A is three times as old as B. C was twice as old as A four years ago. In four years' time, A will be 31 years old. What are the present ages of B and C?

Introduction / Context:
This age problem involves three people: A, B, and C. We know a multiple relationship between the ages of A and B, a past relationship between the ages of C and A, and a future value for A's age. Using these facts, we must determine the current ages of B and C.

Given Data / Assumptions:

- A is currently three times as old as B.
- Four years ago, C was twice as old as A was at that time.
- In four years' time, A will be 31 years old.
- We must find the present ages of B and C.

Concept / Approach:
We first determine A's present age using the information about A's age in four years. With A's present age known, we can immediately find B's age using the "three times as old" relationship. Then we use the statement about C being twice as old as A four years ago to calculate C's present age. The problem is solved with simple linear arithmetic rather than complicated equations.

Step-by-Step Solution:
Step 1: A will be 31 years old in four years. Step 2: Therefore, A's present age = 31 − 4 = 27 years. Step 3: A is three times as old as B now. Let B's age be B years. Step 4: From the relationship, 27 = 3 × B, so B = 27 / 3 = 9 years. Step 5: Four years ago, A's age was 27 − 4 = 23 years. Step 6: At that time, C was twice A's age, so C's age four years ago was 2 × 23 = 46 years. Step 7: Therefore, C's present age = 46 + 4 = 50 years.

Verification / Alternative check:
Check all conditions: A is 27 and B is 9, so A is indeed three times B's age. Four years ago, A was 23 and C was 46; 46 is twice 23, satisfying the second condition. In four years, A will be 27 + 4 = 31 years, matching the given future age. Thus the present ages are B = 9 years and C = 50 years.

Why Other Options Are Wrong:
Pairs such as 45 years, 10 years or 10 years, 50 years do not satisfy all the given relationships when checked carefully. In particular, A must be three times B now and C must be exactly twice A's age four years ago. Only the pair 9 years, 50 years satisfies every condition simultaneously.

Common Pitfalls:
Some students misinterpret the time references, for example using A's age four years from now instead of four years ago when relating to C. Others may incorrectly form equations or forget that age differences remain constant over time. Writing down a timeline and labelling ages at each point in time often helps avoid such confusion.

Final Answer:
The present ages are B = 9 years and C = 50 years.