First slide

Operations on Sets

Question

In a certain town, 25% families own a phone and families 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?

Moderate

A

1 and 2
B

1 and 3
C

2 and 3
D

1,2 and 3

Solution

$\begin{array}{l} n (P) = 25 %, n (C) = 15 %, \\ n (P^{c} \cap C^{c}) = 65 %, n (P \cap C) = 2000 \end{array}$
$\begin{array}{l} n (P^{c} \cap C^{c}) = 65 % \\ \Rightarrow n (P \cup C)^{c} = 65 % \\ \Rightarrow n (P \cup C) = 35 % \\ Now n (P \cup C) = n (P) + n (C) - n (P \cap C) \\ \Rightarrow 35 = 25 + 15 - n (P \cap C) \\ \Rightarrow n (P \cap C) = 40 - 35 = 5 \\ \Rightarrow 5 % of the total = 2000 \end{array}$
$\begin{array}{l} \Rightarrow Total number of families \\ = \frac{2000 \times 100}{5} = 40000 \end{array}$
Therefore, statements 2 and 3 are correct.

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