A person's present age is two fifth of the present age of his mother. Eight years from now, he will be one half of his mother's age at that time. How old is the mother at present?

Introduction / Context:
This age problem gives the current ratio between a person and his mother and another ratio eight years into the future. These two relationships allow us to form and solve a system of equations for their present ages. It is a standard example of age questions using proportion and linear equations.

Given Data / Assumptions:

• The person's present age is two fifth of his mother's present age.
• After 8 years, the person will be one half of his mother's age at that time.
• We must determine the mother's present age.

Concept / Approach:
Let the person's present age be P years and the mother's present age be M years. Then P = (2/5) * M. After 8 years, their ages will be P + 8 and M + 8, and the second condition gives P + 8 = (1/2) * (M + 8). Solving these two equations simultaneously yields M directly.

Step-by-Step Solution:
Let P be the person's present age and M be the mother's present age. We are given P = (2/5) * M. After 8 years, the person's age will be P + 8 and the mother's age will be M + 8. Given that P + 8 = (1/2) * (M + 8). From P = (2/5) * M, substitute into the second equation: (2/5) * M + 8 = (1/2) * (M + 8). Multiply through by 10 to clear denominators: 4M + 80 = 5(M + 8). Expand the right side: 4M + 80 = 5M + 40. Rearrange to get 80 - 40 = 5M - 4M, so 40 = M. Therefore, the mother's present age M is 40 years.

Verification / Alternative check:
If the mother is 40, the person's present age is P = (2/5) * 40 = 16 years. After 8 years, the person will be 24 and the mother will be 48. Indeed, 24 is exactly half of 48. Both original conditions are satisfied, confirming the solution is correct.

Why Other Options Are Wrong:
If the mother were 36, then P would be 14.4 years, which is not an integer and does not fit usual exam expectations. Ages 38, 42 or 44 also lead to future ages that do not satisfy the one half relation after 8 years. Only 40 years makes both the two fifth and half conditions hold simultaneously.

Common Pitfalls:
Mistakes often arise from incorrectly writing P = (5/2) * M instead of (2/5) * M or forgetting to add 8 to both ages when forming the future relation. Some learners also clear denominators incorrectly and lose track of the coefficients. Careful algebra and checking the final ages in both conditions helps avoid these errors.

Final Answer:
Hence, the mother's present age is 40 years.