Pramod and Praveen are the sons of Prajwal . The present age of Prajwal is 4 time the age of Pramod and 6 time the age of Praveen. If the sum of their ages is equal to 51 yr, then the present ages of sons are .

Options

A. 9 yr and 6yr
B. 8 yr and 9 yr
C. 9 yr and 4yr
D. 5 yr and 9 yr

Answer

To solve the problem step by step, we will denote the ages of Prajwal, Pramod, and Praveen using algebraic expressions based on the information given.

### Step 1: Define Variables
Let the present age of Prajwal be \( x \).

### Step 2: Express the Ages of Pramod and Praveen
According to the problem:
- The present age of Pramod is \( \frac{x}{4} \) (since Prajwal's age is 4 times Pramod's age).
- The present age of Praveen is \( \frac{x}{6} \) (since Prajwal's age is 6 times Praveen's age).

### Step 3: Set Up the Equation
The sum of their ages is given to be 51 years. Therefore, we can write the equation:
\[ x + \frac{x}{4} + \frac{x}{6} = 51 \]

### Step 4: Find a Common Denominator
To solve this equation, we need a common denominator for the fractions. The least common multiple (LCM) of 4 and 6 is 12. We will multiply each term by 12 to eliminate the fractions:
\[ 12x + 12 \cdot \frac{x}{4} + 12 \cdot \frac{x}{6} = 12 \cdot 51 \]
This simplifies to:
\[ 12x + 3x + 2x = 612 \]

### Step 5: Combine Like Terms
Now, combine the terms on the left side:
\[ (12x + 3x + 2x) = 17x \]
Thus, we have:
\[ 17x = 612 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 17 to find \( x \):
\[ x = \frac{612}{17} = 36 \]

### Step 7: Calculate the Ages of Pramod and Praveen
Now that we have \( x \), we can find the ages of Pramod and Praveen:
- Age of Pramod:
\[ \text{Age of Pramod} = \frac{x}{4} = \frac{36}{4} = 9 \text{ years} \]
- Age of Praveen:
\[ \text{Age of Praveen} = \frac{x}{6} = \frac{36}{6} = 6 \text{ years} \]

### Final Answer
The present ages of Pramod and Praveen are:
- Pramod: 9 years
- Praveen: 6 years