Two years before the reference stage – A was 4 years older than B at a certain stage. Sixteen years after that stage, A will be thrice his present age and B will be five times his present age. How old were A and B two years before that initial stage?

Introduction / Context:
This problem involves a “reference stage,” future projections from that stage, and then a request to report ages two years before that stage. Careful variable setup keeps the timeline consistent and prevents sign mistakes.

Given Data / Assumptions:

• At the reference stage, A was 4 years older than B.
• Sixteen years after that stage, A will be thrice his age at the stage, and B will be five times his age at the stage.
• We must give A's and B's ages two years before the stage.

Concept / Approach:
Let A and B at the stage be a and b. From “older by 4,” a = b + 4. From “after 16 years,” a + 16 = 3a and b + 16 = 5b. Solve these to find a and b, then subtract 2 for the requested earlier time.

Step-by-Step Solution:
From a + 16 = 3a ⇒ 16 = 2a ⇒ a = 8.From b + 16 = 5b ⇒ 16 = 4b ⇒ b = 4.Check the 4-year difference: a − b = 8 − 4 = 4 (consistent).Two years before the stage: A = 8 − 2 = 6; B = 4 − 2 = 2.

Verification / Alternative check:
Move forward 16 years from the stage: A = 24 which is 3 × 8; B = 20 which is 5 × 4 — both conditions satisfied.

Why Other Options Are Wrong:

• 10 & 6, 12 & 8, 8 & 4 conflict with the “16 years later” multipliers when traced back to the stage.
• “None of these” is unnecessary since 6 & 2 fits all constraints.

Common Pitfalls:
Applying the multipliers to future ages instead of the stage ages, or subtracting 2 at the wrong point in time.

Final Answer:
6 and 2 years