1. **Problem Statement:** An agent sells three magazines: Mwananchi, Nipashe, and Daily News. Given the number of customers buying each and the overlaps, we want to represent this information using a Venn diagram and assign letters to each region. 2. **Given Data:** - $|Mwananchi| = 70$ - $|Nipashe| = 60$ - $|DailyNews| = 50$ - $|Mwananchi \cap Nipashe| = 17$ - $|DailyNews \cap Nipashe| = 15$ - $|DailyNews \cap Mwananchi| = 16$ 3. **Important Note:** The problem does not specify how many customers buy all three magazines, so we will denote that number as $x$. 4. **Assign letters to each region in the Venn diagram:** - Let $a$ be the number of customers who buy only Mwananchi. - Let $b$ be the number who buy only Nipashe. - Let $c$ be the number who buy only Daily News. - Let $d$ be the number who buy Mwananchi and Nipashe but not Daily News. - Let $e$ be the number who buy Nipashe and Daily News but not Mwananchi. - Let $f$ be the number who buy Daily News and Mwananchi but not Nipashe. - Let $x$ be the number who buy all three magazines. 5. **Write equations based on the given data:** - For Mwananchi: $a + d + f + x = 70$ - For Nipashe: $b + d + e + x = 60$ - For Daily News: $c + e + f + x = 50$ - For intersections of two sets: - $d + x = 17$ (Mwananchi and Nipashe) - $e + x = 15$ (Nipashe and Daily News) - $f + x = 16$ (Daily News and Mwananchi) 6. **Express $d, e, f$ in terms of $x$:** - $d = 17 - x$ - $e = 15 - x$ - $f = 16 - x$ 7. **Substitute into the single sets equations:** - $a + (17 - x) + (16 - x) + x = 70 \Rightarrow a + 33 - x = 70 \Rightarrow a = 37 + x$ - $b + (17 - x) + (15 - x) + x = 60 \Rightarrow b + 32 - x = 60 \Rightarrow b = 28 + x$ - $c + (15 - x) + (16 - x) + x = 50 \Rightarrow c + 31 - x = 50 \Rightarrow c = 19 + x$ 8. **Summary of regions:** - Only Mwananchi: $a = 37 + x$ - Only Nipashe: $b = 28 + x$ - Only Daily News: $c = 19 + x$ - Mwananchi and Nipashe only: $d = 17 - x$ - Nipashe and Daily News only: $e = 15 - x$ - Daily News and Mwananchi only: $f = 16 - x$ - All three: $x$ 9. **Interpretation:** Without the exact value of $x$, the number of customers who buy all three magazines, we cannot find exact values for each region, but the relationships above fully describe the Venn diagram. 10. **Venn Diagram Representation:** - Draw three intersecting circles labeled Mwananchi, Nipashe, and Daily News. - Label each region with the corresponding letter and expression above. This completes the problem of representing the data with a Venn diagram and assigning letters to each region.