Problem Setup

We are given that the sum of the present ages of a father and his son is 60 years. Additionally, five years ago, the product of their ages was 525. Our goal is to determine the father's current age.

Let $F$ represent the father's present age and $S$ represent the son's present age.

• From the problem statement, we have the equation for their present ages: $F + S = 60$.
• Five years ago, their ages were $(F - 5)$ for the father and $(S - 5)$ for the son.
• The product of their ages five years ago is given as: $(F - 5)(S - 5) = 525$.

Solving the Age Equations

We will use the given information to set up and solve a system of equations.

1. From the first equation, $F + S = 60$, we can express the son's age ($S$) in terms of the father's age ($F$):

   $S = 60 - F$

2. Substitute this expression for $S$ into the second equation:

   $(F - 5)((60 - F) - 5) = 525$

   Simplify the term inside the second parenthesis:

   $(F - 5)(55 - F) = 525$

3. Expand the equation:

   $55F - F^2 - 275 + 5F = 525$

   Combine like terms:

   $-F^2 + 60F - 275 = 525$

   Rearrange the equation into the standard quadratic form ($aF^2 + bF + c = 0$):

   $F^2 - 60F + 275 + 525 = 0$

   $F^2 - 60F + 800 = 0$

4. Factor the quadratic equation. We need two numbers that multiply to $800$ and add up to $-60$. These numbers are $-20$ and $-40$.

   $(F - 20)(F - 40) = 0$

5. Solve for $F$. This gives two possible values for the father's present age:

   $F = 20$ or $F = 40$

Determining the Father's Age

We must evaluate both possible solutions in the context of the problem.

• Case 1: If the father's present age is $F = 20$. Then the son's present age would be $S = 60 - 20 = 40$. This scenario is impossible because the son cannot be older than the father.
• Case 2: If the father's present age is $F = 40$. Then the son's present age would be $S = 60 - 40 = 20$. This is a valid scenario.

Let's verify this valid scenario using the condition from five years ago:

• Father's age 5 years ago: $40 - 5 = 35$ years.
• Son's age 5 years ago: $20 - 5 = 15$ years.
• Product of their ages 5 years ago: $35 \times 15 = 525$. This matches the given information.

Conclusion

Based on the calculations and verification, the father's present age is 40 years.