Long Division

Date: 08/23/98 at 22:14:25
From: Anna
Subject: Long division lag

Dear Dr. Math,

I am in fifth grade and am having trouble with third grade arithmetic.  
My biggest problem is long division. For example, what is 73/4? 
Please help me.

Date: 08/24/98 at 13:26:18
From: Doctor Peterson
Subject: Re: Long division lag

Hi, Anna. Lots of people need help with long division. Sometimes it 
just takes several different people explaining it before you find an 
explanation that works for you. Here's a nice explanation in our Dr. 
Math archives that you might like to work through:

   Learning Long Division
   http://mathforum.org/library/drmath/view/58816.html   

One thing that bothers many people is that they are told they have to 
"guess" as part of the process. For division by a one-digit number 
like your example, there's really no guessing involved. The guessing 
comes in for bigger divisors, because you haven't memorized (and 
shouldn't!) the multiplication tables for, say, 87 times anything. So 
there are techniques for making a good guess and then fixing it if it 
turns out to be wrong. But as long as we're dealing with a simpler 
problem, I won't go into that. Write back when you're ready for it.

Rather than repeat what others have written about how to divide, I'd 
like to tell you a little about why. Some people can do a task better 
if they know why they do what they do. So let's divide 73 by 4, and 
think about what we are doing.

I have 73 sticks I want to bundle in groups of 4, and I want to know 
how many bundles I will end up with. (That's division.)

Luckily, someone has given them to me already bundled in tens: 7 
bundles of ten, and 3 extra sticks. (That's called place value.)

        70         3
    ||||||||||    |||
    ||||||||||
    ||||||||||
    ||||||||||
    ||||||||||
    ||||||||||
    ||||||||||

Rather than unbundle them all, I can start out by working with the 7 
bundles and divide them by 4. That gives me one group of 4 bundles. 
Those 4 bundles of ten can be easily rebundled into 10 bundles of 4, 
so I've got a big part of the job already done. (I just divided 7 by 
4, getting 1, with 3 left over.)

      10 4's
    ||||||||||
    ||||||||||
    ||||||||||
    ||||||||||
--------------------------
        30         3
    ||||||||||    |||
    ||||||||||
    ||||||||||

Now I have 3 bundles of 10, and 3 more. At this point I untie the 
bundles and find that I have 33 sticks. I check my handy multiplication 
tables and find that 4 times 8 is 32. (I've divided 33 by 4 to get 8 
with a remainder of 1.)

      10 4's
    ||||||||||
    ||||||||||
    ||||||||||
    ||||||||||

      8 4's
    ||||||||
    ||||||||
    ||||||||
    ||||||||

    1
    |

Now look at what I have: 10 bundles of 4 from the first step, and 8 
more from the second. The answer is 18 bundles of 4, and 1 left over.

Do you see how this gave me the answer, one digit at a time, by 
looking at the 73 one digit at a time? 

Let's do the actual work:

Divide 7 by 4, getting 1 rem 3. I write the 1 on top, then multiply it 
by 4 and write down 4 below the 7, subtracting that to find the 
remainder:

      __1__
    4 ) 73
        4
        -
        3

Now combine the remainder of 3 and the next digit, 3, to make 33. 
Divide that by 4 and write everything down the same way:

      __18_
    4 ) 73
        4
        -             ___8_
        33   <--->  4 ) 33
        32              32
        --              --
         1               1

The answer is 18, with a remainder of 1. Look back and see how this 
compares to my stick counting.

I hope that makes a little sense out of the process, so you can 
remember better what you have to do. You're just working with one digit 
of the answer at a time. If you still need more help, you might try 
writing out how you do a problem, and I could see where you have 
trouble. Don't give up. You can do it.

- Doctor Peterson, The Math Forum
 http://mathforum.org/dr.math/