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Long Division


Date: 03/16/99 at 22:18:56
From: Daniel
Subject: Division

Please show some examples in detail about long division and the 
process step by step. 

Thank you.


Date: 03/17/99 at 12:04:28
From: Doctor Peterson
Subject: Re: Division

Hi, Daniel.

I'm not sure how much you have learned about division; I'll show you 
how to divide by a one-digit number. If you need something more 
advanced than this, look in our Dr. Math archives for other problems 
like what you need to do:

  http://mathforum.org/library/drmath/sets/elem_division.html   

I'll give you an example problem, and show you how to keep your mind 
focused on one part of the problem at a time, so all the numbers 
don't get too confusing. You won't write it out in pieces as I will, 
but what you do write is a compact way of saying the same thing. 
Let's divide 175 by 3:

      ______
    3 ) 175

We take the dividend (175) one digit at a time starting at the left. 
If we tried to divide 1 by 3, we would get zero, so let's skip that 
and use the first two digits, 17:

      ___5_
    3 ) 17
        15
        --
         2

What I did was to look through the multiplication table for 3 (in my 
mind), looking for the biggest multiple of 3 that is less than 17: 
3, 6, 9, 12, 15, 18 - since 18 is too big, we use 3 x 5 = 15. 
I wrote the 5 on top, above the "ones" digit of the 17 I'm working on, 
and wrote the 15 under the 17. Then I subtracted, leaving a remainder 
of 2. This says that

    17 = 3 x 5 + 2

What we really did just now was to divide 170 by 3, getting only the 
number of tens in the quotient:

      ___50_
    3 ) 170
        150
        ---
         20

    170 = 3 x 50 + 20

That's not the whole problem, though. What do we do with the next 
digit of 175, the 5 we left out?

I'll use the remainder, 2, as the tens of a new dividend, by 
"bringing down" the next (and last) digit of the dividend:

            175
              |
              v
         2-->25

This is really just adding the missing 5 to the remainder of 20. Now 
I divide that by 3:

           ___8_
         3 ) 25
             24
             --
              1

I made "25" by combining the remainder, 2, with the next digit, 5, 
and divided that by 3 the same way I did before: 25 = 3 x 8 + 1. This 
gives us the ones digit of the answer, 8, and a final remainder of 1.

Putting it all together, here's what you actually write down:

      ___58_
    3 ) 175
        15
        --
         25
         24
         --
          1

Do you see that second problem, 25 divided by 3, hidden in there? 
When you work on it, you focus on that as if it were a separate 
problem by itself, and write the quotient, 8, above the "ones" digit 
of the 25.

Putting it all together, the quotient is 58 and the remainder is 1:

    175 = 3 x 58 + 1

If it would help you to understand WHY we do it this way, here's a 
simple explanation:

   Long Division
   http://mathforum.org/library/drmath/view/58499.html   

You should always check a division. Here's an answer I gave another 
student that should help you:

   Checking Division
   http://mathforum.org/library/drmath/view/58807.html   

Let's check our answer. I'll do the multiplication "upside down" so 
you can see the connection to the way we divided:

       3   <--- divisor
    x 58   <--- quotient
    ----
      24   <--- notice that 24 and 15 were in your division work!
     15    <---
    ----
     174
    +  1   <--- remainder
    ----
     175   <--- this matches the dividend, so it's right

I hope that helps. If the this and the archives don't give you what 
you need, please write back with a specific problem to show where you 
need help.

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

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