Two numbers are in the ratio `3 :4` . If their LCM is 240 , the smaller of the two number is

Options

A. 100
B. 80
C. 60
D. 50

Answer

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

### Step 1: Understand the Ratio
The two numbers are in the ratio of 3:4. This means we can express the two numbers in terms of a variable \( x \):
- First number = \( 3x \)
- Second number = \( 4x \)

### Step 2: Use the Given LCM
We are given that the Least Common Multiple (LCM) of these two numbers is 240. The formula for the LCM of two numbers \( a \) and \( b \) that are expressed in terms of their ratio and a common variable is:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)} \]
For our numbers \( 3x \) and \( 4x \):
- The product \( a \times b = (3x) \times (4x) = 12x^2 \)
- The GCD of \( 3x \) and \( 4x \) is \( x \) (since 3 and 4 are coprime).

Thus, we can express the LCM as:
\[ \text{LCM}(3x, 4x) = \frac{12x^2}{x} = 12x \]

### Step 3: Set Up the Equation
According to the problem, the LCM is given as 240:
\[ 12x = 240 \]

### Step 4: Solve for \( x \)
To find \( x \), we divide both sides of the equation by 12:
\[ x = \frac{240}{12} = 20 \]

### Step 5: Calculate the Numbers
Now that we have \( x \), we can find the two numbers:
- First number = \( 3x = 3 \times 20 = 60 \)
- Second number = \( 4x = 4 \times 20 = 80 \)

### Step 6: Identify the Smaller Number
The smaller of the two numbers is:
\[ \text{Smaller number} = 60 \]

### Final Answer
The smaller of the two numbers is **60**.

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