Dividing

Dividing is when you take the same number away from a bigger number over and over again while keeping count of how many times you've taken it away, and you only stop when you can't take it away any more:-

12 3 = 4 (12 divided by 3 is 4). Here is the slow way to work out that the answer is 4: 12 - 3 = 9 (we've taken 3 away once), 9 - 3 = 6 (we've taken 3 away twice), 6 - 3 = 3 (we've taken 3 away three times), 3 - 3 = 0 (we've taken 3 away four times), 0 - 3 = ... and because we can't take three away any more, the answer's four.

Fortunately division is actually much easier than that, because it's simply multiplication in reverse: 4 x 3 = 12, so 12 3 = 4 and 12 4 = 3. All you have to do is look for the answer in the multiplication table below; look along the row beginning with 3; find the 12 on that line; and then look to see which number's at the top of that column, and there you find the 4. The more often you practise multiplying, the easier you'll find division, because you'll become more familiar with all the big numbers in the table, so when you are asked to divide 56 by 8, you'll recognise the 56 as being the answer to 7 x 8 and you'll instantly know that 56 8 must be 7.

1  2  3  4  5  6  7  8  9 10
2  4  6  8 10 12 14 16 18 20
3  6  9 12 15 18 21 24 27 30
4  8 12 16 20 24 28 32 36 40
5 10 15 20 25 30 35 40 45 50
6 12 18 24 30 36 42 48 54 60
7 14 21 28 35 42 49 56 63 70
8 16 24 32 40 48 56 64 72 80
9 18 27 36 45 54 63 72 81 90
10 20 30 40 50 60 70 80 90 100

See if you can work out the answers to 72 9, 35 7 and 24 6. You should get 8, 5 and 4. You can also get the answers by counting your way along the lines, so for 72 9 you can count your way along the row starting with 9 until you reach 72, and by the time you get there you should have counted up to 8 in your head, and that's the correct answer. You won't always have the multiplication table sitting in front of you when doing division, so you need to be able to work out the answers without it, but to begin with you'll probably find that too hard, so write out the table on a piece of paper and keep it beside you. You will need it less over time, and eventually you won't need it at all.

You should now practise doing this multiplying-in-reverse by taking on the dividing race/game: click on this link. There are clues in green to help you if you don't have the table written out beside you.



So far we have only looked at divisions where they all work out exactly, but in real life it is more common to find that there is an amount left over. If you divide 6 by 2 you get 3, while 8 divided by 2 is 4, but 7 2 = 3 with 1 left over (so if 7 sweets are divided between two people, they'll get three each, plus one left over for them to argue about). We can write this as:-

7 2 = 3 r 1  (with the "r" standing for remainder)

In the same way, 11 3 = 3 r 2 (three threes are nine: that's 2 less than eleven, so the remainder is 2), and 35 4 = 8 r 3 (four eights are thirty two: that's 3 less than thirty five, so the remainder is 3). To work these out the easy way, just look up the table above to find the biggest number in the relevant row that isn't too big: if the sum is 44 7, then you look along the 7 row (7 14 21 28 35 42 49 56 63 70) for the number 44 or the biggest number less than 44. The 49 is too big, so we must choose the 42 which is the biggest one that will fit into 44. 42 is the answer to 6 x 7, so we now know that the answer to 44 7 will be 6, but we also know that there will be a remainder because 42 is not quite 44, and the remainder must be 2 because that's the size of the gap between 42 and 44. So 44 7 = 6 r 2. To practise this, you can tackle the remainder race/game where you have to type in the remainder for each sum (don't type in the main part of the answer at all - it only wants the remainder): click on this link. Again, you should have the multiplication table written out on a piece of paper beside you to begin with, but your job is to train yourself not to need it.



Divisions are written out in a very different way from other sums, using a bracket ")" and a straight line as a framework. The answer is to be written in on top of the straight line, and you'll notice when we start solving one of these sums that we must begin work at the left end rather than the right: you'll understand this better when you see it all in action, so let's do a dividing sum. This first example shows you how to solve 128 4:-

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4 1 2 8 )

Click on these buttons to solve the sum: 1 4 = 0 r 1 (we're off to a slow start because 4 is too big to divide 1 by, so we just have to put 0 in on the top line, and we keep the remainder 1 in play: it will be divided in the second step where it joins forces with the 2 to make 12), 12 4 = 3 (there is no remainder this time, so the 1 and 2 of the 12 have been used up and will play no further part), 8 4 = 2. So, the answer for the whole sum is 32. Click this reset button and go through it again as many times as you need to to understand it. Now look at this next example:-

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4 4 3 6 )

Click on these buttons to solve the sum: 4 4 = 1 (this time there's no remainder to be carried over to the second step), 3 4 = 0 r 3 (the 0 goes in on the top line, and we keep the remainder 3 in play: it will be divided in the third step where it joins forces with the 6 to make 36), 36 4 = 9. So, the answer for the whole sum is 109. Click this reset button and go through it again as many times as you need to to understand it. Go back to the previous sum and click through it again if you need to remind yourself of what happened there. Whenever the answer to part of one of these sums is 0 r something, the something is the number that you've just tried and failed to divide, so it remains active for use in the next step, unless there is no next step still to do, in which case you do this:-

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3 9 6 2 )

Click on these buttons to solve the sum: 9 3 = 3 (there's no remainder to be carried over to the second step), 6 3 = 2 (and there's no remainder to carry over to the third step either), 2 3 = 0 r 2 (but this time there is a remainder, and because there is no next step to pass it on to, we must write in the "r 2" as part of the answer), so the answer to the whole sum is 320 r 2. Click this reset button if you need to go through it again, but don't worry if you're still confused: you'll be given lots of practise later on and it will become easy before long.

Let me remind you: if you're reading this on a computer with a small screen you'll be able to see much more of the page at a time by pressing the F11 key. Pressing that same key again will return things to normal afterwards.

All the remainders above were the result of answers being 0 r something, so the number that we were unable to divide in one step was still sitting there, waiting to be combined with the next number in the next step. But what happens if the answer's something like 1 r 2, or 5 r 3, or 4 r 6? This adds a little complication, but it's not too hard to understand when you see it done, so look at this sum:-

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6 4 5 0 )
    __ _
     
Click on these buttons to solve the sum: 4 6 = 0 r 4 (so the 4 must be carried over to the second step where it will combine with the 5 to make 45), 45 6 = 7 r 3 (because 6 x 7 = 42), but where's the 3? We actually have to take 42 away from 45, and now the remainder 3 appears on a new level, then we copy the 0 down to this new level to make it easier to see what's going to be divided next, 30 6 = 5. So we end up with 450 6 = 75. Click this reset button and go through it again and again until you understand how this sum was done.

Here's another one for you to try for yourself. Notice that because the first answer is 0, it needn't be written in at all: it's shown in red when it first appears, but then it turns gray to show that it can be left out:-

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7 3 1 7 5 )
    __ _
     
    __ _
     
Click on these buttons to solve the sum: 3 7 = 0 r 3, 31 7 = 4 r 3 (because 7 x 4 = 28), 31 - 28 = 3, copy 7 down, 37 7 = 5 r 2 (because 7 x 5 = 35), 37 - 35 = 2, copy 5 down, 25 7 = 3 r 4. Click this reset button and go through it again and again until you understand how this sum was done.

You can now practise what you have just learned by playing another game (click on this link), though you should also go on to make sure you can do division sums on paper with a pencil: you may have difficulty keeping things lined up properly when copying numbers down from the top to the bottom, so you should draw a vertical line between them to help keep track of which column is which - otherwise you may be in danger of copying the wrong number down at a later stage).

Dividing by bigger numbers (e.g. 256 32) will be looked at later.