A man says to his son: "I am five times as old as you were when I was as old as you are now."\nThe sum of their present ages is 64 years.\nWhat are their present ages?

Introduction / Context:
This is a classic age puzzle involving a layered time statement. The man compares his present age to the age of his son at a specific time in the past, when the man was as old as the son is now. The puzzle also gives the sum of their current ages. We need to translate the verbal description into equations and solve.

Given Data / Assumptions:

• Let the present age of the father be F years.
• Let the present age of the son be S years.
• F + S = 64.
• The father says he is five times as old as the son was when the father was as old as the son is now.

Concept / Approach:
When the father was as old as the son is now, his age was S. To reach that moment from the present, we go back in time by F - S years, because F - (F - S) = S. At that time in the past, the sons age would have been S - (F - S) = 2S - F. According to the statement, the father s current age F is five times that past age of the son: F = 5(2S - F). Together with F + S = 64, we can solve for F and S.

Step-by-Step Solution:
From the sum: F + S = 64. When the father was as old as the son is now, time difference = F - S years ago. At that time, son s age = S - (F - S) = 2S - F. Given: F is five times that past age of the son. So F = 5(2S - F). F = 10S - 5F. Bring terms together: 6F = 10S. So 3F = 5S, hence F = (5/3)S. Substitute into F + S = 64: (5/3)S + S = 64. (8/3)S = 64. S = 64 * 3 / 8 = 24. Then F = 64 - 24 = 40. Thus, the father is 40 years old and the son is 24 years old.

Verification / Alternative check:
Check the key condition. When the father was as old as the son is now, we go back F - S = 40 - 24 = 16 years. Sixteen years ago, father was 24 and the son was 24 - 16 = 8. The father now is 40. Five times the son s age then is 5 * 8 = 40, exactly the father s current age. This matches the statement perfectly.

Why Other Options Are Wrong:
Other pairs like 50 and 14 or 48 and 16 do not satisfy both the sum of ages equal to 64 and the five times relationship when checked carefully. Only the pair 40 and 24 satisfies both conditions simultaneously.

Common Pitfalls:
Many learners mis-handle the phrase when I was as old as you are now and either use the wrong time difference or apply the five times condition at the wrong moment. Another common error is to misinterpret the direction of time, subtracting from the wrong age. Drawing a simple time line and labelling ages clearly usually helps to avoid such confusion.

Final Answer:
The present ages are Father = 40 years and Son = 24 years.