The sum of the ages of Raju and Somu is 35 years. The product of their ages is 306. What are their present ages in years?

Options

A. 23 and 13
B. 18 and 17
C. 15 and 20
D. 16 and 19

Answer

To solve the problem, we need to find the ages of Raju and Somu given that their sum is 35 years and their product is 306. Let's denote Raju's age as \( x \) and Somu's age as \( 35 - x \).

### Step-by-Step Solution:

1. **Set Up the Equations**:
We know:
- The sum of their ages:
\[ x + (35 - x) = 35 \]
- The product of their ages:
\[ x(35 - x) = 306 \]

2. **Expand the Product Equation**:
Expanding the product equation gives:
\[ 35x - x^2 = 306 \]

3. **Rearrange the Equation**:
Rearranging this equation to bring all terms to one side results in:
\[ x^2 - 35x + 306 = 0 \]

4. **Factor the Quadratic Equation**:
We need to factor the quadratic equation. We are looking for two numbers that multiply to 306 and add up to 35. The numbers that satisfy these conditions are 18 and 17. Thus, we can factor the equation as:
\[ (x - 18)(x - 17) = 0 \]

5. **Find the Values of \( x \)**:
Setting each factor to zero gives us:
\[ x - 18 = 0 \quad \Rightarrow \quad x = 18 \]
\[ x - 17 = 0 \quad \Rightarrow \quad x = 17 \]

6. **Determine the Ages**:
- If \( x = 18 \), then Somu's age is:
\[ 35 - 18 = 17 \]
- If \( x = 17 \), then Somu's age is:
\[ 35 - 17 = 18 \]

Thus, Raju and Somu are 18 years and 17 years old, respectively.

### Conclusion:
The present ages of Raju and Somu are 18 years and 17 years.