n(T\cup C)\ge 10\Rightarrow n(T\cap C)\ge 10\cdots \left(1\right)$

Now, $n(T\cap C)\le n\left(T\right);n(T\cap C)\le n\left(C\right)$
$\Rightarrow n(T\cap C)\le 45;n(T\cap C)\le 65\Rightarrow n(T\cap C)\le 45\cdots \left(2\right)$

From equations $\left(1\right)$ and $\left(2\right)$,
$10\le x\le 45$