Adding Fractions Without Using the LCD

Date: 11/18/1999 at 09:48:12
From: Steve Russell
Subject: I found a new math formula

Once upon a time I was doing my math homework and I saw a problem. For 
instance, say the problem was 5/6 + 7/12. Well I didn't want to find 
the least common denominator, so I figured out a way to do it without 
having to find the LCD. It takes a little more reducing but it works. 
First you cross-multiply so you do 6 x 7 = 42 and 12 x 5 = 60. Then 
you add them together and it equals 102. Then you multiply the 
denominators 12 x 6 = 72 so you have 60/72. Then you reduce. It works 
out - if you try doing it the other way it equals 5/6, and if you do 
it my way it equals 5/6.

Will you please tell me if I found a new idea or not?
Thanks for your time.

Date: 11/18/1999 at 15:39:37
From: Doctor Rick
Subject: Re: I found a new math formula

Hi, Steve.

Your method works. It's not new, but it's a good observation. When you 
add or subtract fractions, you need to express both fractions with the 
same denominator. It doesn't have to be the *least* common 
denominator; *any* common denominator will do. The product of the two 
denominators works fine, as you have discovered; it is a multiple of 
both denominators, and it's easy to find.

As you have noted, you still have work to do to reduce the result you 
get. This work is the same as what you would do to find the least 
common denominator. One way to find the LCD is to multiply the two 
numbers and then divide by any factors that the two numbers had in 
common: 6 * 12 = 72, but since both 6 and 12 have a factor of 6, you 
divide 72 by 6 to get 12. When you simplified 102/72, you did the same 
thing: you found that the two numbers are both divisible by 6, so you 
divided through by 6 to get 17/12.

Often the LCD method is easier in the long run, because you have 
smaller numbers to factor at the end if the answer needs to be 
simplified. But I will sometimes use your method and put off the 
factoring until the end.

It's good to be aware that there is more than one method to solve a 
problem, so I hope you'll keep looking for methods other than what 
you're taught.

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

Date: 11/18/1999 at 16:56:47
From: Doctor Ian
Subject: Re: I found a new math formula

Hi Steve,

First, let me say that I admire your attitude, not wanting to do 
things by the book all the time! That's the kind of attitude that 
leads to discovery, and even when it doesn't, people with that 
attitude seem to have a lot more fun.

Can I suggest that you go to the library and get a book called _Surely 
You're Joking, Mr. Feynman_?  It's the autobiography of a guy that I 
think you'd really like, Richard Feynman.  He was a great physicist 
(winner of the Nobel Prize) and a great mathematician, but mostly he 
was just a terrifically interesting guy, who practically _never_ did 
anything by the book. As a result, I think he probably had more fun 
than any three other guys I can think of.

Don't worry - it's an easy read, and to be honest, if I could go back 
in a time machine and make one change to my life, it would be to give 
this book to myself when I was your age. I didn't find out about it 
until I was almost 30!

Anyway, on to your new method for adding fractions...

If you write down your method using letters instead of numbers, this 
is what you get:

     a   c
     - + - = ?     original problem
     b   d

     ad + bc       cross-multiply and add

     ad + bc
     -------       multiply denominators, and divide
       bd

What's interesting is that what you're doing _is_ finding a common 
denominator - it just isn't guaranteed to be the smallest one.

Here's another way of doing the same thing:

     a   c
     - + - = ?       original problem
     b   d

     a(d)   c(b)
     ---- + ----     multiply each term by n/n = 1
     b(d)   d(b)

     ad + bc
     -------         simplify
       bd

I think it's only because you're using numbers instead of letters that 
you didn't see what was going on.  Instead of fooling around dividing 
the denominators up front, you're postponing all the division until 
the final simplification step.

Anyway, the bad news is that you haven't really found a new way to add 
fractions - although what really counts is that it was new to you, and 
that no one told you how to do it.

The good news is that the method you're using is much easier than 
finding the lowest common denominator all the time. (But you don't 
need me to tell you that, do you?) I use your method instead of the 
one they teach in the books, and I suspect that a lot of 
mathematicians who have taken the time to play around a little do the 
same. 

Thanks for an interesting question, Steve. Check out the Feynman book, 
and be sure to write back if you make any more discoveries, or if you 
just have a question.

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