The LCM of two numbers is 210 and their ratio is `2:3`. The sum of these numbers is

Options

A. 210
B. 175
C. 315
D. can't be determined

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the two numbers be represented as \(2x\) and \(3x\), based on the given ratio of \(2:3\).

### Step 2: Find the HCF
The highest common factor (HCF) of \(2x\) and \(3x\) is \(x\), since \(2\) and \(3\) are prime numbers and do not have any common factors other than \(1\).

### Step 3: Use the LCM Formula
We know that the product of two numbers is equal to the product of their HCF and LCM. Thus, we can write the equation:
\[ \text{Product of two numbers} = \text{HCF} \times \text{LCM} \]
Substituting the values we have:
\[ (2x) \times (3x) = x \times 210 \]

### Step 4: Simplify the Equation
This simplifies to:
\[ 6x^2 = 210x \]

### Step 5: Divide by x (assuming \(x \neq 0\))
We can divide both sides by \(x\):
\[ 6x = 210 \]

### Step 6: Solve for x
Now, we solve for \(x\):
\[ x = \frac{210}{6} = 35 \]

### Step 7: Calculate the Two Numbers
Now that we have \(x\), we can find the two numbers:
- First number:
\[ 2x = 2 \times 35 = 70 \]
- Second number:
\[ 3x = 3 \times 35 = 105 \]

### Step 8: Find the Sum of the Two Numbers
Finally, we find the sum of the two numbers:
\[ 70 + 105 = 175 \]

### Final Answer
The sum of the two numbers is \(175\).

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