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 will follow these instructions:

### Step 1: Understand the Ratio
The two numbers are in the ratio of 3:4. We can express these two numbers as:
- First number (N1) = 3k
- Second number (N2) = 4k
Where k is a common factor.

**Hint:** The ratio tells us how the two numbers relate to each other in terms of a common multiplier.

### Step 2: Use the LCM Formula
We know that the LCM (Least Common Multiple) of these two numbers is given as 240. The LCM of two numbers in the form of 3k and 4k can be calculated as:
\[ \text{LCM}(3k, 4k) = k \times \text{LCM}(3, 4) \]

**Hint:** The LCM of two numbers can be calculated by multiplying the common factor (k) with the LCM of the coefficients (3 and 4).

### Step 3: Calculate LCM of 3 and 4
The LCM of 3 and 4 is:
\[ \text{LCM}(3, 4) = 12 \]
Thus, we can rewrite the LCM of our numbers:
\[ \text{LCM}(3k, 4k) = k \times 12 \]

**Hint:** Finding the LCM of small numbers can often be done by listing their multiples or using the prime factorization method.

### Step 4: Set Up the Equation
Now, we can set up the equation based on the information given:
\[ k \times 12 = 240 \]

**Hint:** Setting up an equation helps us relate the known value (LCM) to the unknown (k).

### Step 5: Solve for k
To find k, divide both sides of the equation by 12:
\[ k = \frac{240}{12} = 20 \]

**Hint:** When solving for a variable, performing inverse operations (like division) helps isolate the variable.

### Step 6: Find the Two Numbers
Now that we have k, we can find the two numbers:
- First number (N1) = 3k = 3 × 20 = 60
- Second number (N2) = 4k = 4 × 20 = 80

**Hint:** Substitute the value of k back into the expressions for the numbers to find their actual values.

### Step 7: Identify the Smaller Number
The smaller of the two numbers is:
\[ \text{Smaller number} = 60 \]

**Hint:** Comparing the two calculated numbers will help you identify which one is smaller.

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