The product of the ages of Ram and Shyam is 280. If thrice the age of Shyam is more than Ram's age by 2 years, then what is the age of Shyam?

Options

A. 10 years
B. 15 years
C. 12 years
D. 18 years

Answer

To solve the problem, we need to find the ages of Ram and Shyam based on the given conditions. Let's denote Ram's age as \( R \) and Shyam's age as \( S \).

### Step 1: Set up the equations based on the problem statement.
1. The product of their ages is given as:
\[ R \times S = 280 \]

2. The second condition states that thrice Shyam's age is more than Ram's age by 2 years:
\[ 3S = R + 2 \]

### Step 2: Express one variable in terms of the other.
From the first equation, we can express \( S \) in terms of \( R \):
\[ S = \frac{280}{R} \]

### Step 3: Substitute \( S \) in the second equation.
Now, substitute \( S \) into the second equation:
\[ 3\left(\frac{280}{R}\right) = R + 2 \]

### Step 4: Simplify the equation.
Multiply both sides by \( R \) to eliminate the fraction:
\[ 840 = R^2 + 2R \]

### Step 5: Rearrange the equation.
Rearranging gives us a standard quadratic equation:
\[ R^2 + 2R - 840 = 0 \]

### Step 6: Factor the quadratic equation.
To factor the quadratic equation, we need two numbers that multiply to \(-840\) and add to \(2\). The numbers \(30\) and \(-28\) work:
\[ (R + 30)(R - 28) = 0 \]

### Step 7: Solve for \( R \).
Setting each factor to zero gives:
\[ R + 30 = 0 \quad \text{or} \quad R - 28 = 0 \]
Thus, \( R = -30 \) (not valid since age cannot be negative) or \( R = 28 \).

### Step 8: Find \( S \) using \( R \).
Now, substitute \( R = 28 \) back into the equation for \( S \):
\[ S = \frac{280}{28} = 10 \]

### Conclusion:
Shyam's age is \( 10 \) years.