The age of father is twice that of the elder son. After ten years, the age of father will be three times that of the younger son. If the difference of ages of the two sons is 15 years, the age of the father is ?

Options

A. 70 years
B. 55 years
C. 50 years
D. 60 years

Explanation

To solve the problem step by step, we will define variables for the ages of the father and the two sons and set up equations based on the information provided in the question.

### Step 1: Define Variables
Let:
- \( x \) = age of the elder son
- \( 2x \) = age of the father (since the father's age is twice that of the elder son)

### Step 2: Determine Ages After 10 Years
After 10 years:
- Age of the elder son = \( x + 10 \)
- Age of the father = \( 2x + 10 \)

### Step 3: Set Up the Equation for the Younger Son
According to the problem, after 10 years, the father's age will be three times the age of the younger son. Let the age of the younger son be \( y \). Therefore, we can write:
\[ 2x + 10 = 3y \]
From this, we can express \( y \) in terms of \( x \):
\[ y = \frac{2x + 10}{3} \]

### Step 4: Use the Age Difference
The problem states that the difference in ages between the two sons is 15 years:
\[ x - y = 15 \]
Substituting the expression for \( y \) from the previous step:
\[ x - \frac{2x + 10}{3} = 15 \]

### Step 5: Solve the Equation
To eliminate the fraction, multiply the entire equation by 3:
\[ 3x - (2x + 10) = 45 \]
This simplifies to:
\[ 3x - 2x - 10 = 45 \]
\[ x - 10 = 45 \]
Adding 10 to both sides gives:
\[ x = 55 \]

### Step 6: Calculate the Father's Age
Now that we have the elder son's age, we can find the father's age:
\[ \text{Father's age} = 2x = 2 \times 55 = 110 \]

### Step 7: Check the Conditions
1. The elder son is 55 years old.
2. The father's age is \( 110 \) years.
3. The younger son's age can be calculated as:
\[ y = \frac{2(55) + 10}{3} = \frac{110 + 10}{3} = \frac{120}{3} = 40 \]
4. The difference in ages between the two sons is:
\[ 55 - 40 = 15 \] (which is correct)

### Conclusion
The age of the father is \( 110 \) years.