Mother–daughter sum and a past 7× relation: The sum of a mother’s and daughter’s present ages is 50 years. Five years ago, the mother was 7 times the daughter’s age. Find their present ages (mother, daughter).

Introduction / Context:Here we combine a sum constraint with a time-shifted multiplicative relation. Such problems resolve neatly via substitution after expressing one variable in terms of the other using the sum, then enforcing the past 7× condition.

Given Data / Assumptions:

• Let present mother age be M, daughter age be D.
• M + D = 50.
• Five years ago: M − 5 = 7(D − 5).

Concept / Approach:Use M = 50 − D in the second equation and solve for D. Then compute M. This ensures algebra is minimal and transparent.

Step-by-Step Solution:

Verification / Alternative check:Five years ago: 35 and 5 → 35 = 7 × 5; sum now 50; all constraints consistent.

Why Other Options Are Wrong:Other pairs fail the 7× relation or the sum-of-ages constraint.

Common Pitfalls:Mixing present and past values incorrectly; always subtract the same years from both individuals in past relations.

Final Answer:40 years, 10 years