Adding Zeros in a Long Division Problem

Date: 02/26/2006 at 22:39:10
From: Mom
Subject: When do we add zeros in the answer to a division problem

I was trying to remember the long division rules when dividing a small 
number such as 6 by a larger number like 351.  I am teaching my 
daughter and I want to make sure I understand the rules about adding 
zeros correctly.

- Struggling Mom 
 

Date: 02/27/2006 at 11:30:38
From: Doctor Rick
Subject: Re: When do we add zeros in the answer to a division problem

Hi, Mom ...

I prefer not to pile up too many rules, but to help a student see that 
what you do in this case is exactly what you always do in division.  
There are really no new rules; the only thing is to recall that the 
numbers 6, 6.0, 6.00, 6.000, and so on are all names for the same 
number.

I start by looking at the division problem
      ___
  351 ) 6

How many times does 351 go into 6?  None--that's 0, so I put 0 in the 
quotient, above the ones digit of the dividend, multiply 0 by 351, and 
subtract this from 6.

        0
      ___
  351 ) 6
       -0
       --
        6

Then I'm supposed to bring down the next digit.  There is no next 
digit ... but there is, if I write 6 as 6.0.  When I add the decimal 
point to the dividend, I also put one in the quotient directly above 
it.

        0.
      _____
  351 ) 6.0
       -0
       --
        6 0

Now (since I ignore the decimal point at this stage) I'm trying to 
divide 60 by 351, and 60 still isn't big enough; 60 divided by 351 
is 0 again, so ...

        0.0
      _____
  351 ) 6.0
       -0
       --
        6 0
       -  0
        ---
        6 0

It's time to bring down the next digit of the dividend again, but 
again there isn't one ... unless I write the dividend as 6.00:

        0.0
      _______
  351 ) 6.0 0
       -0
       --
        6 0
       -  0
        ---
        6 0 0

This time when I try to divide 600 by 351, it works: it goes once.  I 
put 1 in the quotient, above the ones place of 600; multiply 1 by 351,
and subtract the result; then I bring down the next digit of the
dividend.  To do this, I must of course add another 0 to the 
dividend:

        0.0 1
      _________
  351 ) 6.0 0 0
       -0
       --
        6 0
       -  0
        ---
        6 0 0
       -3 5 1
        -----
        2 4 9 0

I'll just do one more step.  How many times does 351 go into 2490?  I
estimate: 300 goes into 2100 7 times, and since I rounded both numbers
down, that's probably fairly close.  I multiply 351 by 7; that's 2457,
which indeed is smaller than 2490, so I go ahead and write it down.

        0.0 1 7
      _________
  351 ) 6.0 0 0
       -0
       --
        6 0
       -  0
        ---
        6 0 0
       -3 5 1
        -----
        2 4 9 0
       -2 4 5 7
        -------
            3 3

Once a child understands how this works, just following the basic 
rules, I can show some secrets that make it easier!  You don't have 
to write so much, because of what you know about how numbers work. 
Whenever the quotient digit is zero, you know that when you multiply 
zero by the divisor, you'll get zero; and you know that subtracting 
0 from any number leaves it just the same.  You can do those
operations in your head, because they amount to doing nothing!  All 
you need to write down is this:

        0.0 1 7
      _________
  351 ) 6.0 0 0
       -3 5 1
        -----
        2 4 9 0
       -2 4 5 7
        -------
            3 3

The only danger in leaving out those steps is that you might lose 
track of where the next digit goes in the quotient.  That's why I 
emphasize that the digit goes directly above the ones digit of the 
partial dividend.  If you find that this leaves a blank to the left 
of the digit you add, it means that you forgot to write down a zero; 
so put it in.

I hope this helps.

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