Long Division - Organised Guessing

So, how does Long Division work? It works by breaking up the big number and solving it a section at a time.

Thousands, Hundreds, ...

To illustrate, let us try guessing the answer to 7,698 ÷ 6, but following a special method.

We will start at the thousands, then move to the hundreds, then tens, then units.

OK, starting at the thousands: How many 6 lots of a thousand can we fit into 7,698 ? Well, just one, really. Two would be too much:

1,000 x 6 = 6,000	too small
2,000 x 6 = 12,000	too big

We have made progress of sorts. We can guess that the answer is between 1,000 and 2,000.

Now let's move onto the hundreds. Let's add a hundred at a time:

1,100 x 6 = 6,600	too small
1,200 x 6 = 7,200	too small
1,300 x 6 = 7,800	too big

So, somewhere between 1,200 and 1,300. Let's move on to the tens:

1,210 x 6 = 7,260	too small
1,220 x 6 = 7,320	too small
1,230 x 6 = ...

Stop! It is getting to be very tedious doing all this long multiplying ...

...there should be a simpler way

Backtrack:

We already know that 1,200 lots of 6 is close, so why don't we just reduce 7,698 by that:

7,698 - 6 × 1,200 = 7,698 - 7,200 = 498 left to solve

That's better, the problem gets easier the more you work at it!

So, where were we? That's right, working on the tens, but aiming for 498 now:

10 x 6 = 60	too small
20 x 6 = 120	too small
...
80 x 6 = 480	too small
90 x 6 = 540	too big

So, 8 lots of ten.

We can now further reduce the problem:

498 - 480 = 18 left to solve

Aiming for 18 now:

1 x 6 = 6	too small
2 x 6 = 12	too small
3 x 6 = 18	BINGO!

We did it.

So what did we do? We had 1 lot of a thousand, 2 hundreds, 8 tens and 3 units = 1,283.

Better just test it out:

1,283 x 6 = 7,698 ... YES!

So the moral of the story is:

Don't work on the entire number every time,
just work on whatever is left.

Let's try that again, but neatly:

Start with 7,698

Thousands	1,000 x 6 = 6,000	too small
	2,000 x 6 = 12,000	too big

So the answer for thousands is 1,000 x 6.

7,698 - 1,000 x 6 = 1,698 left to figure out:

Hundreds	100 x 6 = 600	too small
	200 x 6 = 1,200	too small
	300 x 6 = 1,800	too big

So the answer for hundreds is 200 x 6.

1,698 - 200 x 6 = 1,698 - 1,200 = 498 left to figure out

Tens	80 x 6 = 480	too small
	90 x 6 = 540	too big

So the answer for tens is 80 x 6

498 - 80 x 6 = 498 - 480 = 18 left to figure out

Units

3 x 6 = 18

perfect!

So the answer for units is 3 x 6

Our Answer: 1,000 x 6 + 200 x 6 + 80 x 6 + 3 x 6 = 7,698, and the answer is 1,283 !

But instead of using all those words, we write it down like this:

What We Did:		Should be Written:

And that is how you do Long Division.

Now look at this Long Division page.