1. **State the problem:** We have 71 guests and three flavors: Mirinda (M), Novida (N), and Fanta (F). We are given information about how many guests prefer combinations of these flavors and need to represent this on a Venn diagram. 2. **Given data:** - Total guests: 71 - Guests who preferred M and F but not other flavor: 10 - Guests who preferred F and N (including those who might prefer all three): 11 - Guests who preferred M and N but not other flavor: 6 - Guests who preferred F (only F): 26 - Guests who preferred M only: 5 - Guests who preferred F only is double those who preferred N only 3. **Define variables:** - Let $x$ = number of guests who preferred N only - Let $y$ = number of guests who preferred all three flavors (M, N, F) 4. **Analyze the data:** - Guests who preferred F and N includes those who preferred all three, so $F \cap N = 11$ means $y +$ guests who preferred only F and N (no M) = 11 - Guests who preferred M and N but not other flavor = 6 means guests who preferred only M and N (no F) = 6 - Guests who preferred M and F but not other flavor = 10 means guests who preferred only M and F (no N) = 10 5. **Express the sets:** - $|M \cap N \cap F| = y$ - $|M \cap N \text{ only}| = 6$ - $|M \cap F \text{ only}| = 10$ - $|F \cap N \text{ only}| = 11 - y$ - $|M \text{ only}| = 5$ - $|N \text{ only}| = x$ - $|F \text{ only}| = 2x$ (since F only is double N only) 6. **Total guests equation:** $$5 + x + 2x + 6 + 10 + (11 - y) + y = 71$$ Simplify: $$5 + x + 2x + 6 + 10 + 11 - y + y = 71$$ $$5 + 3x + 6 + 10 + 11 = 71$$ $$32 + 3x = 71$$ $$3x = 39$$ $$x = 13$$ 7. **Find F only:** $$F \text{ only} = 2x = 2 \times 13 = 26$$ 8. **Find all three flavors (y):** From $F \cap N = 11$, and $F \cap N \text{ only} = 11 - y$, so $y$ is unknown but must be consistent. 9. **Check total with y:** Sum of all parts: $$5 (M only) + 13 (N only) + 26 (F only) + 6 (M \cap N only) + 10 (M \cap F only) + (11 - y) (F \cap N only) + y (M \cap N \cap F) = 71$$ Simplify: $$5 + 13 + 26 + 6 + 10 + 11 - y + y = 71$$ $$71 = 71$$ This holds for any $y$, so $y$ can be any value from 0 to 11. 10. **Conclusion:** - $M$ only = 5 - $N$ only = 13 - $F$ only = 26 - $M \cap N$ only = 6 - $M \cap F$ only = 10 - $F \cap N$ only = $11 - y$ - $M \cap N \cap F$ = $y$ (unknown, between 0 and 11) This information can be represented on a Venn diagram with these values. **Final answer:** $$M \text{ only} = 5, \quad N \text{ only} = 13, \quad F \text{ only} = 26,$$ $$M \cap N \text{ only} = 6, \quad M \cap F \text{ only} = 10, \quad F \cap N \text{ only} = 11 - y, \quad M \cap N \cap F = y$$ where $0 \leq y \leq 11$.