A man is 24 years older than his son. In 2 years, his age will be twice the age of his son. Find their present ages.

Answer

To solve the problem step by step, we will define the variables, set up the equations based on the information given, and then solve for the present ages of the man and his son.

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

### Step 2: Set Up the First Equation
According to the problem, the man is 24 years older than his son. This can be expressed as:
\[ x = y + 24 \]
(Equation 1)

### Step 3: Set Up the Second Equation
In 2 years, the man's age will be twice the age of his son. In 2 years, the man’s age will be \( x + 2 \) and the son’s age will be \( y + 2 \). This gives us the second equation:
\[ x + 2 = 2(y + 2) \]
Expanding this, we get:
\[ x + 2 = 2y + 4 \]
Rearranging gives us:
\[ x = 2y + 2 \]
(Equation 2)

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( x = y + 24 \) (Equation 1)
2. \( x = 2y + 2 \) (Equation 2)

We can set these two expressions for \( x \) equal to each other:
\[ y + 24 = 2y + 2 \]

### Step 5: Rearranging the Equation
Rearranging gives us:
\[ 24 - 2 = 2y - y \]
This simplifies to:
\[ 22 = y \]

### Step 6: Find the Man's Age
Now that we have \( y \) (the son's age), we can find \( x \) (the man's age) using Equation 1:
\[ x = y + 24 \]
Substituting \( y = 22 \):
\[ x = 22 + 24 = 46 \]

### Conclusion
The present ages are:
- The son's age \( y = 22 \) years
- The man's age \( x = 46 \) years