The age of the father 2 years ago was 6 times the age of his son. If 18 years hence his age will be 2 times that of his son, what are their present ages ?

Options

A. 34 years, 9 years
B. 36 years, 11 years
C. 35 years, 7 years
D. 32 years, 7 years

Answer

To solve the problem step by step, we will define the present ages of the father and son and set up equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = present age of the father
- \( y \) = present age of the son

### Step 2: Set Up the First Equation
According to the problem, 2 years ago, the father's age was 6 times the son's age. Therefore, we can express this as:
\[ x - 2 = 6(y - 2) \]
Expanding this equation gives:
\[ x - 2 = 6y - 12 \]
Rearranging it, we get:
\[ x - 6y = -10 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that 18 years from now, the father's age will be twice that of the son's age. This can be expressed as:
\[ x + 18 = 2(y + 18) \]
Expanding this equation gives:
\[ x + 18 = 2y + 36 \]
Rearranging it, we get:
\[ x - 2y = 18 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( x - 6y = -10 \)
2. \( x - 2y = 18 \)

We can solve these equations simultaneously. Start by isolating \( x \) in Equation 2:
\[ x = 2y + 18 \]

### Step 5: Substitute \( x \) in Equation 1
Now substitute \( x \) from Equation 2 into Equation 1:
\[ (2y + 18) - 6y = -10 \]
Simplifying this gives:
\[ 2y + 18 - 6y = -10 \]
\[ -4y + 18 = -10 \]
Subtracting 18 from both sides:
\[ -4y = -28 \]
Dividing by -4:
\[ y = 7 \]

### Step 6: Find \( x \)
Now that we have \( y \), we can find \( x \) using Equation 2:
\[ x = 2(7) + 18 \]
Calculating this gives:
\[ x = 14 + 18 = 32 \]

### Conclusion
The present ages are:
- Father's age \( x = 32 \) years
- Son's age \( y = 7 \) years

### Summary of Steps
1. Define variables for the ages.
2. Set up the first equation based on the age condition 2 years ago.
3. Set up the second equation based on the age condition 18 years hence.
4. Solve the equations simultaneously.
5. Substitute to find the ages.