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### Dividing Fractions to find Decimals

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Date: 03/12/2001 at 19:58:13
From: Erin
Subject: Fractions into decimals and WHY

I have some students who do not understand dividing the numerator
into the denominator to find out the decimal. I cannot give them a
reason why you do this except that it works. Why does this operation
work to give you the decimal form of a fraction?
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Date: 03/12/2001 at 23:01:09
From: Doctor Peterson
Subject: Re: Fractions into decimals and WHY

Hi, Erin.

Let's be careful with our terminology, first: you mean dividing the
numerator BY the denominator! A lot of kids get that wrong.

The basic reason is that division is really what fractions are all
about. In a very real sense, we can say that a fraction is simply a
division we haven't bothered to perform yet, a division problem frozen
in time.

We write "3/4" to mean "I want to divide 3 by 4, but I don't want to
do the work just yet, so I can simplify the work before I finish up."
That's why we use the virgule "/" or the horizontal fraction bar to
signify division in more advanced math (and in computer programming
languages), rather than the old-fashioned obelus.

Let's go back to the basics of fractions to see why I can say this.
What are fractions? The essence of a fraction is a division, a
breaking into pieces. Take a whole object and divide it into 5 (equal)
pieces; each piece is 1/5. We've divided 1 by 5, just as we divide 10
by 5 by dividing a set of 10 things into 5 parts, each of which
consists of 2 objects. So 1 divided by 5 is 1/5; and 10 divided by 5
is the fraction 10/5, which simplifies to 2.

Likewise, we can divide 2 pies into 5 parts by dividing each pie into
5 parts (fifths) and taking 2 of them at a time: 2 divided by 5 is
2/5. The denominator represents the number of parts we divide each
whole into (a divisor); the numerator represents a multiplier, the
number of parts we have.

Let me repeat that, because it's easy to miss: 2/5 MEANS 2 divided by
5. The fraction IS a division.

Now look at the operations on fractions. When we multiply by a
fraction, we're really dividing: 1/2 * 10 is half of ten, or ten
divided by 2, or 10/2 again. Fractions mean division.

So when we want to convert a fraction to an ordinary (decimal) number,
all we're really doing is waking up a division problem that has been
in suspended animation, and letting it continue: "Where was I? Oh,
yeah ... 3 divided by 4 ... that's 0.75."

Here's another approach to the whole thing, focusing more on the
decimals themselves. A (terminating) decimal can be thought of as a
way to represent a fraction whose denominator is a power of ten. So
when we convert 3/4 to a decimal, we are looking for an equivalent
fraction whose denominator is, say, 100:

fraction  decimal

3         ?
---   =   ---
4        100

How do we do that? We have an object divided into 100 parts, and need
to know how many of them make 3/4 of the object. To do that, we can
multiply 100 by 3/4. But that is 300/4, and we can simplify that by
dividing both the numerator and the denominator by 4:

300   300 divided by 4   75
--- = ---------------- = -- = 75
4      4 divided by 4    1

So 3/4 = 75/100, or 0.75 - and we found the 75 by dividing 300 by 4,
which is exactly what you are doing when you divide 3 by 4 and put in
the decimal point where it belongs, two places from the right.

I hope one of these ideas will help.

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

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