Two times the age of a daughter is added to the age of her mother and the total becomes 70 years. If instead two times the age of the mother is added to the age of the daughter and the total becomes 95 years, what is the present age of the mother in years?

Introduction / Context:
This is a classic problems-on-ages question where the age of the mother and the daughter are related by two linear equations. Such questions test your ability to translate language statements into algebraic expressions and then solve simultaneous equations to find the unknown ages.

Given Data / Assumptions:

• Let the present age of the mother be M years.
• Let the present age of the daughter be D years.
• Two times the age of the daughter added to the age of the mother gives a total of 70 years.
• Two times the age of the mother added to the age of the daughter gives a total of 95 years.
• All ages are assumed to be positive real numbers and represent present ages.

Concept / Approach:
The key idea is to convert each English statement into a linear equation in terms of M and D. Once we have two linear equations with two unknowns, we can solve them using substitution or elimination. Problems on ages almost always reduce to solving such small systems of equations.

Step-by-Step Solution:
From the first statement: 2 times daughter age + mother age = 70.So the first equation is: 2 * D + M = 70.From the second statement: 2 times mother age + daughter age = 95.So the second equation is: 2 * M + D = 95.Rewrite the first equation as: M = 70 - 2 * D.Substitute this into the second equation: 2 * (70 - 2 * D) + D = 95.Simplify: 140 - 4 * D + D = 95.Combine like terms: 140 - 3 * D = 95.Rearrange: 140 - 95 = 3 * D, so 45 = 3 * D.Therefore D = 45 / 3 = 15 years.Now substitute back to find M: M = 70 - 2 * 15 = 70 - 30 = 40 years.

Verification / Alternative check:
First condition: 2 * D + M = 2 * 15 + 40 = 30 + 40 = 70, which matches the given information.Second condition: 2 * M + D = 2 * 40 + 15 = 80 + 15 = 95, which also matches the given information.Both conditions are satisfied, so the solution M = 40 years and D = 15 years is correct.

Why Other Options Are Wrong:
30 years: If M = 30, the equations do not hold, and the computed daughter age becomes inconsistent with the second equation.38 years: If M = 38, substituting into either equation gives a non-integer or inconsistent daughter age.41 years: If M = 41, 2 * M + D cannot be exactly 95 while also satisfying the first equation.

Common Pitfalls:
Students sometimes reverse the roles of the mother and daughter while forming equations, leading to swapped variables and incorrect answers.Another frequent mistake is mishandling the coefficients when using substitution, especially when expanding expressions like 2 * (70 - 2 * D).Some learners try to guess-and-check ages mentally without forming equations, which is error-prone when the numbers are not very small.

Final Answer:
The present age of the mother that satisfies both conditions is 40 years.