Question

In the survey, it is found that 21 people read English newspaper, 26 people read Hindi newspaper, and 29 people read regional language newspaper. If 14 people read both English and Hindi newspapers; 15 people both Hindi and regional language newspapers; 12 people read both English and regional language newspaper and 8 read all types of newspapers, using Venn diagram fine:
(a) How many people were surveyed?
(b) How many people read only reginal language paper?

Answer 1

To solve the problem, let's define the sets for clarity:

- Let \( E \) be the set of people who read English newspapers.
- Let \( H \) be the set of people who read Hindi newspapers.
- Let \( R \) be the set of people who read Regional Language newspapers.

From the problem, we have the following values:

- \( |E| = 21 \) (read English newspapers)
- \( |H| = 26 \) (read Hindi newspapers)
- \( |R| = 29 \) (read Regional Language newspapers)
- \( |E \cap H| = 14 \) (read both English and Hindi newspapers)
- \( |H \cap R| = 15 \) (read both Hindi and Regional Language newspapers)
- \( |E \cap R| = 12 \) (read both English and Regional Language newspapers)
- \( |E \cap H \cap R| = 8 \) (read all three newspapers)

To find the number of people surveyed and those who read only the regional language paper, we'll apply the principle of inclusion-exclusion and then subtract the overlaps properly.

### Part (a): Find the total number of people surveyed

1. **Using the Inclusion-Exclusion Principle**:

The total number of people who read at least one type of newspaper is given by:

\[
|E \cup H \cup R| = |E| + |H| + |R| - |E \cap H| - |H \cap R| - |E \cap R| + |E \cap H \cap R|
\]

Substituting the values:

\[
|E \cup H \cup R| = 21 + 26 + 29 - 14 - 15 - 12 + 8
\]

Calculating step-by-step:

\[
= 21 + 26 + 29 = 76
\]
\[
= 76 - 14 - 15 - 12 + 8
\]
\[
= 76 - 41 + 8 = 43
\]

So, the total number of people surveyed is \( \boxed{43} \).

### Part (b): Find how many people read only the regional language paper

To find how many people read only the regional language newspaper \( |R_{\text{only}}| \):

\[
|R_{\text{only}}| = |R| - (|H \cap R| + |E \cap R| - |E \cap H \cap R|)
\]

Substituting known values:

\[
|H \cap R| = 15, \quad |E \cap R| = 12, \quad |E \cap H \cap R| = 8
\]

Calculating

\[
|R_{\text{only}}| = 29 - (15 + 12 - 8)
\]

Calculating the overlaps:

\[
= 29 - (15 + 12 - 8) = 29 - 19 = 10
\]

So, the number of people who read only the regional language newspaper is \( \boxed{10} \).