If two numbers are in the ratio `2:3` and the product of their HCF and LCM is 33750 , then the sum of the numbers is

Options

A. 250
B. 425
C. 325
D. 375

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Numbers
Given the ratio of two numbers is 2:3, we can express the two numbers as:
- First number = 2x
- Second number = 3x

### Step 2: Use the Relationship Between HCF, LCM, and Product of 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. Therefore, we can write:
\[ \text{HCF} \times \text{LCM} = \text{First number} \times \text{Second number} \]

Substituting the expressions for the first and second numbers:
\[ \text{HCF} \times \text{LCM} = (2x) \times (3x) \]
\[ \text{HCF} \times \text{LCM} = 6x^2 \]

### Step 3: Set Up the Equation
According to the problem, the product of the HCF and LCM is given as 33750. Therefore, we can set up the equation:
\[ 6x^2 = 33750 \]

### Step 4: Solve for x²
To find \( x^2 \), we divide both sides of the equation by 6:
\[ x^2 = \frac{33750}{6} \]
\[ x^2 = 5625 \]

### Step 5: Solve for x
Now, we take the square root of both sides to find x:
\[ x = \sqrt{5625} \]
\[ x = 75 \]

### Step 6: Find the Two Numbers
Now that we have the value of x, we can find the two numbers:
- First number = \( 2x = 2 \times 75 = 150 \)
- Second number = \( 3x = 3 \times 75 = 225 \)

### Step 7: Calculate the Sum of the Numbers
Finally, we can find the sum of the two numbers:
\[ \text{Sum} = 150 + 225 = 375 \]

### Final Answer
The sum of the numbers is **375**.

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