What is the age of father, if the sum of the ages of a father and his son in years is 65 and twice the difference of their ages in years is 50?

Options

A. 40 years
B. 45 years
C. 55 years
D. 65 years

Answer

To solve the problem, we will set up equations based on the information given and then solve for the father's age.

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

### Step 2: Set Up the Equations
From the problem, we have two pieces of information:
1. The sum of their ages is 65:
\[ x + y = 65 \quad \text{(Equation 1)} \]
2. Twice the difference of their ages is 50:
\[ 2(x - y) = 50 \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
We can simplify Equation 2:
\[ x - y = 25 \quad \text{(Divide both sides by 2)} \]

### Step 4: Write the System of Equations
Now we have the following system of equations:
1. \( x + y = 65 \)
2. \( x - y = 25 \)

### Step 5: Solve the System of Equations
We will use the elimination method to solve these equations. We can add both equations together:
\[ (x + y) + (x - y) = 65 + 25 \]
This simplifies to:
\[ 2x = 90 \]
Now, divide both sides by 2:
\[ x = 45 \]

### Step 6: Find the Age of the Son
Now that we have \( x \), we can find \( y \) using Equation 1:
\[ 45 + y = 65 \]
Subtract 45 from both sides:
\[ y = 20 \]

### Conclusion
The age of the father is \( x = 45 \) years.