Subtracting Improper Fractions

Date: 3/15/96 at 19:57:40
From: Anonymous
Subject: Math!

  Hi, I'm Jake. Here's a problem I don't get.

3 and 4 9ths - 2 and 8 7ths?

I would like some help.

Also 4 and 5 3rds + 3 and 6 7ths.

I am in 6th grade and would like help on those.

We have not started dividing fractions yet but
 please tell me how.   Thanks a ton!

Date: 3/19/96 at 15:33:39
From: Doctor Aaron
Subject: Re: Math!

Hi Jake,

You asked me to do 

3 and 4 9ths - 2 and 8 7ths.

I'll work through it a little bit.

The biggest problem about subtracting fractions is that the 
columns don't line up.  I'll give you an example:

  35                                       3 x 10 + 5 x 1
- 23                                   -   2 x 10 + 3 x 1
-----                                  ------------------
  12  is easy because it's the same as     1 x 10 + 2 x 1.

We can subtract in columns because the second column is the tens 
column and the first column is the ones column for both numbers.

The trick with fractions is to make the columns line up.

With 3 and 4 9ths, let's make the first column the ones column and 
the second one the ninths column.  

Then 3 and 4 9ths = 3 x 1 + 4 x 1/9.

But the natural second column for 2 and 8 7ths is the 7ths column.
Then 2 and 8 7ths = 2 x 1 + 8 x 1/7

If we try to subtract, we get:

   3 x 1 + 4 x 1/9
-  2 x 1 + 8 x 1/7
---------------------
   1 x 1 + ? x ?        - the columns don't line up so we can't 
subtract.

But we can rewrite the fractions to make them line up.  This is 
what finding the least common denominator is all about.  

I'm going to talk about fractions for a little while. If you 
already know this stuff, I apologize; otherwise I hope that it is 
useful.

You've probably heard that a fraction is like talking about the 
number of slices in an equally cut pizza or cake.  Well, if we 
take a pizza and cut it a few more times, we're going to get the 
same amount of pizza.

So one side of a pizza that has been cut in half can be 
represented as 1/2 where the 1 in the numerator stands for the 
number of pieces in the area that we are interested in, and the 2 
in the denominator stands for the total number of pieces.

If we cut the pizza in fourths, half of the pizza can be 
represented by 2/4, where 2 is the number of pieces in the area 
that we are interested in and 4 is the total number of pieces.

Another way to think about getting from a pizza cut in half to a 
pizza cut into fourths, is that we are cutting the pizza in half 
again, just instead of making the cut from side to side, we make 
it from top to bottom.  By doing this, we are doubling both the 
number of total pieces and the number of pieces in the region that 
we are interested in.  

We can think of this mathematically as multiplying 1/2 (the 
original representation of one side of the pizza) by 2/2 to get 
2/4 (the new representation of one side of the pizza).  Since 
2/2 = 1, and multiplying by 1 doesn't change the value of the 
number, it makes sense that multiplying by 1 doesn't change how 
much pizza we have, only the way we think about it.

Let's look at the problem again.

   3 x 1 + 4 x 1/9
-  2 x 1 + 8 x 1/7
---------------------
   1 x 1 + ? x ?       

The first thing that we have to take care of is that 8/7 is more 
than one.  We wouldn't write 12 in the ones column (unless we 
borrowed from the tens column) so we could re-write 8/7 as 1 + 
1/7. Then we have:

   3 x 1 + 4 x 1/9
-  3 x 1 + 1 x 1/7
---------------------
   0 x 1 + ? x ?       

It's a little bit of a jump, but you might see how we can use 
thepizza example to write 4/9 as 7/7 x 4/9 = 28/63, since 7/7 is 
just 1. Then we also write 1/7 as 9/9 x 1/7 = 9/63.  Our problem 
becomes much nicer:

   3 x 1 + 28 x 1/63
-  3 x 1 + 9 x 1/63
---------------------
   0 x 1 + 19 x 63 and we are done.   

If you understand this, you will also be able to get 4+5/3 + 
3+6/7, so I'll leave that one up to you.

Good luck, 

-Doctor Aaron,  The Math Forum