Question:
A survey shows that $63\%$ of the Americans like cheese where as $76\%$ like apples. If $x\%$ of the Americans like both cheese and apples, then the value of $x$ is
A survey shows that $63\%$ of the Americans like cheese where as $76\%$ like apples. If $x\%$ of the Americans like both cheese and apples, then the value of $x$ is
Updated On: Jun 23, 2024
- $39 \leq x \leq 63 $
- $63$
- $39$
- $139 \geq x$
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The Correct Option is C
Solution and Explanation
Let total number of persons = 100
Let $A = \{x : x$ like cheese$\}$
$B = \{x:x$ like apples $\}$
$\therefore A \, \cap B= \{x: x$ likes both cheese and apples $\}$
Given $n(A) = 63, n(B) = 76$
$ n(A \,\cap\,B)=x$.
We know that
$ n(A\,\cup\,B)=n(A)+n(B)-n(A\,\cap\,B)$
$\therefore 100 =63+76-x$
$\Rightarrow x = 139 - 100 = 39$.
Let $A = \{x : x$ like cheese$\}$
$B = \{x:x$ like apples $\}$
$\therefore A \, \cap B= \{x: x$ likes both cheese and apples $\}$
Given $n(A) = 63, n(B) = 76$
$ n(A \,\cap\,B)=x$.
We know that
$ n(A\,\cup\,B)=n(A)+n(B)-n(A\,\cap\,B)$
$\therefore 100 =63+76-x$
$\Rightarrow x = 139 - 100 = 39$.
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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 "|".