Reading Newspaper.. ?

Number Theory Level 3

Times, Mirror and Sun are three newspapers

All readers of the Times read the Sun.
Every person either reads the Sun or does not read the Mirror.
11 people read the Sun but does not read the Mirror.
8 people read either the Times or the Mirror but not both.
10 people read the Sun and either read the Mirror or do not read the Times.
14 people either read the Sun and not the Mirror or read both the Sun and Mirror.
9 people neither read the Times nor the Mirror.

Answer the following questions:

(a.) Number of people who read only sun are

(1)6

(2)5

(3)16

(4)18

(b.)Number of people who read Times and Mirror both are

(1)3

(2)2

(3)4

(4)5

(c.)Number of people who do not read any of the newspaper are

(1)0

(2)2

(3)4

(4)5

Details and Assumptions: Give you answer as $\overline{A_{a}A_{b}A_{c}}$ , where $A_{a} , A_{b} , A_{c}$ are the option number which you have chosen as the answers of question a,b and c.

The answer is 213.

1 solution

Rishabh Tripathi
May 3, 2015

First of all we have to keep in mind that 'and' has the meaning of Intersection of sets and 'or' means Union of sets.

Let the set of people reading Times be T, Sun be S and Mirror be M and U be the universal set. Here $n(A)$ represents the cardinal number of that set and A' represents compliment of that set.

We will solve the statements one by one:

Statement 1: $T \subset S$

Statement 2: $n(S \bigcup M') = n(U)$ therefore $M \subset S$

Statement 3: $n(S-M) = 11$

Statement 4: $n((T-M) \bigcup (M-T)) = 8$

From these statements we can construct the following Venn Diagram:

Statement 5: $n(S \bigcap M) \bigcup (S \bigcap T')) = 10$

It represents the following shaded area

Therefore, $y+8-x+11-x=10$

$2x-y=9...... (1)$

Statement 6: $n((S \bigcap M') \bigcup (S \bigcap T)) = 14$

It represents the following shaded area

$y+x+11-x=14$

$y=3$

By equation (1)

$x=6$

Statement 7: $n(T \bigcup M)' = 9$

Question (a) : Number of people who read only Sun $= 11-x = 11-6 =\boxed{5}$

Question (b): Number of people who read Times and Mirror both $= n(T \bigcap M) = y =\boxed{3}$

Question (c): Number of people who do not read any newspaper $= n(M \bigcup T)' -n(only S)$

$= n(M \bigcup T)' - n((S-M ) \bigcap (S-T))$

$= 9-5 = \boxed{4}$

Therefore the answer is Option 2, 1 and 3 and hence $\boxed{213}$

Reading Newspaper.. ?

The answer is 213.

1 solution

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