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.

Thanks for your help.

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 
your work, so I can help you out. And do not forget to check our 
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/