Question

Out of $32$ persons, $30$ invest in national savings certificates and $17$ invest in shares. What is the number of persons who invest in both?
A. $13$
B. $15$
C. $17$
D. $19$

Answer 1

Hint: In this question, we are to find the number of persons who invest in both national savings certificates and shares. For this, we use the formula and by substituting the given values into it, we get the required value. Here the total persons represent the union and the persons who invest in both represent the intersection

Formula Used: Set: A collection of objects in which it is possible to decide whether a given object belongs to the collection or not is said to be a set. Those objects are nothing but the elements in the set.
Sets are represented in two ways: The roaster method and the set builder form
Roaster method:
The set of vowels – \[\left\{ a,\text{ }e,\text{ }i,\text{ }o,\text{ }u \right\}\]
The set of natural numbers – \[\left\{ 1,\text{ }2,\text{ }3,\text{ }\ldots . \right\}\]
Set builder form:
The set of vowels – \[\left\{ x:x\text{ }is\text{ }a\text{ }vowel\text{ }in\text{ }English\text{ }alphabet \right\}\]
The set of natural numbers – \[\left\{ x:x\text{ }is\text{ }a\text{ }natural\text{ }number \right\}\]
Some of the important mathematical sets:
$N$ - the natural number’s set - $N=\{1,2,3...\}$
$Z$- the set of integers - $Z=\{0,\pm 1,\pm 2,\pm 3,...\}$
If two sets $A,B$ where the set $A$ is said to be the subset of $B$ i.e., $A\subseteq B$ then every element of $A$ is in the set $B$ and the set $A$ is said to be the proper subset of $B$ i.e., \[A\subset B\] then $A\subseteq B$ and $A\ne B$.
Some of the important set operations:
$\begin{align}
  & n(A\cup B)=n(A)+n(B)-n(A\cap B) \\
 & n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C) \\
\end{align}$

Complete step by step solution: Given that, there are $32$ persons. Here $30$ of them invested in national savings certificates and $17$ of them invested in shares.
Consider the set of persons who invested in national certificates as $N$ and the set of persons who invested in shares as $S$.
Then, the number of persons who invested in national savings is $n(N)=30$, the number of persons who invested in shares is $n(S)=17$, and the total number of persons are$n(N\cup S)=32$.
So, the number of persons who invested in both is represented by $n(N\cap S)$
From the theorem we have
 $n(A\cup B)=n(A)+n(B)-n(A\cap B)$
Then,
$\begin{align}
  & n(N\cup S)=n(N)+n(S)-n(N\cap S) \\
 & \Rightarrow n(N\cap S)=n(N)+n(S)-n(N\cup S) \\
\end{align}$
On substituting the given values, we get
$n(N\cap S)=30+17-32=15$
Therefore, the number person who invested in both national savings and in shares is $15$ persons.

Option ‘B’ is correct

Note: Here, we have given all the required values for calculating the number of persons who invested in both. So, we can easily calculate by substituting them in the predefined formula.