To solve this problem, we'll set up a Venn diagram using the information provided and then derive the numbers for the various categories.
A. Venn Diagram Setup
Let:
- \( A \) = readers of the Daily Times
- \( B \) = readers of the Guardian
- \( C \) = readers of the Punch
Given:
- \( |A| = 181 \)
- \( |B| = 141 \)
- \( |C| = 117 \)
- \( |A \cap B| = 75 \) (Daily Times and Guardian)
- \( |A \cap C| = 60 \) (Daily Times and Punch)
- \( |B \cap C| = 54 \) (Guardian and Punch)
Let \( x \) be the number of readers who read all three newspapers (i.e., \( |A \cap B \cap C| = x \)).
Step-by-Step Calculation
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Use Inclusion-Exclusion Principle: The total number of readers who read at least one paper can be expressed as: \[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \] Since we know that \( |A \cup B \cup C| = 200 \): \[ 200 = 181 + 141 + 117 - 75 - 60 - 54 + x \]
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Simplifying the equation: \[ 200 = 181 + 141 + 117 - 189 + x \] \[ 200 = 250 + x \] \[ x = 200 - 250 = -50 \]
It appears there's a calculation mistake. Let's reevaluate the specific values by directly calculating the intersections.
Individual Intersections
Let:
- \( |A \cap B \cap C| = x \)
- Readers who read only Daily Times and Guardian: \( |A \cap B| - x = 75 - x \)
- Readers who read only Daily Times and Punch: \( |A \cap C| - x = 60 - x \)
- Readers who read only Guardian and Punch: \( |B \cap C| - x = 54 - x \)
Now we can find the single paper readers:
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Readers who read only Daily Times: \[ |A| - (|A \cap B| + |A \cap C| - |A \cap B \cap C|) = 181 - [(75 - x) + (60 - x) + x] = 181 - 135 + x = 46 + x \]
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Readers who read only Guardian: \[ |B| - (|A \cap B| + |B \cap C| - |A \cap B \cap C|) = 141 - [(75 - x) + (54 - x) + x] = 141 - 129 + x = 12 + x \]
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Readers who read only Punch: \[ |C| - (|A \cap C| + |B \cap C| - |A \cap B \cap C|) = 117 - [(60 - x) + (54 - x) + x] = 117 - 114 + x = 3 + x \]
Sum these all together: \[ (46 + x) + (12 + x) + (3 + x) + (75 - x) + (60 - x) + (54 - x) + x = 200 \]
Combine and simplify: \[ 200 + 3x - 4x = 200 \Rightarrow -x = 0 \Rightarrow x = 0 \]
So, \( |A \cap B \cap C| = 0 \), which means no reader reads all three.
B. Answers
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i. All 3 papers: 0 readers read all three.
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ii. Exactly two of the papers:
- Only Daily Times and Guardian: \( 75 = 75 - 0 \)
- Only Daily Times and Punch: \( 60 = 60 - 0 \)
- Only Guardian and Punch: \( 54 = 54 - 0 \)
Total = 75 + 60 + 54 = 189.
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iii. Exactly one of the papers:
- Only Daily Times: \( 46 + 0 = 46 \)
- Only Guardian: \( 12 + 0 = 12 \)
- Only Punch: \( 3 + 0 = 3 \)
Total = 46 + 12 + 3 = 61.
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The Guardian alone: \[ = 12 \]
Summary:
- Readers of all three papers: 0
- Readers of exactly two papers: 189
- Readers of exactly one paper: 61
- Readers of the Guardian alone: 12
Venn Diagram:
You can illustrate with three overlapping circles labeled A (Daily Times), B (Guardian), and C (Punch) showing:
- In the overlap of A and B but not C, put 75.
- In the overlap of A and C but not B, put 60.
- In the overlap of B and C but not A, put 54.
- The outer sections would contain the counts of those reading only one paper.