Two-time conditions, find a ratio today: Two years ago a mother was four times as old as her son. Six years from now her age will exceed twice her son's age by 10 years. What is their present age ratio (mother : son)?

Introduction / Context:
We are given two timed conditions to pin down two unknown present ages, then asked for their ratio. As usual, convert each sentence into an equation in present-day variables.

Given Data / Assumptions:

• M = mother's current age; S = son's current age.
• Two years ago: M − 2 = 4(S − 2).
• Six years from now: M + 6 = 2(S + 6) + 10.

Concept / Approach:
Solve the system. From the first, M = 4S − 6. Substitute into the second to find S, then M, and finally reduce M : S.

Step-by-Step Solution:
1) Substitute: (4S − 6) + 6 = 2S + 22 ⇒ 4S = 2S + 22.2) 2S = 22 ⇒ S = 11.3) M = 4S − 6 = 44 − 6 = 38.4) Ratio M : S = 38 : 11.

Verification / Alternative check:
Two years ago: 36 vs 9 → 36 = 4 × 9. In six years: 44 vs 17 → 44 = 2 × 17 + 10, OK.

Why Other Options Are Wrong:
They do not match both time-shifted conditions simultaneously.

Common Pitfalls:
Dropping the +10 in the future condition or mishandling the −2 on both sides in the past condition.

Final Answer:
38 : 11