In a certain town 25% families own a phone and 15% own a car, 65% families own neither a phone nor a

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1.Set Theory

hard

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

$1$ and $2$

$1$ and $3$

$2$ and $3$

$1, 2$ and $3$

$n\,({P^c} \cap {C^c}) = 65\% ,\,\,n(P \cap C) = 2000$

Since, $n\,({P^c} \cap {C^c}) = 65\% $

$\therefore$ $n\,{(P \cup C)^c} = 65\% $ and $n(P \cup C) = 35\% $

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

$35 = 25 + 15 – n(P \cap C)$

$\therefore$ $n(P \cap C) = 40 – 35 = 5$. Thus $n\,(P \cap C) = 5\% $

But $n\,(P \cap C) = 2000$

$\therefore$ Total number of families $ = \frac{{2000 \times 100}}{5} = 40,000$

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

and total number of families = $40,000$

and $n(P \cap C) = 5\% $. $\therefore$ $(2)$ and $(3)$ are correct.

Standard 11

Mathematics

easy

easy

medium

easy

medium