Sure! Let's go step by step to solve the problem.
We start by summarizing the given information:
1. The total population of the town is 5000 people.
2. The number of people who like only eggs is 3200.
3. The number of people who like only meat is 500.
4. The number of people who like both eggs and meat is 1500.
Now, we want to find out how many people are pure vegetarians, i.e., people who do not like either eggs or meat.
First, we need to determine the total number of people who like eggs, meat, or both. To do this, we use the principle of inclusion and exclusion:
- The total number of people who like eggs or meat or both is equal to the sum of the number of people who like only eggs and the number of people who like only meat, plus the number of people who like both:
[tex]\[ \text{people who like eggs or meat or both} = \text{people who like only eggs} + \text{people who like only meat} + \text{people who like both} \][/tex]
- Plugging in the given values, we get:
[tex]\[ \text{people who like eggs or meat or both} = 3200 + 500 + 1500 = 5200 \][/tex]
However, this total includes the people who like both eggs and meat twice (once in the egg group and once in the meat group). Therefore, we must subtract the number of people who like both to get the correct total:
[tex]\[ \text{people who like eggs or meat or both} = \text{people who like eggs or meat or both} - \text{people who like both} \][/tex]
[tex]\[ \text{people who like eggs or meat or both} = 5200 - 1500 = 3700 \][/tex]
Now, we know the total number of people who like either eggs, meat, or both (3700 people). To find the number of pure vegetarians (people who like neither eggs nor meat), we subtract this number from the total population:
[tex]\[ \text{pure vegetarians} = \text{total population} - \text{people who like eggs or meat or both} = 5000 - 3700 = 1300 \][/tex]
So, the number of pure vegetarians in the town is:
[tex]\[ \boxed{1300} \][/tex]
