11003 ÷ 11 
The aim is to understand the frequency of 11 in 11003.

Step 1:
When dividing 1 by 11, we get 0 with a remainder of 1.
Record the quotient 0 as the next digit in the result. 
Result so far: 0.0
If we take 0 away from 1, we end up with 1.
Include the next digit (1) from the dividend after 1, then repeat: 11 / 11

Step 2:
The number 11 fits into 11 1 times, leaving a remainder of 0.
The next digit of our result is 1. 
Result so far: 1.0
The remainder is 0 after subtracting 11 from 11.
Bring next digit (0) of the dividend behind the 0 and repeat the process: 0 / 11

Step 3:
11 can be fit into 0 0 times, resulting in a remainder of 0.
Use 0 as the next digit of our solution. 
Result so far: 10.0
Deduct 0 from 0 and we're left with 0.
Include the next digit (0) from the dividend after 0, then repeat: 0 / 11

Advancing to step 4:
11 can be fit into 0 0 times, resulting in a remainder of 0.
Use 0 as the next digit of our solution. 
Result so far: 100.0
Subtract 0 from 0 to get 0.
Grab the next digit (3) from the dividend, add it to 0, then carry on: 3 / 11

Advancing to step 5:
11 can be fit into 3 0 times, resulting in a remainder of 3.
Put 0 as the next digit of the answer. 
Result so far: 1000.0
Subtract 0 from 3 to get 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

Advancing to step 6:
When dividing 30 by 11, we get 2 with a remainder of 8.
Write down 2 as next digit of the result. 
Result so far: 1000.2
Subtract 22 from 30 to get 8.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Step 7:
If we divide 80 by 11, we get 7 and a remainder of 3.
Put 7 as the next digit of the answer. 
Result so far: 1000.27
If we subtract 77 from 80, we get 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

Step 8:
The number 11 fits into 30 2 times, leaving a remainder of 8.
Record the quotient 2 as the next digit in the result. 
Result so far: 1000.272
Deduct 22 from 30 and we're left with 8.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Step 9:
The number 11 fits into 80 7 times, leaving a remainder of 3.
Put 7 as the next digit of the answer. 
Result so far: 1000.2727
Deduct 77 from 80 and we're left with 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

Let's proceed to step 10:
If we divide 30 by 11, we get 2 and a remainder of 8.
Write down 2 as next digit of the result. 
Result so far: 1000.27272
If we subtract 22 from 30, we get 8.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Let's proceed to step 11:
When dividing 80 by 11, we get 7 with a remainder of 3.
Put 7 as the next digit of the answer. 
Result so far: 1000.272727
Subtracting 77 from 80 leaves us with 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

On to step 12:
If we divide 30 by 11, we get 2 and a remainder of 8.
The number 2 becomes the next digit in our result. 
Result so far: 1000.2727272
The remainder is 8 after subtracting 22 from 30.
Since there are no more digits in the dividend, we add a zero to the remainder making it 80, and continue the process.

Step 13:
80 divided by 11 is 7 with a remainder of 3.
The number 7 becomes the next digit in our result. 
Result so far: 1000.27272727
Subtract 77 from 80 to get 3.
Since there are no more digits in the dividend, we add a zero to the remainder making it 30, and continue the process.

In conclusion, the division gives 1000.27272727 with a balance of 3e-09.