The sum of the ages of a father and his son is 45 years. Five years ago, the product of their was 34. The ages of the son and the father are respectively in years

Options

A. 6 and 39
B. 7 and 36
C. 9 and 36
D. 11 and 34

Answer

To solve the problem, we need to find the present ages of the father and the son based on the information given. Let's break it down step by step.

### Step 1: Define 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 45 years:
\[ F + S = 45 \quad \text{(Equation 1)} \]

2. Five years ago, the product of their ages was 34:
- Five years ago, the father's age would be \( F - 5 \) and the son's age would be \( S - 5 \).
- Therefore, the equation for their ages five years ago is:
\[ (F - 5)(S - 5) = 34 \quad \text{(Equation 2)} \]

### Step 3: Expand Equation 2
Expanding Equation 2:
\[ FS - 5F - 5S + 25 = 34 \]
Rearranging gives:
\[ FS - 5F - 5S = 34 - 25 \]
\[ FS - 5F - 5S = 9 \quad \text{(Equation 3)} \]

### Step 4: Substitute Equation 1 into Equation 3
From Equation 1, we can express \( S \) in terms of \( F \):
\[ S = 45 - F \]
Now substitute \( S \) into Equation 3:
\[ F(45 - F) - 5F - 5(45 - F) = 9 \]
Expanding this gives:
\[ 45F - F^2 - 5F - 225 + 5F = 9 \]
Simplifying:
\[ 45F - F^2 - 225 = 9 \]
\[ -F^2 + 45F - 234 = 0 \]
Multiplying through by -1 gives:
\[ F^2 - 45F + 234 = 0 \quad \text{(Equation 4)} \]

### Step 5: Solve the Quadratic Equation
Now we will solve the quadratic equation using the quadratic formula:
\[ F = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 1, b = -45, c = 234 \):
\[ F = \frac{45 \pm \sqrt{(-45)^2 - 4 \cdot 1 \cdot 234}}{2 \cdot 1} \]
Calculating the discriminant:
\[ F = \frac{45 \pm \sqrt{2025 - 936}}{2} \]
\[ F = \frac{45 \pm \sqrt{1089}}{2} \]
\[ F = \frac{45 \pm 33}{2} \]

Calculating the two possible values for \( F \):
1. \( F = \frac{78}{2} = 39 \)
2. \( F = \frac{12}{2} = 6 \)

### Step 6: Find Corresponding Values of \( S \)
Using \( F + S = 45 \):
1. If \( F = 39 \):
\[ S = 45 - 39 = 6 \]
2. If \( F = 6 \):
\[ S = 45 - 6 = 39 \]

### Conclusion
Thus, the present ages of the son and the father are:
- Son's age \( S = 6 \) years
- Father's age \( F = 39 \) years

### Final Answer
The ages of the son and the father are respectively 6 years and 39 years.

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