The LCM and HCF of two positive numbers are 175 and 5 respectively. If the sum of the numbers is 60 what is the difference between them?

Answer

To solve the problem step by step, we will use the information provided about the LCM, HCF, and the sum of the two numbers.

### Step-by-Step Solution:

1. **Identify the given values**:
- LCM (Least Common Multiple) = 175
- HCF (Highest Common Factor) = 5
- Sum of the two numbers (x + y) = 60

2. **Use the relationship between LCM, HCF, and the product of two numbers**:
The product of the two numbers can be calculated using the formula:
\[ \text{Product of numbers} = \text{HCF} \times \text{LCM} \]
Substituting the given values:
\[ \text{Product of numbers} = 5 \times 175 = 875 \]

3. **Set up the equations**:
Let the two numbers be \( x \) and \( y \).
From the given information, we have:
\[ x + y = 60 \quad \text{(1)} \]
\[ x \times y = 875 \quad \text{(2)} \]

4. **Express one variable in terms of the other**:
From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 60 - x \]

5. **Substitute \( y \) in equation (2)**:
Substitute \( y \) in equation (2):
\[ x \times (60 - x) = 875 \]
Expanding this gives:
\[ 60x - x^2 = 875 \]
Rearranging this into standard quadratic form:
\[ x^2 - 60x + 875 = 0 \]

6. **Solve the quadratic equation**:
We will use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -60 \), and \( c = 875 \).
First, calculate the discriminant:
\[ b^2 - 4ac = (-60)^2 - 4 \times 1 \times 875 = 3600 - 3500 = 100 \]
Now, substitute into the quadratic formula:
\[ x = \frac{60 \pm \sqrt{100}}{2 \times 1} = \frac{60 \pm 10}{2} \]
This gives us two possible values for \( x \):
\[ x = \frac{70}{2} = 35 \quad \text{or} \quad x = \frac{50}{2} = 25 \]

7. **Find the corresponding values of \( y \)**:
If \( x = 35 \):
\[ y = 60 - 35 = 25 \]
If \( x = 25 \):
\[ y = 60 - 25 = 35 \]

8. **Calculate the difference between the two numbers**:
The difference between the two numbers \( |x - y| \):
\[ |35 - 25| = 10 \]

### Final Answer:
The difference between the two numbers is **10**.