The present age of a man is two fifths of the age of his father. After 8 years, the man will be one half of the age of his father. What is the present age of the father?

Introduction / Context:
This age problem uses fractional relationships between the ages of a man and his father at different times. At present, the man age is a fixed fraction of his father age, and after some years, the man age will be another fraction of his father age. This leads to a pair of equations that can be solved to find the father present age.

Given Data / Assumptions:

• The present age of the man is two fifths of the age of his father.
• After 8 years, the man will be one half of the age of his father.
• We are asked to find the current age of the father.
• All ages are in years and positive.

Concept / Approach:
We introduce variables for the present ages of the man and the father. Using the fraction relationships, we derive two equations: one for the present and one for the future. Solving these equations simultaneously yields the exact ages. This technique is standard for problems involving fractional comparisons at different times.

Step-by-Step Solution:
Let the father present age be F years.Then the man present age is (2 / 5) * F years.After 8 years, the father age will be F + 8 years.After 8 years, the man age will be (2 / 5) * F + 8 years.The condition says that after 8 years, the man will be one half of his father age, so:(2 / 5) * F + 8 = (1 / 2) * (F + 8).Multiply every term by 10 to clear the fractions: 10 * ((2 / 5) * F + 8) = 10 * (1 / 2) * (F + 8).This gives 4 * F + 80 = 5 * (F + 8).Expand the right side: 4 * F + 80 = 5 * F + 40.Rearrange: 80 - 40 = 5 * F - 4 * F, so 40 = F.Therefore the present age of the father is 40 years.

Verification / Alternative check:
If F = 40, man present age is (2 / 5) * 40 = 16 years.After 8 years, father age will be 40 + 8 = 48 years, man age will be 16 + 8 = 24 years.Check the future condition: 24 is exactly one half of 48.Both the present and future fractional relationships are satisfied, confirming that F = 40 years is correct.

Why Other Options Are Wrong:
50 years: If the father were 50, the man would be 20 now and 28 after 8 years, while the father would be 58, and 28 is not half of 58.33 years: This leads to non-integer or inconsistent values when the future half relationship is applied.30 years: This would give a present man age of 12 and future ages of 20 and 38, which do not satisfy the one half condition.

Common Pitfalls:
Misinterpreting the fractions as F = (2 / 5) * man or swapping numerator and denominator is a common error.Some students forget to add 8 years to both ages for the future condition.Algebraic mistakes when clearing fractions, such as incorrect multiplication, often result in wrong values.

Final Answer:
The present age of the father is 40 years.