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?

To tackle this survey problem, let's use some neat mathematical reasoning and logic! Let's define: - D = Readers of Daily Times - G = Readers of Guardian - P = Readers of Punch Given: - |D| = 181 - |G| = 142 - |P| = 117 - |D ∩ G| = 75 - |D ∩ P| = 60 - |G ∩ P| = 54 Using the principle of inclusion-exclusion, we can find the number of readers who read all three papers (|D ∩ G ∩ P|): 1. From the total number of readers and using the inclusion-exclusion formula: \[ |D \cup G \cup P| = |D| + |G| + |P| - |D \cap G| - |D \cap P| - |G \cap P| + |D \cap G \cap P| \] \[ 290 = 181 + 142 + 117 - 75 - 60 - 54 + |D \cap G \cap P| \] Simplifying gives: \[ 290 = 440 + |D \cap G \cap P| - 189 \] \[ 290 = 251 + |D \cap G \cap P| \] Thus, we find: \[ |D \cap G \cap P| = 290 - 251 = 39 \] Now we can find the number of readers for each section: i. Readers of all three papers = 39. ii. For readers of exactly two papers, we use: \[ |D \cap G| - |D \cap G \cap P| = 75 - 39 = 36 \text{ (D and G only)} \] \[ |D \cap P| - |D \cap G \cap P| = 60 - 39 = 21 \text{ (D and P only)} \] \[ |G \cap P| - |D \cap G \cap P| = 54 - 39 = 15 \text{ (G and P only)} \] Thus, total readers of exactly two papers = 36 + 21 + 15 = 72. iii. For readers of exactly one paper: Using: \[ |D| - (|D \cap G| + |D \cap P| - |D \cap G \cap P|) = 181 - (75 + 60 - 39) = 181 - 96 = 85 \text{ (only D)} \] \[ |G| - (|G \cap D| + |G \cap P| - |D \cap G \cap P|) = 142 - (75 + 54 - 39) = 142 - 90 = 52 \text{ (only G)} \] \[ |P| - (|P \cap D| + |P \cap G| - |D \cap G \cap P|) = 117 - (60 + 54 - 39) = 117 - 75 = 42 \text{ (only P)} \] Thus, total readers of exactly one paper = 85 + 52 + 42 = 179. iv. For readers of the Guardian alone: The calculation is: \[ |G| - ( |D \cap G| + |G \cap P| - |D \cap G \cap P|) = 142 - (75 + 54 - 39) = 142 - 90 = 52. \] Summarizing: - Readers of all three papers: 39 - Exactly two papers: 72 - Exactly one paper: 179 - Guardian alone: 52 Now grab a pencil and doodle that Venn diagram; it’s both fun and helpful! 🎨