Eighteen years ago, a father was three times as old as his son. Now the father is only twice as old as his son. Then the sum of the present ages of the son and the father is

Options

A. 32 years
B. 72 years
C. 105 years
D. 108 years

Answer

To solve the problem, we need to establish the present ages of the father and son based on the information provided. Let's denote the present age of the son as \( S \) and the present age of the father as \( F \).

### Step 1: Establish the equations based on the information given.

1. **Eighteen years ago**, the father was three times as old as his son:
- Father's age 18 years ago: \( F - 18 \)
- Son's age 18 years ago: \( S - 18 \)
- According to the problem:
\[ F - 18 = 3(S - 18) \]

2. **Now**, the father is twice as old as his son:
- According to the problem:
\[ F = 2S \]

### Step 2: Substitute the second equation into the first equation.

Now we can substitute \( F \) from the second equation into the first equation:
\[ 2S - 18 = 3(S - 18) \]

### Step 3: Simplify the equation.

Expanding the equation:
\[ 2S - 18 = 3S - 54 \]

Now, rearranging the equation to isolate \( S \):
\[ 2S - 3S = -54 + 18 \]
\[ -S = -36 \]
\[ S = 36 \]

### Step 4: Find the father's present age.

Now that we have the son's age, we can find the father's age using the second equation:
\[ F = 2S = 2 \times 36 = 72 \]

### Step 5: Calculate the sum of their present ages.

Now, we can find the sum of the present ages of the son and the father:
\[ \text{Sum} = S + F = 36 + 72 = 108 \]

### Final Answer:
The sum of the present ages of the son and the father is **108 years**.

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