Question

In a city three daily newspaper x, y and z are published. 40% of the people of the city read x, 50% of the people read y, 30% read z, 20% read both x and y, 15% read both x and z, 10% read y and z and 24% read all of the three papers.
Can this be true ? Yes=9 No=6

Answer 1

Let the number of people in the city be $100x$ (where $x\ne 0$, a constant).

Furher assume that $A,B$ and $C$ be respectively the sets of people of that city who reads the newspaper $x,y$ and $z$.

And let $S$ denote the set of people in that city.

Then we have,

$\left|S\right|=100x,\left|A\right|=40x,\left|B\right|=50x,$$\left|C\right|=30x,|A\cap B|=20x,|B\cap C|=15x,$$|C\cap A|=10x$

Assume $\left|{(A\cup B\cap C)}^{\prime }\right|=0$

(i.e., everyone read atleast one paper)

And assume that $|A\cap B\cap C|=P$

(i.e., number of people who read all the three newspapers is $P$).

Now, we know that

$\left|{(A\cup B\cup C)}^{\prime }\right|=\left|S\right|\left|(A\cup B\cup C)\right|=\left|S\right|-|A\cup B\cup C|$

Hence we have,

$\left|S\right|-|A\cup B\cup C|=0\Rightarrow \left|S\right|=|A\cup B\cup C|$

$\Rightarrow \left|A\right|+\left|B\right|+\left|C\right|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|=100x$

$\Rightarrow 40x+50x+30x-20x-15x-10x+P=100x$

$\Rightarrow P=25x$

i.e., people who read all newspaper is $25$%.