Two-timepoint constraints for mother and daughter: Two years ago, a mother was four times her daughter’s age. Eight years from now, the mother will be 12 years older than her daughter. What is the ratio of their present ages (mother : daughter)?

Introduction / Context:
Two linear relations at different time points determine both present ages uniquely. Express both conditions in present variables M and D, then solve the simultaneous equations. The final ratio is simply M : D after solving.

Given Data / Assumptions:

• Two years ago: M − 2 = 4(D − 2).
• Eight years ahead: M + 8 = (D + 8) + 12 ⇒ M = D + 12.

Concept / Approach:
Use the future relation to express M in terms of D, substitute into the past relation, and solve for D. Then compute M, and reduce the ratio M : D.

Step-by-Step Solution:

Verification / Alternative check:
Two years ago: 16 and 4 (4×). In eight years: 26 and 14, difference 12 (as stated).

Why Other Options Are Wrong:
4:1, 3:2, 5:1 are not equal to 3:1 given both constraints.

Common Pitfalls:
Forgetting to add/subtract from both ages when shifting timeframes or misreading “exceeds by 12” as a multiplicative relation.

Final Answer:
3 : 1