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### Why Decimal Division Works

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Date: 04/24/2001 at 03:14:31
From: Marc
Subject: WHY division by decimals

For my college math education class I have to write a paper on why
(not how) the procedure of division with decimals works. The question
is, "All elementary school students learn how to divide with decimals
such as in the problem 551.2/1.06. Explain as if talking to 5th and
6th graders why this procedure works." If you could help me it would
help very much!

Thanks,
Marc
```

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Date: 04/24/2001 at 08:51:51
From: Doctor Peterson
Subject: Re: WHY division by decimals

Hi, Marc.

If you'd like to look over our shoulders as we actually explain
division of decimals to a 5th or 6th grader, visit our archives:

Fractions and Decimals - Elementary School Level
http://mathforum.org/dr.math/tocs/fractions.elem.html

Division - Elementary School Level
http://mathforum.org/dr.math/tocs/division.elem.html

Here's a sample:

How to Divide Decimals by Decimals
http://mathforum.org/dr.math/problems/wright3.7.98.html

That should give you some good ideas for your own explanation.

The basic idea is that we first learn a method for dividing whole
numbers. When we introduce decimals, we find that the work can be
divided into two parts, just as in multiplication: we can first ignore
the decimal point entirely, and get the DIGITS of the answer using the
methods we've already learned; then we can determine where in that
answer to put the decimal point. This is because, for example, we can
rewrite a division of decimals as a division of whole numbers times a
division of powers of ten:

1.95   195 * 0.01   195   0.01   195
---- = ---------- = --- * ---- = --- * 0.01 * 10 = 3 * 0.01 * 10
6.5    65 * 0.1      65   0.1     65

This says that, after dividing 195 by 65, we put the decimal point in
the same place it was in the dividend, 1.95 (multiplying 3 by 0.01 to
get 0.03), and then shift it one place to the right (multiplying by
10, which is the same as dividing by 1/10, to get 0.3).

The traditional way to make this memorable is to first move the
decimal points in the divisor and dividend until the dividend is
whole, and then put the decimal point in the quotient directly above
the new position in the dividend. This effectively shifts the decimal
point in the quotient first to the left, accounting for the decimal in
the dividend, and then the to right, accounting for the divisor.

So you see, it's not that we MUST divide only by a whole number;
rather, this is just one easily remembered way to decide where to put
the decimal point in the quotient. It's not much different from what
we do in multiplication of decimals, where we first ignore the decimal
point, and then add the number of decimal places in the two numbers.
In division, we are really SUBTRACTING the number of decimal places in
the two numbers; but explaining it in terms of moving the decimal
point ensures that we won't accidentally subtract in the wrong order.

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

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