If I understand your question clearly, the reason why you can use subtraction in a division problem is that, in order for you to find out what a remainder or mixed number is, you need to subtract.
Example: Tamira has 60 bags of cookies to give out at her local community center, each person receives 5 cookies in a bag, how many people got a bag of cookies? Is there a remainder left?
So, after reading that problem, you already know you need to divide.
5 )‾‾60‾‾‾‾
How many times can 5 go into 6? 1 time. So, you put the 1 over the 6.
1
5 )‾60‾‾‾‾‾‾
- 5 ↓
______
1 0
How many times can 5 go into 10? 2 times.
12
5)‾‾60‾‾‾
-5↓
_________
10
-10
_______
0
So, Tamira gave 12 people cookies with no remainder. And for when you need to find out the mixed number, it's the same process, but you go in the order of: quotient, remainder, dividend.
I hope I helped :)