Step-by-step Solution:

Quick Solution

The best way to solve this is by setting up two algebraic equations based on the information given and solving for the mother's age.

1. Set Up the Equations

- Let the person's present age be P and the mother's present age be M.
- From the first sentence ("A person's present age is two-fifth of the age of his mother"):
P = (2/5)M

- In 8 years, their ages will be (P + 8) and (M + 8). From the second sentence ("After 8 years, he will be one half of the age of his mother"):
P + 8 = (1/2)(M + 8)

2. Solve for the Mother's Age (M)

- Substitute the first equation into the second one to have only one variable (M):
(2/5)M + 8 = (1/2)(M + 8)
- Now, solve for M:
(2/5)M + 8 = (1/2)M + 4
8 - 4 = (1/2)M - (2/5)M
4 = (5/10 - 4/10)M
4 = (1/10)M
M = 4 × 10 = 40.

3. Verification (Optional)

- If the mother's present age is 40, the person's age is (2/5) × 40 = 16.
- In 8 years, they will be 48 and 24.
- 24 is indeed half of 48. The answer is correct.

Final Answer: The present age of the mother is 40.