Question

In a certain town, $25$% families own a phone and $15$% own a car, $65$ % families own neither a phone nor a car. $2000$ families own both a car and a phone. Consider the following statements in this regard:
1. $10$% families own both a car and a phone.
2. $35$% families own either a car or a phone.
3. $40,000$ families live in the town.
Which of the above statements are correct?

• A. $2$
• B. $3$
• C. $1$ and $3$
• D. $1,2$ and $3$

Answer 1

Answer: (A), (B)

$3$
$2$

$n\left(P\right)=25$%, $n\left(C\right)=15$%, $n(P^c\cap C^c)=65$%,
$n(P\cap C)=2000$
Since $n(P^c\cap C^c)=65$%
$\therefore n{(P\cup C)}^c=65$% $\therefore n(P\cup C)=35$% (2 is correct)
Now $n(P\cup C)=n\left(P\right)+n\left(C\right)-(P\cap C)$
$\therefore 35=25+15-n(P\cap C)\Rightarrow n(P\cap C)=40-35=5$
Thus $n(P\cap C)=5$%
But $n(P\cap C)=2000$
$\therefore$ $5$% of total $=2000$
$\therefore$ total number of families $=\frac{2000\times 100}{5}=40000$
$\therefore n(P\cup C)=35$%
Total number of families $=40,000$ (3 is correct)
and $n(P\cap C)=5$%