1. **State the problem:** We have 100 students with the following data: - 72 eat meat (M) - 60 eat chicken (C) - 58 eat beans (B) - Number who eat both chicken and meat only (C\u2227M only) is twice the number who eat both chicken and beans only (C\u2227B only) - 8 eat only beans - 5 eat only chicken - 15 eat only meat and beans - Number who eat none equals number who eat only chicken We need to find the number who eat all three meals (C\u2227M\u2227B). 2. **Define variables:** Let: - $x$ = number who eat all three (C\u2227M\u2227B) - $a$ = number who eat chicken and meat only (C\u2227M only) - $b$ = number who eat chicken and beans only (C\u2227B only) - $c$ = number who eat meat and beans only (M\u2227B only) = 15 (given) - $d$ = number who eat only chicken = 5 (given) - $e$ = number who eat only beans = 8 (given) - $f$ = number who eat only meat (M only) - $g$ = number who eat none = number who eat only chicken = 5 (given) 3. **Use given relations:** - $a = 2b$ 4. **Use total counts for each food:** - Meat total: $72 = f + a + c + x$ - Chicken total: $60 = d + a + b + x$ - Beans total: $58 = e + b + c + x$ 5. **Substitute known values:** - Meat: $72 = f + a + 15 + x$ - Chicken: $60 = 5 + a + b + x$ - Beans: $58 = 8 + b + 15 + x$ 6. **Simplify:** - Meat: $72 = f + a + 15 + x \Rightarrow f + a + x = 57$ - Chicken: $60 = 5 + a + b + x \Rightarrow a + b + x = 55$ - Beans: $58 = 8 + b + 15 + x \Rightarrow b + x = 35$ 7. **Use $a = 2b$ in chicken equation:** $$2b + b + x = 55 \Rightarrow 3b + x = 55$$ 8. **From beans equation:** $$b + x = 35$$ 9. **Subtract beans from chicken equation:** $$(3b + x) - (b + x) = 55 - 35 \Rightarrow 2b = 20 \Rightarrow b = 10$$ 10. **Find $x$ from beans equation:** $$10 + x = 35 \Rightarrow x = 25$$ 11. **Find $a$ from $a=2b$:** $$a = 2 \times 10 = 20$$ 12. **Find $f$ from meat equation:** $$f + a + x = 57 \Rightarrow f + 20 + 25 = 57 \Rightarrow f = 12$$ 13. **Check total students:** Sum all groups: $$f + d + e + g + a + b + c + x = 12 + 5 + 8 + 5 + 20 + 10 + 15 + 25 = 100$$ Everything checks out. **Final answer:** The number of students who eat all three meals is **$25$**.