Question

13. A certain town with a population of 100,000 has 3 newspapers: I, II, and III. The proportions of townspeople who read these papers are as follows: I: 10 percent I and II: 8 percent I and II and III: 1 percent II: 30 percent I and III: 2 percent III: 5 percent II and III: 4 percent (The list tells us, for instance, that 8000 people read newspapers I and II.) Find the number of people who read only one newspaper. How many people read at least two newspapers? If I and III are morning papers and II is an evening paper, how many people read at least one morning paper plus an evening paper? How many people do not read any newspapers? How many people read only one morning paper and one evening paper?

Answer 1

Answer:

24,000 people read only one newspaper

12,000 people read at least two newspapers

11,000 people read at least one morning paper plus an evening paper.

64,000 people do not read any newspapers.

10,000 people read only one morning paper and one evening paper.

Step-by-step explanation:

The problem states that:

- 10 percent of the population reads the newspaper I = 10,000

- 8 percent of the population reads the newspaper I and II = 8,000

- 1 percent of the population reads the newspaper I and II an III = 1,000

- 30 percent of the population reads the newspaper III = 30,000

- 2 percent of the population reads the newspaper I and III = 2,000

- 5 percent of the population reads the newspaper III = 5,000

- 4 percent of the population reads the newspaper II and III = 4,000.

Then we have to build the Venn diagram for this: I am going to call

- A the people that read the newspaper I

- B the people that read the newspaper II

- C the people that read the newspaper III

We start this from the people that read each newspaper

$A \cap B \cap C = 1,000$ - 1,000 people read each newspaper

The people that read only I and II are the number of people that read I and II minus the number of people that read I, II and III. So:

$A \cap B = 8,000 - A \cap B \cap C = 8,000 - 1,000 = 7,000$

It applies for I and III

$A \cap C = 2,000 - A \cap B \cap C = 2,000 - 1,000 = 1,000$

And for II and III

$B \cap C = 4,000 - A \cap B \cap C = 4,000 - 1,000 = 3,000$

The problem states that 10,000 people read the newspaper I, so:

$A + (A \cap B) + (A \cap C) + (A \cap B \cap C) = 10000$ where A is the number of people that read only the newspaper I. So:

A + 7,000 + 1,000 + 1,000 = 10,000

A + 9,000 = 10,000

A = 1,000

30,000 read the newspaper II, so:

$B + (A \cap B) + (B \cap C) + (A \cap B \cap C) = 30,000$

B + 7,000 + 3,000 + 1,000 = 30,000

B + 11,000 = 30,000

B = 19,000

5,000 read the newspaper III, so:

$C + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 5,000$

C + 1,000 + 1,000 + 1,000 = 5,000

C = 2,000

The answers:

a) Find the number of people who read only one newspaper

A + B + C = 1,000 + 21,000 + 2,000 = 24,000

24,000 people read only one newspaper

b) How many people read at least two newspapers

$(A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C) = 7,000 + 1,000 + 3,000 + 1,000 = 12,000$

12,000 people read at least two newspapers

If I and III are morning papers and II is an evening paper, how many people read at least one morning paper plus an evening paper?

$(A \cap B) + (B \cap C) + (A \cap B \cap C) = 7,000 + 3,000 + 1,000 = 11,000$

11,000 people read at least one morning paper plus an evening paper.

How many people do not read any newspapers?

There is 100,000 people in the town.

So, the number of people that does not read any newspaper is:

$100,000 - (A + B + C + (A \cap B) + (A \cap C) + (B \cap C) + (A \cap B \cap C)) = 100,000 - 36,000 = 64,000$

64,000 people do not read any newspapers.

How many people read only one morning paper and one evening paper?

$(A \cap B) + (B \cap C) = 7,000 + 3,000 = 10,000$

10,000 people read only one morning paper and one evening paper.