Two years ago, the average age of A and B was 26 years. The age of A five years from now will be 40 years, and B is 5 years younger than C. What is the difference between the present ages of A and C?

Introduction / Context:
This age problem connects the ages of three people: A, B and C. We are given the past average age of A and B, the future age of A, and an age difference between B and C. Using this information, we must find the difference between the present ages of A and C. The question tests your ability to handle average age information and relative age differences across time.

Given Data / Assumptions:

- Two years ago, the average age of A and B was 26 years.
- A's age five years from now will be 40 years.
- B is 5 years younger than C at present.
- All ages are in years and positive.

Concept / Approach:
We first use A's future age to compute A's present age. Then, from the past average of A and B, we compute B's present age. Using the information that B is 5 years younger than C, we derive C's present age. Finally, we calculate the absolute difference between A's and C's present ages. This requires careful tracking of time shifts and differences.

Step-by-Step Solution:
Step 1: Let A's present age be A years and B's present age be B years. Step 2: We are told that A's age five years from now will be 40 years, so A + 5 = 40. Step 3: Solve for A: A = 40 − 5 = 35 years. Step 4: Two years ago, the average age of A and B was 26 years. Step 5: Two years ago, their ages were A − 2 and B − 2, and their average was (A − 2 + B − 2) / 2 = 26. Step 6: Simplify: (A + B − 4) / 2 = 26 ⇒ A + B − 4 = 52 ⇒ A + B = 56. Step 7: Substitute A = 35 into A + B = 56 to find B: 35 + B = 56 ⇒ B = 21 years. Step 8: B is 5 years younger than C, meaning C's present age = B + 5 = 21 + 5 = 26 years. Step 9: Now we have A = 35 years and C = 26 years. The difference between their present ages is |35 − 26| = 9 years.

Verification / Alternative check:
We can quickly verify all the given conditions. If A is 35 years old now, then in 5 years A will be 40 years old, matching the future age statement. Two years ago, A was 33 and B was 19. Their average then was (33 + 19) / 2 = 52 / 2 = 26 years, which matches the given average. With B at 21 and C at 26 now, B is indeed 5 years younger than C. Therefore, all given conditions are satisfied by these values, and the calculated difference of 9 years is consistent.

Why Other Options Are Wrong:
Differences such as 6, 12, 15 or 18 years would imply different present ages for C that would either break the two-years-ago average or the "5 years younger" relationship between B and C. Only a 9-year difference keeps all conditions consistent when A is 35 years old and B is 21 years old.

Common Pitfalls:
Students may misread the time references, for example treating the 26-year average as the current average rather than the average two years ago. Others might forget to adjust both A and B by 2 years when using that average, or misapply the "5 years younger" relationship. Writing down a small timeline and methodically converting averages into total ages helps to avoid such mistakes.

Final Answer:
The difference between the present ages of A and C is 9 years.