The sum of two numbers is 392 and the sum and difference of the LCM and the HCF is equal to 1288 and 1232 respectively. Find difference between these two numbers ?

Options

A. 96
B. 140
C. 104
D. 112

Answer

To solve the problem step by step, we will follow the given information and derive the necessary values.

### Step 1: Define the Variables
Let the two numbers be \( x \) and \( y \).

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of the two numbers:
\[ x + y = 392 \quad \text{(1)} \]
2. The sum of the LCM and HCF:
\[ \text{LCM} + \text{HCF} = 1288 \quad \text{(2)} \]
3. The difference of the LCM and HCF:
\[ \text{LCM} - \text{HCF} = 1232 \quad \text{(3)} \]

### Step 3: Solve for LCM and HCF
We can add equations (2) and (3) to eliminate HCF:
\[ (\text{LCM} + \text{HCF}) + (\text{LCM} - \text{HCF}) = 1288 + 1232 \]
This simplifies to:
\[ 2 \cdot \text{LCM} = 2520 \]
Thus, we find:
\[ \text{LCM} = \frac{2520}{2} = 1260 \quad \text{(4)} \]

Now, we can substitute the value of LCM back into equation (2) to find HCF:
\[ 1260 + \text{HCF} = 1288 \]
This gives us:
\[ \text{HCF} = 1288 - 1260 = 28 \quad \text{(5)} \]

### Step 4: Use the Relationship Between LCM, HCF, and the Two Numbers
We know from the properties of LCM and HCF that:
\[ x \cdot y = \text{LCM} \cdot \text{HCF} \]
Substituting the values from (4) and (5):
\[ x \cdot y = 1260 \cdot 28 \]
Calculating this gives:
\[ x \cdot y = 35280 \quad \text{(6)} \]

### Step 5: Solve the System of Equations
Now we have two equations:
1. \( x + y = 392 \) (from (1))
2. \( x \cdot y = 35280 \) (from (6))

We can express \( y \) in terms of \( x \):
\[ y = 392 - x \]
Substituting this into equation (6):
\[ x(392 - x) = 35280 \]
Expanding this gives:
\[ 392x - x^2 = 35280 \]
Rearranging leads to:
\[ x^2 - 392x + 35280 = 0 \]

### Step 6: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -392, c = 35280 \):
\[ x = \frac{392 \pm \sqrt{(-392)^2 - 4 \cdot 1 \cdot 35280}}{2 \cdot 1} \]
Calculating the discriminant:
\[ (-392)^2 = 153664 \]
\[ 4 \cdot 1 \cdot 35280 = 141120 \]
Thus:
\[ b^2 - 4ac = 153664 - 141120 = 14464 \]
Now, taking the square root:
\[ \sqrt{14464} = 120 \]
Substituting back into the formula:
\[ x = \frac{392 \pm 120}{2} \]
Calculating the two possible values for \( x \):
1. \( x = \frac{512}{2} = 256 \)
2. \( x = \frac{272}{2} = 136 \)

### Step 7: Find the Corresponding Values of \( y \)
Using \( y = 392 - x \):
1. If \( x = 256 \), then \( y = 392 - 256 = 136 \)
2. If \( x = 136 \), then \( y = 392 - 136 = 256 \)

### Step 8: Find the Difference
The difference between the two numbers \( x \) and \( y \) is:
\[ |x - y| = |256 - 136| = 120 \]

### Final Answer
The difference between the two numbers is \( 120 \).