Why it works
Comparing the division to the continued subtraction probably has already let
you see why it works. In the conventional way of writing the long
division, it is not so easy to see the process. The key is that in each
step, one does NOT actually divide by the actual divisor but by a multiple of
it. Just like in the apples/bags examples, you don't start out by
subtracting 3 apples each time, but first 'hit it hard' by subtracting multiples
of 300 apples if possible, then multiples of 30, then 3. In essence, you
first divide by 300, then by 30, then by 3.
Also, in the conventional long division, you only place one digit into the
quotient in each step, not with all the zeros. The digits shown in gray
are not usually written out in the conventional long division algorithm.
To get the hundreds digit in the quotient, one asks the question: "How many times does
300 go into 789", or the
division 789 ÷ 300! You are not dividing by 3 because you try to 'hit it
hard' and subtract as many multiples of 300 as possible. Since 300 is a whole hundred, the tens and ones
digits in the 789 won't matter when you are finding how many times 300 goes into
789. So the thing can be done easier by calculating
7 ÷ 3, or thinking "How many times does 3 go into 7".
The remainder from first step (what is left after subtraction) is in reality
189. But since the ones digit (9) won't be important in the next step
(which deals with the tens digit), in the traditional way, you only subtract 76 and
then you 'drop' down the tens digit 8 from the dividend.
To get the tens digit, similarly one asks the question: "How
many times does 30 go into 189", or does the division 189 ÷ 30.
Again, since you're dividing by a multiple of ten, the ones digit '9' in the 189
does not affect the division at all. The important thing is to look at the
whole tens in the number 189, which is 180. So to find the answer to the
division 189 ÷ 30, you can think of the division 180 ÷ 30, which is the same
as thinking 18 ÷ 3: "How many times does 3 go into
18?"
The last step is simple since it is dealing with ones digits, how many times
does 3 go into 9.
