Age Calculation: Father and Son Problem

This solution details the steps to find the present ages of a father and son using algebra based on the given conditions.

Setting up the Age Equations

Let the present age of the father be $F$ years$ and the present age of the son be $S$ years$.

• Nine years ago:
  Father's age was $F - 9$.
  Son's age was $S - 9$.
  Given: $F - 9 = 9(S - 9)$.
• Ten years hence:
  Father's age will be $F + 10$.
  Son's age will be $S + 10$.
  Given: $F + 10 = 2(S + 10) + 2$.

Solving the System of Equations

We need to solve the two equations simultaneously:

1. Simplify the first equation:
   $F - 9 = 9S - 81$
   $F = 9S - 72$ (Equation 1)
2. Simplify the second equation:
   $F + 10 = 2S + 20 + 2$
   $F + 10 = 2S + 22$
   $F = 2S + 12$ (Equation 2)
3. Equate Equation 1 and Equation 2:
   Substitute the expression for $F$ from Equation 2 into Equation 1 (or vice versa):
   $9S - 72 = 2S + 12$
   $9S - 2S = 12 + 72$
   $7S = 84$
   $S = \frac{84}{7}$
   $S = 12$
4. Find the father's age:
   Substitute the value of $S = 12$ into Equation 2:
   $F = 2(12) + 12$
   $F = 24 + 12$
   $F = 36$

Conclusion: The present age of the father is 36 years and the present age of the son is 12 years.