 ## Long Division Made Simple

Long division seems to strike fear into people's hearts, and when there is normally a calculator to hand it may be that you don't often need to know how to do it. But from time to time - perhaps when you want to work out  how to divide a bill evenly between 17 people, using a pen and paper - long division is the key.

### Short Division

Some divisions can be done quickly in the head, if you know your multiplication tables. If you recall that six times seven is forty-two, then you can quickly work out that forty-three divided by six is seven with remainder one.
42 ÷ 6 = 7 R 1
Short division comes into play when dividing larger numbers by a small one, where you know the multiplication tables for the small number but the larger number is bigger than anything you came across in the tables. For example, 197 divided by 3. How do we work this out? Normally, the sum is written down something like this:
3197
We start at the left of the long number, working to the right, writing the answer below the line as we go. The first step is to see if the leftmost digit (1) is less than 3.
3197                1 is less than 3
It is, so we bring in the next digit, and see if this number (19) is less than 3. It is not, so we now work out
19÷3, which is 6 remainder 1. We write the 6 under the last digit that we brought in from the long number, and we write the remainder in small numbers before the next digit:
31917            19÷3=6 R 1
6
Now moving on to the next digit (7) in the long number we read it together with the small 1, as 17, and ask ourselves what is 17÷3. The answer is 5 remainder 2, so we write 5 under the 7, and since there are no more digits in the large number we write the remainder as part of our answer:
31917            17÷3=5 R 2
6 5 Remainder 2
And now we have the final answer:  197÷3=65 remainder 2

### Long Division

But what do we do if instead of dividing by a nice small number like 3, we need to divide by a number  like 129? Chances are we didn't learn the 129 times table in school! This is where long division comes into play.

As an example, let's work out 330267÷129 using long division. The procedure is actually pretty much the same as for short division, but we normally write it down in a different way because some of the steps are more easily done on paper instead of in our heads. We will write this long division so:
129330267
Our answer will now appear above the line. Like before, we start with the first digit (3) of the long number and ask if it is less than 129. Obviously it is! So we bring in the next digit to get 33, and ask the same question - the answer is still yes, 33 is less than 129, so we bring in the following digit to get 330 and again ask if 330 is less than 129. This time the answer is no:
129330267        330 is more than 129
So we now work out how many times 129 will divide into 330. The answer will always be less than 10, so we can use trial-and-error, multiplying 129 by likely-looking numbers to see which is the biggest number we can multiply 129 by and still get an answer no greater than 330. Maybe the first number we try is 3 - but 3×129=387 which is too big. Let's try 2... 2×129=258 which is less than 330. So we write 2 and 258 as shown:
2
129330267        2×129=258  258 is not greater than 330
258
Now we subtract the number 258 from the number above it (330):
2
129330267        330 - 258 = 72
-258
72
As with short division we now move on to the next digit (2) of the long number - but this time we 'bring it down' and write it next to the 72:
2
129330267
-258
722
Now we need to find how many times 129 will divide into 722, again by trial-and-error. This time the answer is 5, because 5×129=645 ( 6×129=774 which would be too big). So we write 5 in the next position in our answer, and 645 underneath the 722:
25
129330267        5×129=645
-258
722
645
Once again we perform a subtraction, this time 722-645, and then bring down the next digit (6) from the long number:
25
129330267        5×129=645    722-645=77
-258
722
-645
776
How many times will 129 divide into 776? We worked that out previously, the answer is 6, because 6×129=774. So we write 6 in our answer, and subtract 774 from 776, then bring down the final digit (7) from the top:
256
129330267        6×129=774    776-774=2
-258
722
-645
776
-774
27

Now we have to ask how many times will 129 divide into 27? The answer is 0, since 27 is less than 129, so we write a 0 in the answer space and subtract 0×129 from 27, leaving 27 of course. And since there are no more digits to bring down, we know that this is our final remainder.
2560
129330267        0×129=0
-258
722
-645
776
-774
27

- 0
27
Answer: 330267 ÷ 129 = 2560 Remainder 27