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,
- Only (a) and (b) are correct
- Only (a) and (c) are correct
- Only (b) and (c) are correct
- All (a), (b) and (c) are correct
The Correct Option is D
Solution and Explanation
$n(P) = 25%$
$n(C)= 15%$
$n\left(P' \cup C'\right) = 65\%$
$\Rightarrow n\left(P \cup C\right)' = 65\%$
$n\left(P\cup C\right) = 35\%$
$n\left(P\cap C\right) = n\left(P\right) + n\left(C\right)-n\left(P\cup C\right)$
$25+15-35 = 5\%$
$x \times 5\% = 2000$
$x=40,000$
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".