Two numbers are in the ratio 3 : 4. The product of their HCF and LCM is 2028. The sum of the numbers is:

Options

A. 68
B. 72
C. 86
D. 91

Answer

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

### Step 1: Understand the ratio of the two numbers
The two numbers are in the ratio 3:4. We can express these numbers in terms of a variable \( k \):
- Let the first number \( N_1 = 3k \)
- Let the second number \( N_2 = 4k \)

### Step 2: Use the relationship between HCF, LCM, and the numbers
We know that the product of the HCF (Highest Common Factor) and LCM (Least Common Multiple) of two numbers is equal to the product of the numbers themselves:
\[ \text{HCF} \times \text{LCM} = N_1 \times N_2 \]
Given that the product of the HCF and LCM is 2028, we can write:
\[ N_1 \times N_2 = 2028 \]

### Step 3: Substitute the values of \( N_1 \) and \( N_2 \)
Substituting \( N_1 \) and \( N_2 \) into the equation:
\[ (3k) \times (4k) = 2028 \]
This simplifies to:
\[ 12k^2 = 2028 \]

### Step 4: Solve for \( k^2 \)
To find \( k^2 \), divide both sides by 12:
\[ k^2 = \frac{2028}{12} \]
Calculating the right side:
\[ k^2 = 169 \]

### Step 5: Find the value of \( k \)
Now, take the square root of both sides to find \( k \):
\[ k = \sqrt{169} = 13 \]

### Step 6: Calculate the values of \( N_1 \) and \( N_2 \)
Now that we have \( k \), we can find the values of \( N_1 \) and \( N_2 \):
- \( N_1 = 3k = 3 \times 13 = 39 \)
- \( N_2 = 4k = 4 \times 13 = 52 \)

### Step 7: Find the sum of the two numbers
Now, we can find the sum of the two numbers:
\[ \text{Sum} = N_1 + N_2 = 39 + 52 = 91 \]

### Final Answer
The sum of the two numbers is **91**.