Question

In a certain city two newspapers $A$ and $B$ are published. It is known that $25$% of the city population reads $A$ and $20\%$ of the population reads $B$. $8\%$ of the population reads both $A$ and $B$. It is known that $30$% of those who read $A$ but not $B$ look into advertisements and $40$% of those who read $B$ but not $A$ look advertisements while $50$% of those who read both $A$ and $B$ look into advertisements . What is the percentage of the population who reads an advertisement?

• A. $\frac{139}{500}$
• B. $\frac{361}{500}$
• C. $\frac{139}{1000}$
• D. $\frac{861}{1000}$

Answer 1

Answer: (C)

$\frac{139}{1000}$

Let $P\left(A\right)$ and $P\left(B\right)$ denote the precentage of city population who reda newspapers $A$ and $B$.
Then from given data ,we have
$P\left(A\right)=25\%=$$=\frac{1}{4}$,P(B)=20%=$=\frac{1}{5}.$
$P(A\cap B)=8$

$\therefore$ Percentage of those who read A but not B $P(A\cap \overline{B})=P\left(A\right)-P(A\cap B)$ [see theorem 2 & 3] $=\frac{1}{4}-\frac{2}{25}=\frac{17}{100}=17$
Similarly,$P(\overline{A}\cap B)=P\left(B\right)-P(A\cap B)$ $=\frac{1}{5}-\frac{2}{25}=\frac{3}{25}=12$
If $P\left(C\right)$ denotes the percentage of those who look into advertisement , then from the given data we obtain $P\left(C\right)30$ of
$P(A\cap \overline{B})+50ofP(A\cap B)$
$=\frac{3}{10}\times \frac{17}{100}+\frac{2}{5}\times \frac{3}{25}+\frac{1}{2}\times \frac{2}{25}$
$=\frac{51+48+40}{1000}=\frac{139}{1000}=13.9\%$.
Thus the percentage of population who read an advertisement is $13.9$%