Long Division - Organized 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

At the start, when we were working on the thousands, why not chop off what we know already.

We could chop off 1,000 x 6 = 6,000 and get:

Answer so far Number left
1,000 x 6 = 6,000 7,698 - 6,000 = 1,698

Now (remembering we have chopped off 1,000 x 6) we can continue, but we work on 1,698 now.

Guesses Compared to 1,698
100 x 6 = 600 too small
200 x 6 = 1,200 too small
300 x 6 = 1,800 too big

Now we can take 200 x 6 = 1,200 from 1,698: 1,698-1200 = 498.

Now we are only working on 498 ...

We can continue like this, following this idea:

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

But let's try the whole thing from the start, working 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 !

Check: 1,283 x 6 = 7,698


 

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.