Present Age of Father and Son

The question asks us to find the present age of the father based on two conditions relating his age and his son's age. Let's define variables for their present ages.

• Let the present age of the son be \(s\) years.
• Let the present age of the father be \(f\) years.

Setting Up Algebraic Equations

We are given two pieces of information, which we can translate into mathematical equations:

1. The present age of the father is the square of the age of his son.
   This can be written as: \(f = s^2\) (Equation 1)
2. After six years, the age of the father would be \(3\frac{1}{2}\) times the age of the son.
   After six years, the son's age will be \(s+6\) years, and the father's age will be \(f+6\) years.
   \(3\frac{1}{2}\) can be written as an improper fraction: \(3 + \frac{1}{2} = \frac{6}{2} + \frac{1}{2} = \frac{7}{2}\).
   So, the second condition is: \(f+6 = \frac{7}{2} (s+6)\)
   Multiplying both sides by 2 to remove the fraction: \(2(f+6) = 7(s+6)\)
   \(2f + 12 = 7s + 42\)
   Rearranging the terms to form a linear equation: \(2f - 7s = 42 - 12\)
   \(2f - 7s = 30\) (Equation 2)

Solving the System of Equations

We now have a system of two equations with two variables:

• \(f = s^2\)
• \(2f - 7s = 30\)

We can use the substitution method. Substitute the expression for \(f\) from Equation 1 into Equation 2:

\(2(s^2) - 7s = 30\)

\(2s^2 - 7s = 30\)

To solve this quadratic equation, we set it equal to zero:

\(2s^2 - 7s - 30 = 0\)

We can solve this quadratic equation by factoring. We look for two numbers that multiply to \(2 \times -30 = -60\) and add up to -7. These numbers are -12 and 5.

Rewrite the middle term using these numbers:

\(2s^2 - 12s + 5s - 30 = 0\)

Group terms and factor:

\((2s^2 - 12s) + (5s - 30) = 0\)

\(2s(s - 6) + 5(s - 6) = 0\)

Factor out the common binomial factor \((s - 6)\):

\((s - 6)(2s + 5) = 0\)

This gives two possible solutions for \(s\):

1. \(s - 6 = 0 \implies s = 6\)
2. \(2s + 5 = 0 \implies 2s = -5 \implies s = -\frac{5}{2}\)

Since age cannot be a negative value, \(s = 6\) is the only valid solution for the son's present age.

Calculating Father's Present Age

Now that we have the son's present age (\(s=6\)), we can find the father's present age using Equation 1 (\(f = s^2\)):

\(f = 6^2\)

\(f = 36\)

So, the present age of the father is 36 years.

Verification

Let's check if these ages satisfy the second condition.

• Present ages: Son = 6, Father = 36. (Father's age \(36\) is the square of son's age \(6\), \(6^2 = 36\). This checks out).
• Ages after 6 years: Son = \(6+6 = 12\) years, Father = \(36+6 = 42\) years.
• Is father's age \(3\frac{1}{2}\) times the son's age after 6 years?
  \(3\frac{1}{2} \times 12 = \frac{7}{2} \times 12 = 7 \times 6 = 42\).
  Yes, the father's age (42) is \(3\frac{1}{2}\) times the son's age (12) after 6 years.

Both conditions are satisfied with these ages.

The present age of the father is 36 years, which matches one of the given options.