Dividing Decimals with a Remainder

Date: 03/30/2001 at 16:33:56
From: Genna
Subject: Dividing Decimals

Dear Dr. Math, 

I am a student in an elementary school and just got accepted to be in 
accelerated math next year in Jr. High. I have a question about 
dividing decimals.

I know how to divide decimals, but when there is a remainder my 
teacher tells me I have to add zeros to the inside number. He says you 
can either add 2 or 3.  I usually add 2 zeros, but if you add 2 you 
will get a totally different answer than if you add 3 zeros.  

Also, sometimes when you have a remainder and you bring it out 2 or 3 
places (or however many it tells you to), you still have a remainder.  
My teacher says to just leave the remainder and round the answer, but 
I don't understand. 

If you could explain to me what to do when the directions only say 
DIVIDE: and you get a remainder, and also if the directions say ROUND 
TO THE HUNDRETHS PLACE: what to do when there is still a remainder, I 
would be very happy. 

Thank you very much,
Genna

Date: 03/30/2001 at 23:29:15
From: Doctor Peterson
Subject: Re: Dividing Decimals

Hi, Genna.

Let's take an example, so we have something specific to talk about:

       ___1.9_
    23 ) 45.6
         23
         --
         22 6
         20 7
         ----
          1 9

Okay, we have a remainder, and we want to round our answer to the 
hundredths place, so we'll add two zeros to the dividend and continue. 
(I'll deal with how many zeros to add in a moment.)

       ___1.982_
    23 ) 45.600
         23
         --
         22 6
         20 7
         ----
          1 90
          1 84
          ----
             60
             46
             --
             14

Now what do we do? Since we were told to round to two decimal places, 
we just look at the 2 in the quotient, drop it, and leave the rest as 
it is. The answer is 1.98.

I didn't really have to add that last zero. I could have stopped with 
the remainder 6, and seen that the next digit of the quotient would be 
less than 5, because 6 is less than half of 23. See if you can see why 
that is true.

(I don't know what you mean when you say that you get a totally 
different answer if you add two zeros or three. I just see one extra 
decimal place in the answer, which is not very different. Can you give 
me an example of what you mean?)

Now let's get back to the big question: what is this all about? Why 
should you just ignore the remainder?

The problem with decimals is that most division problems never end. 
You can keep adding more zeros to the dividend, and you'll just get 
more zeros in the quotient. If you've learned about repeating 
decimals, you know why that is: when you change a fraction to a 
decimal, you get a terminating decimal only if the denominator (in 
lowest terms) has only 2 and 5 as prime factors, so that it can be 
converted to a fraction with a power of ten in the denominator. So 
when we work with decimals, we expect to have to approximate. 

The reason we can approximate without losing anything important is 
that each decimal place we get is worth a tenth of the previous one, 
so eventually they get small enough not to affect whatever we are 
going to do with them. At that point we can just drop the remainder 
(and therefore all the rest of the digits), and round if we wish.

An important topic related to this is "significant digits." You may 
want to search our archives for this phrase and read about it; it 
explains how we can decide how many digits we need, and why digits 
after a certain point don't matter and can be ignored.

When we need to be precise, we use fractions rather than decimals, 
because we never have to drop anything. When we use decimals, we KNOW 
that we are going to be rounding, so it doesn't bother us.

Write back if you have any further questions.

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