The sum of the ages of father and son at present is 33 years. Two years ago the product of their ages was 28 years. What is the age of the father and the son? (in years)

Options

A. 26.7
B. 30.3
C. 29.4
D. 32.1

Answer

To solve the problem step by step, we can follow these steps:

### Step 1: Define the Variables
Let the present age of the father be \( F \) and the present age of the son be \( S \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of their ages is 33 years:
\[ F + S = 33 \quad \text{(Equation 1)} \]
2. Two years ago, the product of their ages was 28 years:
\[ (F - 2)(S - 2) = 28 \quad \text{(Equation 2)} \]

### Step 3: Expand Equation 2
Expanding Equation 2 gives:
\[ FS - 2F - 2S + 4 = 28 \]
Rearranging this, we have:
\[ FS - 2F - 2S = 24 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 1 into Equation 3
From Equation 1, we can express \( S \) in terms of \( F \):
\[ S = 33 - F \]
Now substitute \( S \) into Equation 3:
\[ F(33 - F) - 2F - 2(33 - F) = 24 \]
Expanding this gives:
\[ 33F - F^2 - 2F - 66 + 2F = 24 \]
Simplifying it results in:
\[ 33F - F^2 - 66 = 24 \]
Rearranging gives:
\[ F^2 - 33F + 90 = 0 \quad \text{(Equation 4)} \]

### Step 5: Solve the Quadratic Equation
We can solve Equation 4 using the quadratic formula:
\[ F = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -33 \), and \( c = 90 \):
\[ F = \frac{33 \pm \sqrt{(-33)^2 - 4 \cdot 1 \cdot 90}}{2 \cdot 1} \]
Calculating the discriminant:
\[ F = \frac{33 \pm \sqrt{1089 - 360}}{2} \]
\[ F = \frac{33 \pm \sqrt{729}}{2} \]
\[ F = \frac{33 \pm 27}{2} \]
Calculating the two possible values for \( F \):
1. \( F = \frac{60}{2} = 30 \)
2. \( F = \frac{6}{2} = 3 \)

### Step 6: Find the Corresponding Age of the Son
Using \( F = 30 \):
\[ S = 33 - F = 33 - 30 = 3 \]
Using \( F = 3 \):
\[ S = 33 - F = 33 - 3 = 30 \]
Thus, the ages are:
- Father: 30 years, Son: 3 years

### Conclusion
The present age of the father is 30 years and the present age of the son is 3 years.

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