The present age of a man is four times the present age of his son.\nFive years ago, the man was nine times as old as his son was at that time.\nWhat is the present age of the man?

Introduction / Context:
This age problem uses a present age ratio combined with a past age ratio. The goal is to form simultaneous equations from the verbal description and then solve for the present ages, specifically for the age of the man.

Given Data / Assumptions:

• Let the present age of the son be S years.
• The present age of the man (father) is four times the son, so it is 4S years.
• Five years ago, the man was nine times as old as the son was at that time.
• We must find the present age of the man.

Concept / Approach:
We convert the past relationship into an equation. Five years ago, the father s age was 4S - 5 and the son s age was S - 5. The given condition is that the father s age at that time was nine times the son s age at that time. This provides a linear equation in S, which we can solve and then use to get the man s age.

Step-by-Step Solution:
Present son age = S. Present man age = 4S. Five years ago, ages were: son = S - 5, man = 4S - 5. Given: 4S - 5 = 9(S - 5). 4S - 5 = 9S - 45. Rearrange: 4S - 9S = -45 + 5. -5S = -40. S = 8. Therefore present man age = 4S = 4 * 8 = 32 years. So the man is 32 years old now.

Verification / Alternative check:
Check the condition with these ages. Five years ago the son was 8 - 5 = 3 years old. The man was 32 - 5 = 27 years old. Nine times the son s age then is 9 * 3 = 27, which matches the man s age at that time. This confirms the solution.

Why Other Options Are Wrong:
If the man were 24, 40 or 44 years old now, and his age is four times the son s age, the resulting past ages would not satisfy the nine-times relationship. Those values do not simultaneously satisfy both conditions.

Common Pitfalls:
Frequent errors include mixing the time reference, such as using present ages directly instead of subtracting five years, or misreading the phrase four times as four more. Always distinguish clearly between multiplicative relationships and additive differences and keep track of the time shifts.

Final Answer:
The present age of the man is 32 years.