Question

Aaron, Bryce, and Craig are all distinct integer ages. They had the following conversation.

i.Bryce to Craig: You're the youngest one.
ii. Aaron to Bryce: Your age is exactly 70% greater than mine.
iii. Aaron to Craig: Your age is the average of my age and Bryce's age.
iv. Craig to Aaron: I'm at least 8 years older than you.
However, not all of these statements are true. When speaking to someone older, the speaker always tells the truth. Otherwise when speaking to someone younger, the speaker always lies. How old is Aaron?

Answer 1

Aaron is 24 years old. The given statements lead to contradictory conclusions, so none of them are true. By considering the possibilities and eliminating options, we find that Aaron's age must be 24.

Step-by-step explanation:

Aaron is 24 years old.

Craig to Aaron: I'm at least 8 years older than you.

We can solve this problem using a process of elimination. Let's consider the statements one by one:

From statement i, we know that Craig is the youngest.

If Craig is the youngest, statement ii cannot be true because it would mean Bryce's age is 70% greater than Aaron's age, which is not possible.

Now, let's switch to statement iii. If Aaron's age is the average of his and Bryce's age, and Craig is the youngest, it would mean that Aaron's age is the average of Bryce's age and Craig's age. However, this would require Aaron's age to be the same as Bryce's age, which contradicts statement ii. Therefore, statement iii cannot be true.

Lastly, let's consider statement iv. If Craig is at least 8 years older than Aaron, it means Aaron's age must be less than or equal to Craig's age minus 8. Since Aaron's age cannot be equal to Craig's age, it must be less. However, this contradicts statement ii where Bryce's age is 70% greater than Aaron's age. Therefore, statement iv cannot be true.

Based on the process of elimination, none of the statements are true. Therefore, the only consistent conclusion is that Aaron must be the oldest. Since Aaron is distinct from Bryce and Craig, his age must be greater than both of theirs. Let's consider some possibilities:

If Aaron is 25 years old, Bryce's age would need to be 70% greater, which is 42.5 years. However, this is not possible since ages must be integers.

If Aaron is 24 years old, Bryce's age would need to be 70% greater, which is 40.8 years. Again, this is not possible as ages must be integers.

If Aaron is 23 years old, Bryce's age would need to be 70% greater, which is 39.1 years. This is still not possible.

If Aaron is 22 years old, Bryce's age would need to be 70% greater, which is 37.4 years. Once again, this is not possible.

Finally, if Aaron is 21 years old, Bryce's age would need to be 70% greater, which is 35.7 years. Again, this is not possible.

Based on these deductions, we can conclude that Aaron's age cannot be 21, 22, 23, 24, or 25. Thus, the only remaining option is that Aaron must be 24 years old.

Answer 2

Aaron's age is 22.

Step-by-step explanation:

To determine Aaron's age, we need to analyze each statement and its truthfulness.

Statement ii says that Bryce's age is exactly 70% greater than Aaron's age. Let's represent Aaron's age as A and Bryce's age as B. So, according to this statement, B = A + 70% of A. Simplifying this equation, we get B = 1.7A.

Statement iii says that Craig's age is the average of Aaron's age and Bryce's age. Let's represent Craig's age as C. So, according to this statement, C = (A + B) / 2. Substituting the value of B from statement ii, we get C = (A + 1.7A) / 2 = 1.35A.

Statement iv says that Craig is at least 8 years older than Aaron. So, C ≥ A + 8. Substituting the value of C from the previous equation, we get 1.35A ≥ A + 8. Solving this inequality, we find that A ≤ 8 / 0.35 ≈ 22.86. Since A must be an integer, the possible values for A are 22 and 23. However, in statement i, Bryce tells Craig that he is the youngest one, which means Craig must be older than Bryce. Therefore, Craig's age cannot be 23. Hence, Aaron's age is 22.