420471 ÷ 513 
Let's see how many times 513 fits into 420471.

On to step 1:
If we divide 4 by 513, we get 0 and a remainder of 4.
Use 0 as the next digit of our solution. 
Result so far: 0.0
If we subtract 0 from 4, we get 4.
Append the next digit (2) from the dividend to 4 and continue with: 42 / 513

On to step 2:
513 goes into 42 0 times with a remainder of 42.
The number 0 becomes the next digit in our result. 
Result so far: 0.0
If we take 0 away from 42, we end up with 42.
Include the next digit (0) from the dividend after 42, then repeat: 420 / 513

On to step 3:
420 divided by 513 is 0 with a remainder of 420.
Write down 0 as next digit of the result. 
Result so far: 0.0
The remainder is 420 after subtracting 0 from 420.
Grab the next digit (4) from the dividend, add it to 420, then carry on: 4204 / 513

Step 4:
The number 513 fits into 4204 8 times, leaving a remainder of 100.
The next digit of our result is 8. 
Result so far: 8.0
If we subtract 4104 from 4204, we get 100.
Include the next digit (7) from the dividend after 100, then repeat: 1007 / 513

Going ahead to step 5:
513 can be fit into 1007 1 times, resulting in a remainder of 494.
The next digit of our result is 1. 
Result so far: 81.0
Deduct 513 from 1007 and we're left with 494.
Include the next digit (1) from the dividend after 494, then repeat: 4941 / 513

Moving on to step 6:
If we divide 4941 by 513, we get 9 and a remainder of 324.
Use 9 as the next digit of our solution. 
Result so far: 819.0
If we take 4617 away from 4941, we end up with 324.
Since there are no more digits in the dividend, we add a zero to the remainder making it 3240, and continue the process.

On to step 7:
3240 divided by 513 is 6 with a remainder of 162.
Write down 6 as next digit of the result. 
Result so far: 819.6
If we take 3078 away from 3240, we end up with 162.
Since there are no more digits in the dividend, we add a zero to the remainder making it 1620, and continue the process.

Moving on to step 8:
513 goes into 1620 3 times with a remainder of 81.
The number 3 becomes the next digit in our result. 
Result so far: 819.63
Subtract 1539 from 1620 to get 81.
Since there are no more digits in the dividend, we add a zero to the remainder making it 810, and continue the process.

Let's proceed to step 9:
The number 513 fits into 810 1 times, leaving a remainder of 297.
The next digit of our result is 1. 
Result so far: 819.631
The remainder is 297 after subtracting 513 from 810.
Since there are no more digits in the dividend, we add a zero to the remainder making it 2970, and continue the process.

Step 10:
513 goes into 2970 5 times with a remainder of 405.
Record the quotient 5 as the next digit in the result. 
Result so far: 819.6315
If we take 2565 away from 2970, we end up with 405.
Since there are no more digits in the dividend, we add a zero to the remainder making it 4050, and continue the process.

Step 11:
If we divide 4050 by 513, we get 7 and a remainder of 459.
Write down 7 as next digit of the result. 
Result so far: 819.63157
Subtract 3591 from 4050 to get 459.
Since there are no more digits in the dividend, we add a zero to the remainder making it 4590, and continue the process.

Step 12:
The number 513 fits into 4590 8 times, leaving a remainder of 486.
Write down 8 as next digit of the result. 
Result so far: 819.631578
Deduct 4104 from 4590 and we're left with 486.
Since there are no more digits in the dividend, we add a zero to the remainder making it 4860, and continue the process.

Advancing to step 13:
4860 divided by 513 is 9 with a remainder of 243.
Write down 9 as next digit of the result. 
Result so far: 819.6315789
If we take 4617 away from 4860, we end up with 243.
Since there are no more digits in the dividend, we add a zero to the remainder making it 2430, and continue the process.

Advancing to step 14:
513 can be fit into 2430 4 times, resulting in a remainder of 378.
Record the quotient 4 as the next digit in the result. 
Result so far: 819.63157894
Subtracting 2052 from 2430 leaves us with 378.
Since there are no more digits in the dividend, we add a zero to the remainder making it 3780, and continue the process.

The final outcome is 819.63157894, with a leftover of 3.78e-07.