Final answer:
After assessing the conditions and applying them, it is determined that John is 8 years old. Angie's age as a teenager must be a prime number, which leaves us with the options of 13, 17, or 19. John's age must satisfy the condition that one more than half his age is a prime number, eliminating all options but 8 for his age.
Step-by-step explanation:
We need to find John's age based on the given conditions:
- Angie's age + John's age = Michael's age
- Angie's age is a prime number
- John's age and Michael's age are not prime numbers
- John's age + Michael's age = prime number
- Angie's age - John's age = prime number
- 1 more than half of John's age is a prime number
- Angie is a teenager older than John
Let's take into account the age constraints:
- Since Angie is the only teenager and older than John, her age could be any prime number between 13 and 19: 13, 17, or 19.
- Since 1 more than half of John's age is prime, and knowing that he is younger than a teenager, possible ages for John are ages that result in prime numbers when halved and increased by 1. This limits our options to the ages of 2 (prime when halved +1 is 2), 4 (half +1 is 3, prime), and 8 (half +1 is 5, prime).
- John cannot be 2, since that's not reasonable for the context, so we're left with 4 and 8.
- If Angie is 13 and John is 4, Michael would be 17, which is prime, so that doesn't fit the non-prime requirement for Michael. Therefore, John cannot be 4.
- If Angie is 17 or 19 and John is 8, then Michael would be 25 or 27, neither of which are prime, satisfying all the conditions.
- We can then conclude that John is 8 years old.