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. Now 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 to guess.
Let's try that again, 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:
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What We Did:
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Should be Written:
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Onto: Long Division >>>
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