The sum of the present ages of a father and his son is 70 years. After 10 years, the son age will be exactly half of the father age. What are their present ages now?

Introduction / Context:
This family based age problem involves a fixed sum of present ages and a simple relationship between their ages in the future. The condition that one person age will be exactly half of another person age after some years is a typical pattern in problems on ages. It helps to practice forming equations from such verbal statements and understanding how age relationships evolve over time.

Given Data / Assumptions:

• The sum of the present ages of a father and his son is 70 years.
• After 10 years, the son age will be exactly half the age of the father at that time.
• We need to find their current ages.
• Ages grow in a normal way, so both father and son age by 10 years over the period.

Concept / Approach:
Let the father age and the son age be two variables. The first statement gives a simple sum equation. The second statement gives a proportional relationship between their future ages. By expressing the future ages in terms of the current ones and using the half relationship, we obtain a second equation. Solving the two linear equations together yields the present ages. This is a standard two variable, two equation algebra problem derived from an age context.

Step-by-Step Solution:
Step 1: Let the father present age be F years and the son present age be S years. Step 2: From the sum condition, F + S = 70. Step 3: After 10 years, the father age will be F + 10 and the son age will be S + 10. Step 4: The future condition says S + 10 is half of F + 10, so S + 10 = (F + 10) / 2. Step 5: Use these two equations to solve for F and S. You obtain F = 50 years and S = 20 years. Step 6: Thus the present ages are 50 years for the father and 20 years for the son.

Verification / Alternative check:
First verify the sum: 50 + 20 = 70 years, which matches the given total. After 10 years, the father will be 60 years old and the son will be 30 years old. In that future scenario, the son age 30 is exactly half of the father age 60. Both conditions are satisfied perfectly, so the ages 50 years and 20 years are correct and consistent with the problem statement.

Why Other Options Are Wrong:
Option 45 years, 25 years: The sum is 70, but after 10 years the ages will be 55 and 35, and 35 is not half of 55.
Option 47 years, 23 years: The sum is 70, but after 10 years they will be 57 and 33, and 33 is not half of 57.
Option 50 years, 25 years: After 10 years the ages would be 60 and 35, and 35 is not half of 60.
Option 40 years, 30 years: The sum is 70, but after 10 years they will be 50 and 40, and 40 is not half of 50.

Common Pitfalls:
A frequent mistake is to misread the condition and think that the son is half the father age at present rather than after 10 years. Another common error is to forget to add 10 years to both ages when forming the half equation. Always check that time shifts apply equally to all people involved and that you use future ages when the statement clearly refers to a future time. Revisiting the wording carefully reduces such misunderstandings.

Final Answer:
The present ages are 50 years for the father and 20 years for the son.