In a certain town, 25% of families own a phone and 15% own a car; 65% families own neither a phone n

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1.Set Theory

hard

In a certain town, $25\%$ of the families own a phone and $15\%$ own a car; $65\%$ families own neither a phone nor a car and $2,000$ 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 or a phone
$(C)\,\,\,40,000$ families live in the town
Then,

A

Only $(A)$ and $(C)$ are correct.

B

Only $(B)$ and $(C)$ are correct

C

All $(A),$ $(B)$ and $(C)$ are correct.

D

Only $(A)$ and $(B)$ are correct

(JEE MAIN-2015)

Solution

$n(P)\, = \,25\,\% $

$n(C)\, = \,15\,\% $

$n(P'\, \cup \,C')\, = \,65\,\% $

$ \Rightarrow n(P \cup \,C')\, = \,65\,\% $

$n(P \cup \,C)\, = \,35\,\% $

$n(P \cap \,C)\, = \,n(P)\, + n(C)\, – \,n(P \cup \,C)$

$25\, + \,15\, – 35\, = \,5\% $

$x\, \times \,5\,\% \, = \,2000$

$x\, = \,40,000$

Standard 11

Mathematics

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