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 ?

To solve this problem, we use the relationship between the LCM, HCF, and the product of two numbers. The formula is:

**LCM × HCF = Product of the two numbers.**

Let the two numbers be $a$ and $b$. Given:
- LCM = 175
- HCF = 5
- $a+b=60$

### Step 1: Use the formula for the product of the numbers
From the formula:
$$\text{LCM}\times \text{HCF}=a\times b$$

Substituting the given values:
$$175\times 5=a\times b$$
$$a\times b=875$$

### Step 2: Represent the numbers in terms of their HCF
Since the HCF of the numbers is 5, the two numbers can be expressed as:
$$a=5m\text{ and }b=5n,$$
where $m$ and $n$ are coprime integers (they have no common factors other than 1).

### Step 3: Substitute into the product equation
$$a\times b=(5m)\times (5n)=25mn$$

From Step 1:
$$25mn=875$$
$$mn=35$$

### Step 4: Use the sum of the numbers
The sum of the numbers is:
$$a+b=5m+5n=5(m+n)$$

Given $a+b=60$:
$$5(m+n)=60$$
$$m+n=12$$

### Step 5: Solve for $m$ and $n$
We now have two equations:
1. $mn=35$
2. $m+n=12$

These are the sum and product of the roots of a quadratic equation:
$$x^2-(m+n)x+mn=0$$
$$x^2-12x+35=0$$

Solve this quadratic equation using factorization:
$$x^2-12x+35=0$$
$$(x-5)(x-7)=0$$

So, $m=5$ and $n=7$ (or vice versa).

### Step 6: Calculate the numbers and their difference
The two numbers are:
$$a=5m=5\times 5=25$$
$$b=5n=5\times 7=35$$

The difference between the numbers is:
$$b-a=35-25=10$$

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