Two numbers are in the ratio of 5 : 7. The product of their LCM and HCF is 12635. then the sum of the numbers will be.

Options

A. 252
B. 228
C. 304
D. 380

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Ratio**:
The two numbers are in the ratio of 5:7. We can express these numbers as:
- First number = \( 5x \)
- Second number = \( 7x \)

2. **Using the LCM and HCF Relationship**:
We know from the problem that the product of the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the two numbers is given as 12635. The relationship between the LCM, HCF, and the two numbers can be expressed as:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]
Therefore, we can write:
\[ \text{LCM} \times \text{HCF} = (5x) \times (7x) = 35x^2 \]

3. **Setting Up the Equation**:
From the above relationship, we can set up the equation:
\[ 35x^2 = 12635 \]

4. **Solving for \( x^2 \)**:
To find \( x^2 \), we divide both sides of the equation by 35:
\[ x^2 = \frac{12635}{35} \]
Now, calculating the right-hand side:
\[ x^2 = 361 \]

5. **Finding \( x \)**:
To find \( x \), we take the square root of both sides:
\[ x = \sqrt{361} = 19 \]

6. **Calculating the Numbers**:
Now that we have the value of \( x \), we can find the two numbers:
- First number = \( 5x = 5 \times 19 = 95 \)
- Second number = \( 7x = 7 \times 19 = 133 \)

7. **Finding the Sum of the Numbers**:
Finally, we calculate the sum of the two numbers:
\[ \text{Sum} = 95 + 133 = 228 \]

### Final Answer:
The sum of the two numbers is **228**.