Sum and difference of two ages – The combined age of a mother and her daughter is 60 years, and their age difference is 30 years. What is the mother’s age?

Introduction / Context:
When a sum and a difference of two numbers are known, each number can be found using the standard half-sum formulas. This technique appears frequently in age problems involving a parent and child.

Given Data / Assumptions:

• Total age (mother + daughter) = 60.
• Age difference = 30 (mother is older).

Concept / Approach:
Let M = mother, D = daughter. Then M + D = 60 and M − D = 30. Solve by adding the equations or using half-sum: M = ( (M + D) + (M − D) ) / 2.

Step-by-Step Solution:
M = (60 + 30) / 2 = 90 / 2 = 45.D = (60 − 30) / 2 = 30 / 2 = 15.Therefore, mother = 45 years.

Verification / Alternative check:
M + D = 45 + 15 = 60 and M − D = 45 − 15 = 30: both constraints satisfied.

Why Other Options Are Wrong:
Any value other than 45 fails one of the two simultaneous conditions.

Common Pitfalls:
Using 30 directly as the mother’s age or halving the wrong quantity. Always apply the half-sum formulas carefully.

Final Answer:
45 years