The sum of the ages of father and a son presently is 70 years. After 10 years, the son's age is exactly half that of the father. What are their ages now?

Options

A. (a) 45 years, 25 years
B. (b) 50 years, 20 years
C. (c) 47 years, 23 years
D. (d) 50 years, 25 years

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the variables
Let:
- \( F \) = age of the father
- \( S \) = age of the son

### Step 2: Set up the equations
From the problem, we know:
1. The sum of their ages is 70 years:
\[ F + S = 70 \quad \text{(Equation 1)} \]

2. After 10 years, the son's age will be half of the father's age:
\[ S + 10 = \frac{1}{2}(F + 10) \quad \text{(Equation 2)} \]

### Step 3: Simplify Equation 2
Multiply both sides of Equation 2 by 2 to eliminate the fraction:
\[ 2(S + 10) = F + 10 \]
This simplifies to:
\[ 2S + 20 = F + 10 \]
Rearranging gives us:
\[ F - 2S = 10 \quad \text{(Equation 3)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( F + S = 70 \) (Equation 1)
2. \( F - 2S = 10 \) (Equation 3)

We can solve these equations simultaneously. Let's subtract Equation 1 from Equation 3:
\[ (F - 2S) - (F + S) = 10 - 70 \]
This simplifies to:
\[ -3S = -60 \]
Dividing both sides by -3 gives:
\[ S = 20 \]

### Step 5: Find the father's age
Now that we have the son's age, we can substitute \( S = 20 \) back into Equation 1 to find \( F \):
\[ F + 20 = 70 \]
Subtracting 20 from both sides gives:
\[ F = 50 \]

### Conclusion
The present ages are:
- Father's age \( F = 50 \) years
- Son's age \( S = 20 \) years

### Final Answer
The father is 50 years old and the son is 20 years old.

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