If the ratio of two numbers is 2 : 3 and their L.C.M. is 54 then the sum of the two numbers is

Options

A. 5
B. 15
C. 45
D. 270

Answer

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

### Step 1: Understand the Ratio
The ratio of the two numbers is given as 2:3. This means we can express the two numbers in terms of a common variable \( x \):
- First number = \( 2x \)
- Second number = \( 3x \)

### Step 2: Use the LCM
We know that the LCM of the two numbers is 54. The LCM of two numbers in the ratio \( a:b \) can be expressed as:
\[ \text{LCM} = \frac{a \times b}{\text{GCD}(a, b)} \]
In our case, since the numbers are \( 2x \) and \( 3x \), the GCD of \( 2x \) and \( 3x \) is \( x \) (as 2 and 3 are co-prime). Therefore, we can write:
\[ \text{LCM}(2x, 3x) = \frac{(2x) \times (3x)}{x} = 6x \]

### Step 3: Set Up the Equation
Since we know the LCM is 54, we can set up the equation:
\[ 6x = 54 \]

### Step 4: Solve for \( x \)
To find \( x \), we divide both sides of the equation by 6:
\[ x = \frac{54}{6} = 9 \]

### Step 5: Find the Two Numbers
Now that we have \( x \), we can find the two numbers:
- First number = \( 2x = 2 \times 9 = 18 \)
- Second number = \( 3x = 3 \times 9 = 27 \)

### Step 6: Calculate the Sum
Finally, we calculate the sum of the two numbers:
\[ \text{Sum} = 18 + 27 = 45 \]

### Final Answer
The sum of the two numbers is **45**.

---