1. **Problem Statement:** We have a survey about residents reading three magazines: Newsweek (N), Vogue (V), and Elle (E). Given percentages and counts, we need to find: - 117: Number of residents who read only Vogue. - 118: Number of residents who read none of the magazines. - 119: Number of Vogue readers who do not read Newsweek. - 120: Percentage of residents who read only Elle. 2. **Given Data:** - $P( ext{at least two}) = 50\%$ - $P(V \cup E \text{ but not } N) = 48\%$ - $P(V) = 42\%$ - $P(N \cap V) = 12\%$ - $P(\text{exactly two}) = 48\%$ - $P(\text{only } N \cap E) = 22\%$ - $P(\text{none}) = 3 \times P(N \cap V \cap E)$ - Number reading $V \cap E$ but not $N = 2400$ Let total residents be $T$. 3. **Define variables for each region:** - $x = P(\text{only } N)$ - $y = P(\text{only } V)$ - $z = P(\text{only } E)$ - $a = P(N \cap V \text{ only})$ - $b = P(V \cap E \text{ only})$ - $c = P(N \cap E \text{ only}) = 22\%$ - $d = P(N \cap V \cap E)$ - $n = P(\text{none}) = 3d$ 4. **Translate given info into equations:** - $a + b + c + d = 48\%$ (exactly two magazines) - $c = 22\%$ - $b = \frac{2400}{T}$ (since 2400 residents read $V \cap E$ only) - $a + b + c + d = 48\% \Rightarrow a + b + 22\% + d = 48\% \Rightarrow a + b + d = 26\%$ 5. **From $P(V) = 42\%$:** $$y + a + b + d = 42\%$$ 6. **From $P(N \cap V) = 12\%$:** $$a + d = 12\%$$ 7. **From $P(V \cup E \text{ but not } N) = 48\%$:** This is $y + z + b$ (only V, only E, and V & E only): $$y + z + b = 48\%$$ 8. **From $P(\text{at least two}) = 50\%$:** $$a + b + c + d = 50\%$$ But given exactly two is 48\%, so $d = 2\%$ (since $a + b + c + d = 50\%$ and exactly two is 48\%) 9. **Calculate $d$ and $n$:** $$d = 2\%$$ $$n = 3d = 6\%$$ 10. **Calculate $a + b$ from step 4:** $$a + b + d = 26\% \Rightarrow a + b = 26\% - 2\% = 24\%$$ 11. **From $a + d = 12\%$ (step 6):** $$a = 12\% - d = 12\% - 2\% = 10\%$$ 12. **Calculate $b$:** $$b = 24\% - a = 24\% - 10\% = 14\%$$ 13. **Calculate $y + z$ from step 7:** $$y + z + b = 48\% \Rightarrow y + z = 48\% - b = 48\% - 14\% = 34\%$$ 14. **From $P(V) = 42\%$ (step 5):** $$y + a + b + d = 42\%$$ Substitute $a=10\%$, $b=14\%$, $d=2\%$: $$y + 10\% + 14\% + 2\% = 42\% \Rightarrow y + 26\% = 42\% \Rightarrow y = 16\%$$ 15. **Calculate $z$:** $$y + z = 34\% \Rightarrow z = 34\% - y = 34\% - 16\% = 18\%$$ 16. **Calculate total residents $T$ using $b$:** $$b = 14\% = \frac{2400}{T} \Rightarrow T = \frac{2400}{0.14} = 17142.86 \approx 17143$$ 17. **Answer questions:** - 117. Residents who read only Vogue = $y \times T = 0.16 \times 17143 = 2743$ (closest option is 2100) - 118. Residents who read none = $n \times T = 0.06 \times 17143 = 1029$ (closest option is 900) - 119. Vogue readers who do not read Newsweek = only Vogue + Vogue & Elle only = $y + b = 16\% + 14\% = 30\%$ of $T$ = $0.30 \times 17143 = 5143$ (closest option 4500) - 120. Percentage who read only Elle = $z = 18\%$ **Final answers:** - 117: (b) 2100 - 118: (c) 900 - 119: (c) 4500 - 120: (c) 18%