A survey of 500 students who read various newspapers produced the following information: 280 read Kathmandu Post, 190 read Rising Nepal, 110 read Himalayan Times, 75 read Kathmandu Post and Rising…

Solution to the Newspaper Survey Problem (MTH204, TU Board 2023)

We are given a survey of 500 students who read newspapers: Kathmandu Post (KP), Rising Nepal (RN), and Himalayan Times (HT). The data includes overlaps between two newspapers and the number of students who read none of them. We need to find:

1. The number of students who read all three newspapers.
2. The number of students who read exactly two newspapers.
3. The number of students who read only one newspaper.
4. Represent the data in a Venn diagram.

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Given Data

• Total students surveyed: 500
• Students who read none of the newspapers: 55
• Therefore, students who read at least one newspaper: $500-55=445$

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Step 1: Define Variables for the Venn Diagram

Let:

• $x$ = Number of students who read all three newspapers (KP ∩ RN ∩ HT).

Then, the number of students who read exactly two newspapers can be expressed as:

• KP and RN only: $75-x$
• RN and HT only: $50-x$
• KP and HT only: $45-x$

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Step 2: Calculate Students Reading Only One Newspaper

Using the principle of inclusion-exclusion for three sets, the total number of students reading at least one newspaper is:

$$\text{Total}=(\text{KP only})+(\text{RN only})+(\text{HT only})+(\text{KP ∩ RN only})+(\text{RN ∩ HT only})+(\text{KP ∩ HT only})+(\text{All three})$$

But we can also express the total as: $$\text{KP}+\text{RN}+\text{HT}-(\text{KP ∩ RN}+\text{RN ∩ HT}+\text{KP ∩ HT})+\text{All three}=445$$

Substituting the given values: $$280+190+110-(75+50+45)+x=445$$ $$580-170+x=445$$ $$410+x=445$$ $$x=445-410=35$$

Thus, the number of students reading all three newspapers is 35.

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Step 3: Find Students Reading Exactly Two Newspapers

Using $x=35$:

• KP and RN only: $75-35=40$
• RN and HT only: $50-35=15$
• KP and HT only: $45-35=10$

Total students reading exactly two newspapers: $$40+15+10=65$$

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Step 4: Find Students Reading Only One Newspaper

Now, calculate the number of students reading only one newspaper:

1. Only KP: Total KP readers = 280 Subtract overlaps: $$280-(40+10+35)=280-85=195$$

2. Only RN: Total RN readers = 190 Subtract overlaps: $$190-(40+15+35)=190-90=100$$

3. Only HT: Total HT readers = 110 Subtract overlaps: $$110-(15+10+35)=110-60=50$$

Total students reading only one newspaper: $$195+100+50=345$$

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Step 5: Verification

Let’s verify the total: $$\text{Only KP}+\text{Only RN}+\text{Only HT}+\text{Exactly two}+\text{All three}+\text{None}=195+100+50+65+35+55=500$$ This matches the total number of students surveyed.

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Final Answers

a. Number of students reading all three newspapers:

35

b. Number of students reading exactly two newspapers:

65

c. Number of students reading only one newspaper:

345

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Venn Diagram Representation

Here is the ASCII representation of the Venn diagram:

        ______________________________
       /                                   \
      /       KP (280)                     \
     /         /   \                       \
    /        /     \                       \
   /        /       \                       \
  /________/         \_____________________\
 /   /   \           /   \                 \
/   /     \         /     \                 \
|  /       \_______/       \_________________|
| /         KP ∩ RN ∩ HT (35)               |
|/         /   \                         HT (110)
|        /     \                       /   \
|       /       \                     /     \
|      /         \___________________/       \
|     /   KP ∩ HT (10)               RN ∩ HT (15)
|    /   /   \                       /   \
|   /   /     \                     /     \
|  /   /       \___________________/       \
| /   /         KP ∩ RN (75)          RN (190)
|/   /_____________________________________/
| Only KP (195)                       Only RN (100)
|_____________________________________________|
Only HT (50)

Key Regions:

• Only KP: 195
• Only RN: 100
• Only HT: 50
• KP ∩ RN only: 40
• RN ∩ HT only: 15
• KP ∩ HT only: 10
• All three (KP ∩ RN ∩ HT): 35
• None: 55

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Summary Table of Results