The ratio of two numbers are 3:4 and their L.C.M is 84,then which number is greater number of the following ?

Options

A. 21
B. 24
C. 28
D. 84

Answer

To solve the problem, we need to find the two numbers based on the given ratio and LCM. Let's break it down step by step.

### Step 1: Understand the Ratio
The ratio of the two numbers is given as 3:4. This means we can represent the two numbers as:
- First number = 3x
- Second number = 4x
where x is a common multiplier.

### Step 2: Use the LCM
We are also given that the LCM of these two numbers is 84. The formula for the LCM of two numbers in terms of their product and GCD (Greatest Common Divisor) is:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{GCD}(a, b)} \]

### Step 3: Calculate the Product of the Numbers
The product of the two numbers can be expressed as:
\[ (3x) \times (4x) = 12x^2 \]

### Step 4: Relate LCM and Product
Since we know the LCM is 84, we can set up the equation:
\[ \frac{12x^2}{\text{GCD}(3x, 4x)} = 84 \]

### Step 5: Find the GCD
The GCD of 3x and 4x is x (since 3 and 4 are coprime). Thus, we can rewrite the equation:
\[ \frac{12x^2}{x} = 84 \]
This simplifies to:
\[ 12x = 84 \]

### Step 6: Solve for x
Now, we can solve for x:
\[ x = \frac{84}{12} = 7 \]

### Step 7: Find the Two Numbers
Now that we have x, we can find the two numbers:
- First number = 3x = 3 * 7 = 21
- Second number = 4x = 4 * 7 = 28

### Step 8: Identify the Greater Number
Comparing the two numbers:
- 21 and 28

The greater number is **28**.

### Final Answer
The greater number is **28**.

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