Question

In a certain town, $25\%$ families own a cell phone, $15\%$ families own a scooter and $65\%$ families own neither a cell phone nor a scooter. If $1500$ families own both a cell phone and a scooter, then the total number of families in the town is
A. 10000
B. 20000
C. 30000
D. None of these

Answer 1

Hint: The problem that we are given can be solved using the concept of set theories. At the first step we can assume the total number of families to be $x$ . Upon assuming that we get the number of families who own cell phones, scooter, both cell phone and scooter and also the number of families who own neither of them. Using the concept of set theories, we form an equation with the said parameters and find the value of $x$ which gives us the total number of families.

Complete step-by-step answer:
Let’s say the number of families that own a cell phone is $A$ , number of families that own a scooter $B$ , the number of families that own neither of them is \[C\] and the total number of families in that town is $x$.
According to set theory we can equate an equation as
$A+B-\left( A\cap B \right)+C=x$
Here, $A=\dfrac{25x}{100}$ , $B=\dfrac{15x}{100}$ , $C=\dfrac{65x}{100}$ and $\left( A\cap B \right)=1500$
Substituting the above values in the formed equation we get
$\Rightarrow \dfrac{25x}{100}+\dfrac{15x}{100}-1500+\dfrac{65x}{100}=x$
Subtracting $x$ from both the sides of the above equation we get
$\Rightarrow \dfrac{25x}{100}+\dfrac{15x}{100}-1500+\dfrac{65x}{100}-x=0$
Adding $1500$ to both the sides of the above equation we get
$\Rightarrow \dfrac{25x}{100}+\dfrac{15x}{100}+\dfrac{65x}{100}-x=1500$
Further simplifying the above equation, we get
$\Rightarrow \dfrac{105x}{100}-x=1500$
$\Rightarrow \dfrac{105x-100x}{100}=1500$
$\Rightarrow \dfrac{5x}{100}=1500$
Multiplying both the sides of the above equation to $100$ we get
$\Rightarrow 5x=150000$
Dividing both the sides of the above equation by $5$ we get
$\Rightarrow x=30000$
Therefore, we conclude that the total number of families in that town is $30000$ .

So, the correct answer is “Option c”.

Note: While solving problems using the set theories, we must keep in mind that the equations are formed correctly so that it can give the exact solution of the problem. Also, we must do the calculations properly to get an error free solution of the problem.