Long Division

Long division isn't all that different from the kind of dividing you've already done, but because you're dividing by a bigger number, you may need to work out a few extra numbers before you can start dividing it in. For example, if you're dividing by 17, you may need to work out the answers to 2 x 17, 3 x 27, 4 x 17, 5 x 17, 6 x 17, 7 x 17, 8 x 17 and 9 x 17 before you can tell how many times it can be divided into a large number. You usually don't need to work all of them out, however, and I'll show you a trick you can use to minimise the amount of work you have to do, but first, let's just run through a sum and see how it is solved. The most important thing to notice is that we treat the 17 as if it's a single digit (so we don't need to split it into a 1 and a 7 to divide them in separately):-

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17 8 6 7 )
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Click on these buttons to solve the sum: 8 17 = 0 r 8 (so the 8 must be carried over to the second step where it will combine with the 6 to make 86), 86 17 = 5 r 1 (because 17 x 5 = 85), We take 85 away from 86 as normal, and now the remainder 1 appears on a new level, then we copy the 7 down to this new level, 17 17 = 1. So we end up with 867 17 = 51. Click this reset button and go through it again if you need to.

So there is nothing really new in the way you do the dividing, but it certainly isn't obvious that 5 x 17 = 85. The trick with long division is to work out your 17 times table for this sum in the following way. First double 17 to get 2 x 17. You can probably do this in your head, because doubling a number simply means adding it to itself. The answer is of course 34. Now double the 34 to get the answer to 4 x 17: that'll be 68. Now double the 68 to get the answer to 8 x 17: you can probably do that in your head too (two eights are sixteen, write down the six and hold the one in your head, add the two sixes to get 12, add the one that was stored in your head to get 13, put that in front of the 6, and the answer's 136). We can now put these into a table:-

1     2     3     4     5     6     7     8     9
17    34    __    68    __    __    __    138   ___

If we're trying to divide 17 into 86, we look at our table and try to guess which number might come closest to it, and it looks as if it might be 5 x 17, one of the ones we haven't worked out yet. Well, any of the answers that haven't been worked out yet can be calculated easily enough just by adding 17 to the number to the left of it. The only answers you can't do that way are 6 x 17 and 7 x 17, but the former can be calculated by adding 2 x 17 to 4 x 17 (or 34 + 68), and again that's easy enough to do in your head (102). The latter (7 x 17) can then be calculated by adding another 17, or alternatively you can take 17 away from 8 x 17. Sometimes you will have to work out the whole lot, but usually you can get away with just doing the few that you actually need for solving a particular sum. This is the only difficulty with long division, so you are now ready to go straight into the game: click here. Again you should also go on to make sure you can do long division sums on paper with a pencil. Make sure you come back here afterwards, because we've still got decimals to deal with:-


So far we have always left all the remainders undivided, but it is easy enough just to carry on dividing them into tenths, hundredths, thousandths, etc. We can just put a decimal point in and write as many zeros after it as we like, depending on how accurate we need the answer to be. If you want to divide 8 into 12, you will get the answer 1, and then you'll take 8 away from the 12 to get the remainder 4, but now you don't write the 4 after an "r" at the top: instead you write ".0" after the 12. You can then copy the 0 down next to the 4 to make 40, before dividing the 40 by 8. This gives you a 5, and you can write that in at the top after another point (a dot), giving the answer 1.5 for the whole sum. Let's do it all properly to make it easier to follow:-

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 8 1 2   )
    __ _
      . .
Click on these buttons to solve the sum: 1 8 = 0 r 1 (so the 1 must be carried over to the second step where it will combine with the 2 to make 12), 12 8 = 1 r 4, We take 8 away from 12 as normal, and now the remainder 4 appears on a new level, then we create a 0 after a point and copy it down next to the 4, extending the line at the same time, 40 8 = 5, and this goes at the top after the point. So we end up with 12 8 = 1.5 (meaning 1 and a half). Click this reset button if you want to go through it again.

In many cases there will be another remainder, but all you need to do is create another zero and bring it down next to the remainder as before: you can go on dividing forever, as in the case of 10 3, where the answer is 3.3333333r, and in this case the "r" means "repeat" rather than "remainder". I'm not going to send you to a game to sharpen your skills this time because there isn't enough new to merit that, but you should try doing a few sums on paper, and you can check your results with a calculator. Here are some interesting ones to start with:-

1 4       1 8       3 4       5 7

If you keep on getting remainders and the sum goes on forever, the digits probably won't all be the same (as they were in 3.3333333r), so you can't just put an "r" at the end to signal infinite repeats. Instead, you have to decide how accurate you need the answer to be, and then you can round it off to the nearest digit once you feel you've got enough digits to do the job, so 0.7142857 can be shortened to 0.714286, or to 0.71429, or 0.7143, or 0.714, or 0.71, or even just to 0.7 which may still be accurate enough for some tasks. Sometimes you will be asked to give an answer to a certain number of "significant figures", so if it's "three significant figures", that means you only need to work out the first four digits of the answer (not including zero if the first result is zero), and you then you have to round the last digit up (add 1 to it) if the fourth digit is 5 or more (and you'll do this instead of writing the fourth digit in). This means that 0.8825 would be rounded up to 0.883, but if the third digit is pushed up to 10, then it becomes a zero and you'll have to add 1 to the second digit of the answer as well (e.g. 0.1195 becomes 0.120, and the final zero is written in to show that we have rounded it off to three significant digits rather than just two). In some cases, this can ripple on through even further: e.g. 0.9995 gets rounded up to 1.000 (and then it can simply become 1.00 because we only need three significant digits), but 0.9994 is left as 0.999. If a question in an exam doesn't say how many significant digits are required for the answer, it may ask instead for a certain number of digits after the point instead, and if it doesn't specify that, then you can assume that three digits after the point will do, with the last one rounded up if necessary.