Eighteen years ago, a mother was 3 times as old as her daughter. At present, the mother is 2 times as old as her daughter. What is the sum of their present ages (mother + daughter) in years?

Introduction:
This is a standard “mother-daughter ages” question involving two time-based ratio conditions. The main skill tested is forming equations correctly for “years ago” and “at present.” The best method is to use variables for present ages, write both conditions as equations, and solve. Finally, the question asks for the sum of present ages, not the individual ages alone.

Given Data / Assumptions:

• Let mother's present age = M years
• Let daughter's present age = D years
• 18 years ago: M - 18 = 3 * (D - 18)
• Now: M = 2D
• We need M + D

Concept / Approach:
Use the present ratio equation M = 2D, substitute into the “18 years ago” equation, solve for D, then compute M and the sum M + D.

Step-by-Step Solution:
From present condition: M = 2DUse 18 years ago condition: M - 18 = 3(D - 18)Substitute M = 2D: 2D - 18 = 3D - 54Move terms: -18 + 54 = 3D - 2D36 = DSo M = 2D = 72Sum = M + D = 72 + 36 = 108

Verification / Alternative check:
18 years ago: mother 72-18 = 54, daughter 36-18 = 18. 54 is 3 times 18, correct. Present: 72 is 2 times 36, correct. So the solution is consistent.

Why Other Options Are Wrong:
84 and 96: result from incorrect subtraction of 18 or mixing the ratio conditions.116: typically comes from adding wrong ages or using 3 times relation in the present instead of 18 years ago.100: does not satisfy both equations together.

Common Pitfalls:
Writing M - 18 = 3D instead of 3(D - 18).Assuming the same ratio applies at present and in the past.Solving for individual ages but forgetting the question asks for the sum.

Final Answer:
108 years