Family ages puzzle: A father is three times his daughter's age. The mother is 9 years younger than the father. The son's age is half that of his mother, and the brother is 7 years older than his sister. What is the mother's age?

Introduction / Context:
This is a multi-equation age problem. We translate relationships into algebra and solve simultaneously. Such questions teach consistent variable assignment and systematic elimination.

Given Data / Assumptions:

• Daughter’s age = d.
• Father’s age = 3d.
• Mother’s age = Father − 9 = 3d − 9.
• Son’s age = Mother / 2 = (3d − 9) / 2.
• Brother is 7 years older than sister: Son = d + 7.

Concept / Approach:
Use the last relation to connect son and daughter, substitute the son’s expression from the mother’s relation, and solve for d. Then compute the mother’s age.

Step-by-Step Solution:
Equation from siblings: (3d − 9) / 2 = d + 7.Multiply by 2: 3d − 9 = 2d + 14.Solve: 3d − 2d = 14 + 9 → d = 23.Mother’s age = 3d − 9 = 69 − 9 = 60 years.

Verification / Alternative check:
Compute all ages: Daughter 23; Father 69; Mother 60; Son = Mother/2 = 30; Brother older than sister by 7 → 23 + 7 = 30; all relations satisfied.

Why Other Options Are Wrong:

• 40, 45, 50, 54: None fit the system while keeping all relationships true.

Common Pitfalls:

• Mistaking “mother is 9 years younger than father” as father = mother − 9.
• Forgetting to double both sides when clearing the fraction.

Final Answer:
60 years