Dividing Fractions to find Decimals

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?

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/