Mother–Daughter Ages — “Mother and daughter sum to 56. After 4 years, the mother will be three times the daughter. Find both present ages.”

Introduction / Context:
We are given a present-day sum and a future multiplicative relation. Converting both into equations in two variables lets us solve for the exact present ages of mother and daughter.

Given Data / Assumptions:

• D + M = 56 (present sum).
• M + 4 = 3(D + 4) (future relation after 4 years).
• All ages are nonnegative integers.

Concept / Approach:
From the future condition get M in terms of D, then substitute into the present sum to find D. Compute M from D and cross-check the future condition.

Step-by-Step Solution:

Verification / Alternative check:
After 4 years: daughter 16, mother 48; 48 = 3 * 16 ✓. Sum today is 12 + 44 = 56 ✓.

Why Other Options Are Wrong:
10 & 46, 11 & 45, 13 & 43, 14 & 42 do not satisfy the future “three times” relation when advanced 4 years.

Common Pitfalls:
Forgetting to add 4 to both ages in the future relation; substituting 3D directly into the present sum without the +8 shift.

Final Answer:
12 years and 44 years