The LCM of two numbers is 495 and their HCF is 5.If sum of the numbers is 100,then find the difference of the numbers

Options

A. 10
B. 546
C. 70
D. 90

Answer

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

### Step 1: Define the variables
Let the first number be \( x \). Since the sum of the two numbers is 100, the second number will be \( 100 - x \).

**Hint:** Remember that if the sum of two numbers is known, you can express one number in terms of the other.

### Step 2: Use the relationship between LCM, HCF, and the numbers
We know that:
\[ \text{LCM} \times \text{HCF} = \text{Product of the two numbers} \]
Given that the LCM is 495 and the HCF is 5, we can write:
\[ 495 \times 5 = x \times (100 - x) \]

**Hint:** This formula is crucial for relating the LCM and HCF to the numbers.

### Step 3: Calculate the product
Calculate \( 495 \times 5 \):
\[ 495 \times 5 = 2475 \]
So, we have:
\[ 2475 = x \times (100 - x) \]

**Hint:** Make sure to perform the multiplication correctly to avoid errors.

### Step 4: Rearrange the equation
Rearranging gives us:
\[ x(100 - x) = 2475 \]
Expanding this, we get:
\[ 100x - x^2 = 2475 \]
Rearranging further, we have:
\[ x^2 - 100x + 2475 = 0 \]

**Hint:** This is a standard quadratic equation format.

### Step 5: Solve the quadratic equation
To solve the quadratic equation \( x^2 - 100x + 2475 = 0 \), we can factor it. We need two numbers that multiply to 2475 and add up to 100. The numbers are 45 and 55.

Thus, we can write:
\[ (x - 45)(x - 55) = 0 \]

**Hint:** Look for pairs of factors that meet the conditions of the problem.

### Step 6: Find the values of \( x \)
Setting each factor to zero gives us:
1. \( x - 45 = 0 \) → \( x = 45 \)
2. \( x - 55 = 0 \) → \( x = 55 \)

**Hint:** Each solution represents one of the two numbers.

### Step 7: Determine the two numbers
If \( x = 45 \), then the second number is:
\[ 100 - 45 = 55 \]
If \( x = 55 \), then the second number is:
\[ 100 - 55 = 45 \]

**Hint:** The two numbers are interchangeable, so you can find the second number easily.

### Step 8: Calculate the difference
Now, we find the difference between the two numbers:
\[ |55 - 45| = 10 \]

**Hint:** The absolute value ensures the difference is always positive.

### Conclusion
The difference of the two numbers is \( 10 \).

**Final Answer:** The difference of the numbers is \( 10 \).