Question

In a city 20 percent of the population travel by a car, 50 percent travel by bus and 10 percent travel by both car and bus .Then persons travelling by a car or a bus is
A. 80 percent
B. 40 percent
C. 60 percent
D. 70 percent

Answer 1

Hint: Here we need to find the percentage of the population travelling by a car or a bus. For that, we will first assume the total population is any variable. Then we will use the set formula to solve the given problem further. We will substitute the values in the set formula and then we will find the value of the variable from there. Using the value of the variable, we will get the resultant value of the percentage of the population travelling by a car or a bus.

Formula used:
According to set formula: When \[A\] and \[B\] are two sets then \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\].

Complete step by step solution:
Here we need to find the percentage of the population travelling by a car or a bus.
Let the total number of populations be \[x\].
According to question:-
\[20\% \] of the population travel by a car. So we can write it as
\[n\left( {{\rm{car}}} \right) = 20\% \times x\]
On further simplification, we get
\[ \Rightarrow n\left( {{\rm{car}}} \right) = \dfrac{{20}}{{100}} \times x = \dfrac{x}{5}\]
Similarly, it is given that \[50\% \] of the population travel by a bus. So we can write it as
\[n\left( {{\rm{bus}}} \right) = 50\% \times x\]
On further simplification, we get
\[ \Rightarrow n\left( {{\rm{bus}}} \right) = \dfrac{{50}}{{100}} \times x = \dfrac{x}{2}\]
It is also given that 10 percent of the population travels by both car and bus. So we can write it as
\[ \Rightarrow n\left( {{\rm{car}} \cap {\rm{bus}}} \right) = 10\% \times x\]
On further simplification, we get
\[ \Rightarrow n\left( {{\rm{car}} \cap {\rm{bus}}} \right) = \dfrac{{10}}{{100}} \times x = \dfrac{x}{{10}}\]
Now, we will use the set formula.
According to the set formula,
When \[A\] and \[B\] are two sets then \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\].
Using this formula, we can write
\[n\left( {{\rm{car}} \cup {\rm{bus}}} \right) = n\left( {{\rm{car}}} \right) + n\left( {{\rm{bus}}} \right) - n\left( {{\rm{car}} \cap {\rm{bus}}} \right)\]
Now, we will substitute the values in the above equation, we get
\[ \Rightarrow n\left( {car \cup bus} \right) = \dfrac{x}{5} + \dfrac{x}{2} - \dfrac{x}{{10}}\]
On adding and subtracting the terms, we get
\[ \Rightarrow n\left( {car \cup bus} \right) = \dfrac{{3x}}{5}\]
Now, we will find the percentage of the population travelling by a car or a bus and this will be equal to the multiplication of the ratio of the required number of population to the total population and the number 100.
Required percentage \[ = \dfrac{{\dfrac{{3x}}{5}}}{x} \times 100\]
On further simplification, we get
\[ \Rightarrow \] Required percentage \[ = 60\% \]

Hence, the correct option is option (c).

Note:
Here \[n\left( {A \cup B} \right)\] shows the number of elements present in any one of the sets and \[n\left( {A \cap B} \right)\] shows the number of elements present in both the sets. Percentage is defined as the ratio or number expressed as a fraction of 100. We need to keep in mind that if the given values are in percentage, then the total of the value becomes 100 automatically, since the percentage represents a fraction by 100. The percentage can be represented in fraction or in decimal numbers.