The H.C.F and L.C.M of two numbers are 21 and 84 respectively. If the ratio of the two numbers is `1: 4`, then the larger of the two numbers is:

Options

A. 48
B. 12
C. 84
D. 108

Answer

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

### Step 1: Understand the relationship between HCF, LCM, and the two numbers
The relationship between the HCF (Highest Common Factor), LCM (Lowest Common Multiple), and the two numbers (let's call them N1 and N2) is given by the formula:
\[ \text{HCF} \times \text{LCM} = N1 \times N2 \]

### Step 2: Assign the values
From the question, we know:
- HCF = 21
- LCM = 84
- Ratio of N1 to N2 = 1:4

### Step 3: Express the numbers in terms of a variable
Let:
- N1 = x (the smaller number)
- N2 = 4x (the larger number, since the ratio is 1:4)

### Step 4: Substitute into the relationship formula
Using the relationship from Step 1:
\[ 21 \times 84 = N1 \times N2 \]
Substituting N1 and N2:
\[ 21 \times 84 = x \times 4x \]
This simplifies to:
\[ 21 \times 84 = 4x^2 \]

### Step 5: Calculate the left side
Now, calculate \( 21 \times 84 \):
\[ 21 \times 84 = 1764 \]
So we have:
\[ 1764 = 4x^2 \]

### Step 6: Solve for x
Now, divide both sides by 4:
\[ x^2 = \frac{1764}{4} \]
Calculating the right side:
\[ x^2 = 441 \]
Now, take the square root of both sides:
\[ x = \sqrt{441} \]
Thus:
\[ x = 21 \]

### Step 7: Find N2
Now that we have x, we can find N2:
\[ N2 = 4x = 4 \times 21 = 84 \]

### Conclusion
The larger of the two numbers is:
\[ \text{Larger number (N2)} = 84 \]

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