The sum of the ages of father and son is 50 years. Eight years ago, the product of their ages was two time the father's age at that time, then the present ages (in years) of the father and son respectively are

Options

A. A)`39,6`
B. B)`35,10`
C. C)`36,9`
D. D)`40,10`

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information given.

### Step 1: Define Variables
Let:
- \( f \) = Father's current age
- \( s \) = Son's current age

### Step 2: Set Up the First Equation
According to the problem, the sum of the ages of the father and son is 50 years:
\[ f + s = 50 \]

### Step 3: Set Up the Second Equation
Eight years ago, the father's age was \( f - 8 \) and the son's age was \( s - 8 \). The problem states that the product of their ages at that time was two times the father's age at that time:
\[ (f - 8)(s - 8) = 2(f - 8) \]

### Step 4: Substitute \( s \)
From the first equation, we can express \( s \) in terms of \( f \):
\[ s = 50 - f \]

### Step 5: Substitute \( s \) into the Second Equation
Now, substitute \( s \) into the second equation:
\[ (f - 8)((50 - f) - 8) = 2(f - 8) \]
This simplifies to:
\[ (f - 8)(42 - f) = 2(f - 8) \]

### Step 6: Expand the Equation
Expanding both sides:
\[ f \cdot 42 - f^2 - 8 \cdot 42 + 8f = 2f - 16 \]
This simplifies to:
\[ 42f - f^2 - 336 + 8f = 2f - 16 \]
Combining like terms gives:
\[ 50f - f^2 - 336 = 2f - 16 \]

### Step 7: Rearranging the Equation
Rearranging gives us:
\[ -f^2 + 50f - 2f - 336 + 16 = 0 \]
This simplifies to:
\[ -f^2 + 48f - 320 = 0 \]
Multiplying through by -1 gives:
\[ f^2 - 48f + 320 = 0 \]

### Step 8: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ (f - 8)(f - 40) = 0 \]

### Step 9: Solve for \( f \)
Setting each factor to zero gives us:
1. \( f - 8 = 0 \) → \( f = 8 \)
2. \( f - 40 = 0 \) → \( f = 40 \)

### Step 10: Find Corresponding \( s \)
Now we will find the corresponding \( s \) values:
1. If \( f = 8 \):
\[ s = 50 - 8 = 42 \quad (\text{Not valid, as son's age cannot be greater than father's}) \]
2. If \( f = 40 \):
\[ s = 50 - 40 = 10 \quad (\text{Valid}) \]

### Conclusion
Thus, the present ages of the father and son are:
- Father's age = 40 years
- Son's age = 10 years

### Final Answer
The present ages of the father and son are \( 40 \) years and \( 10 \) years, respectively.

---