in a survey of 290 newspaper readers,
181 of them read the Daily Times, 142 read
the Guardian, 117 read Punch and each
reads at least one of the three papers. If 75
read the Daily Times and the Guardian, 60
read the Daily Times and Punch, and 54 read
the Guardian and Punch:
a Draw a Venn diagram to illustrate this
information.
b How many readers read
i all three papers,
if exactly two of the papers,
ili exactly one of the papers,
iv the Guardian alone?

Question

UpStudy AI · Accepted Answer

Let's analyze the survey data step by step to answer the questions.

### **Given:**
- **Total newspaper readers:** 290
- **Readers of Daily Times (D):** 181
- **Readers of Guardian (G):** 142
- **Readers of Punch (P):** 117
- **Readers of both Daily Times and Guardian (D ∩ G):** 75
- **Readers of both Daily Times and Punch (D ∩ P):** 60
- **Readers of both Guardian and Punch (G ∩ P):** 54
- **Each reader reads at least one of the three newspapers.**

### **Part a: Venn Diagram Illustration**

While I can't draw the Venn diagram here, I can describe the regions and their values based on the data:

1. **All Three Newspapers (D ∩ G ∩ P):** Let's denote this as **x**.
2. **Only Daily Times and Guardian (D ∩ G) but not Punch:** 75 - x
3. **Only Daily Times and Punch (D ∩ P) but not Guardian:** 60 - x
4. **Only Guardian and Punch (G ∩ P) but not Daily Times:** 54 - x
5. **Only Daily Times (D only):** 181 - (75 + 60 - x) = 181 - 135 + x = 46 + x
6. **Only Guardian (G only):** 142 - (75 + 54 - x) = 142 - 129 + x = 13 + x
7. **Only Punch (P only):** 117 - (60 + 54 - x) = 117 - 114 + x = 3 + x

**Total:** The sum of all these regions should equal 290.

### **Part b: Calculations**

Let's solve for **x** (the number of readers who read all three newspapers).

**Using Inclusion-Exclusion Principle:**
$$
\begin{align*}
|D \cup G \cup P| &= |D| + |G| + |P| - |D \cap G| - |D \cap P| - |G \cap P| + |D \cap G \cap P| \
290 &= 181 + 142 + 117 - 75 - 60 - 54 + x \
290 &= 440 - 189 + x \
290 &= 251 + x \
x &= 39
\end{align*}
$$
So, **39 readers** read all three newspapers.

**i. Number of Readers Who Read All Three Papers:**
- **39 readers**

**ii. Number of Readers Who Read Exactly Two Papers:**
$$
\begin{align*}
\text{Exactly two} &= (|D \cap G| - x) + (|D \cap P| - x) + (|G \cap P| - x) \
&= (75 - 39) + (60 - 39) + (54 - 39) \
&= 36 + 21 + 15 \
&= 72 \text{ readers}
\end{align*}
$$
- **72 readers**

**iii. Number of Readers Who Read Exactly One Paper:**
$$
\begin{align*}
\text{Exactly one} &= \text{Total} - \text{Exactly two} - \text{All three} \
&= 290 - 72 - 39 \
&= 179 \text{ readers}
\end{align*}
$$
- **179 readers**

**iv. Number of Readers Who Read Only the Guardian:**
From earlier:
$$
\text{Guardian only} = 13 + x = 13 + 39 = 52
$$
- **52 readers**

### **Summary of Answers:**

**b.**
i. **39** readers read all three papers.

ii. **72** readers read exactly two papers.

iii. **179** readers read exactly one paper.

iv. **52** readers read only the Guardian.
**b.** i. 39 readers read all three papers. ii. 72 readers read exactly two papers. iii. 179 readers read exactly one paper. iv. 52 readers read only the Guardian.

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