Question

In certain town, $25\%$ of the families own a phone and $15\%$ own a car; $65\%$ families own neither a phone nor a car and $2000$ families own both a car and a phone. Consider the following three statements :
(a) $5\%$ families own both a car and a phone.
(b) $35\%$ families own either a car and a phone.
(c) $40,000$ families live in the town.
Then,

• A. All (a), (b) and (c) are correct.
• B. Only (b) and (c) are correct.
• C. Only (a) and (b) are correct.
• D. Only (a) and (c) are correct.

Answer 1

The correct option is A All (a), (b) and (c) are correct.

Let $A$ be the set of people who own phone and $B$ be who own a car
$p\left(A\right)=\frac{25}{100}$ , $p\left(B\right)=\frac{15}{100}$
$p(A^{\prime }\cap B^{\prime })=\frac{65}{100}$ and $n(A\cap B)=2000$
$\Rightarrow p(A\cup B{)}^{\prime }=\frac{65}{100}$
$\Rightarrow p(A\cup B)=\frac{35}{100}$ ------(1)
$p(A\cup B)=p\left(A\right)+p\left(B\right)-p(A\cap B)$
$\Rightarrow p(A\cap B)=\frac{5}{100}$ ----(2)
if $x$ be the total no. of people then
$\frac{5}{100}x=2000$
$\therefore x=40000$ -----(3)
from (1), (2) and (3)
All (a),(b) and (c) are correct.
Hence, option A.