In a town with population of 4000, 3000 people are egg eaters, 2000 meat eaters and 1500 eat both eggs and meat. How many are pure vegetarians?

### Concept & Logic This is a fundamental Venn diagram problem. "Pure vegetarians" implies people who eat *neither* eggs nor meat. We must find the total number of non-vegetarians (the union of egg and meat eaters) and subtract it from the total population. ### Step-by-Step Solution * **Given:** Total Population = 4000 Egg eaters, $n(E) = 3000$ Meat eaters, $n(M) = 2000$ Both, $n(E \cap M) = 1500$ * **Calculation:** * Find the total number of people who eat eggs, meat, or both using the inclusion-exclusion principle: $$n(E \cup M) = n(E) + n(M) - n(E \cap M)$$ $$n(E \cup M) = 3000 + 2000 - 1500$$ $$n(E \cup M) = 5000 - 1500 = 3500$$ * This means 3500 people in the town are non-vegetarians (they consume animal products). * To find the pure vegetarians, subtract this number from the total town population: $$\text{Vegetarians} = \text{Total Population} - n(E \cup M)$$ $$\text{Vegetarians} = 4000 - 3500 = 500$$ ### Exam Strategy & Shortcut Add the single categories together: $3000 + 2000 = 5000$. Since the town only has 4000 people, the "excess" must represent the overlap (people counted twice). Wait, the overlap is given as 1500. So, distinct eaters = $5000 - 1500 = 3500$. Remaining population = $4000 - 3500 = 500$. ### Common Pitfall The most frequent error is simply subtracting the two single categories from the total: $4000 - 3000 - 2000 = -1000$. This fails because you are heavily double-subtracting the 1500 people who do both. Always establish the precise union before subtracting from the master total. ### Final Answer **Therefore, the correct answer is 500.**