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### Three- and Four-Digit Long Division

```
Date: 03/15/99 at 20:20:25
From: Candace VanHulle
Subject: Long Division

Could you explain how to divide problems such as 67932/918 and
5676308/6532? I just do not understand how to go about finding how
many times a number goes into another number when you are using three
or four digits.

```

```
Date: 03/19/99 at 16:57:30
From: Doctor Peterson
Subject: Re: Long Division

I am going to assume that you can handle smaller divisors and know the
basic techniques of long division, but need help on how to choose
(guess) a digit to try in the quotient. Let us try your first problem,
and see how it goes.
_______
918)67932

The first set of digits bigger than 918 is 6793. We need to estimate
the quotient 6793/918. To get a rough estimate, you can just drop the
last two digits from both numbers: 67/9 = 7. (Do you see why? It is
the same as approximating the fraction 6793/918 by 6700/900 and
simplifying.) So let us try using 7 in the quotient: 7 * 918 = 6426,
so we write

____7__
918)67932
6426
----
367

That looks good: the remainder is positive (always a good sign) but
less then 918. So we continue:

____7__
918)67932
6426
----
3672

Again, we can estimate 3672/918 by 36/9 = 4. Since the result is a
whole number, I will not be surprised if it is wrong; it will not take
much increase in the dividend to push the quotient past 4. But let us
try it: 4 * 918 = 3672. Let us write it down:

____74_
918)67932
6426
----
3672
3672
----
0

That was too easy; it did not let me demonstrate how to recover from a
mistake. Since estimation plays a big role here, you have to be
prepared to make a wrong guess. Let us suppose I had somehow guessed 6
for the first digit:

____6__
918)67932
5508
----
1285

When I see that my remainder (1285) is bigger than my divisor (918), I
know I have to increase the quotient; I will try 7 and it will be
right.

Suppose instead that I guessed 8:

____8__
918)67932
7344
----

I do not even bother subtracting, because 7344 is too big to subtract
from 6793; I stop right here and decrease my quotient to 7.

I can also give you some more ideas on how to guess in the first
place. As I said, for the first digit we approximated the fraction
6793/918 by 6700/900, and found that 7*900 = 6300. This worked out
well, because if we add the dropped 18 back onto the 900, and multiply
it by 7, we are only adding 126 to the quotient, which is not too much
more than the 93 we dropped from 6793. If the digits you are dropping
from the divisor are bigger, you may be better off estimating by
rounding up. For example, to estimate 6793/983, I would probably use
6700/1000 = 6 rather than 6700/900 = 7. Am I right? 6 * 983 = 5898,
while 7 * 983 = 6881, so 7 would have been barely too large. If the
dividend had been smaller, say 6719, I would have definitely been
right to round the denominator up.

If this is where you have trouble, I would recommend two things.
First, practice just estimating single-digit quotients like this to
get a feel for when you should round up or down. Second, get used to
correcting wrong guesses, because that is an unavoidable part of the
process.

I hope this will build your confidence on these problems. If you get
stuck on some specific problem, feel free to write back and show me
archives whenever you have a question; here is the page on division:

http://mathforum.org/dr.math/tocs/division.elem.html

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Division

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