Multiplying/Adding Fractions Gives Same Answer

Date: 03/01/2002 at 10:21:49
From: Ryan
Subject: Obtaining the same answer when multiplying and adding 2 
fractions

I don't know how to begin finding out the answer to this. I need to 
find two fractions that when they are multiplied and added give the 
same answer.  

I have two examples I can't use: 13/4 + 13/9 and 13/4 x 13/9 both have 
the same answer, and 169/30 + 13/15 and 169/30 x 13/15 both have the 
same answer.  

Can you find another occurrence of this?

Date: 03/01/2002 at 12:32:03
From: Doctor Ian
Subject: Re: Obtaining the same answer when multiplying and adding 2 
fractions

Hi Ryan,

If we write the fractions as a/b and c/d, then the condition to be 
satisfied is that

  a   c   a   c
  - + - = - * -  
  b   d   b   d

   ad   cb   ac
   -- + -- = --
   bd   db   bd


   ad + cb = ac


Let's check that with your examples:

1)  13*9 + 13*4 = 13*13

      13(9 + 4) = 13*13

          13*13 = 13*13              Okay.

2)    13*30 + 169*15 = 13*169

    13*30 + 13*13*15 = 13*169

      13(30 + 13*15) = 13*169
  
              13*195 = 13*169

                 195 = 169            Oops!

Just to take a completely different approach, let's check that with a 
calculator:

  169/30 = 5.63

   13/15 = 0.87

  5.63 + 0.87 = 6.50

  5.63 * 0.87 = 4.90

With a little thought, we can see that this pair _couldn't_ work, 
since (if we're working with positive numbers) for the product to be 
equal to the sum, both fractions would have to be greater than 1.  
Otherwise, a/b gets bigger when you add c/d to it; but it gets smaller 
when you multiply it by c/d.

Does this make sense? Anyway, let's forget about your second example 
for now. In particular, let's look at 

    13*9 + 13*4 = 13*13

      13(9 + 4) = 13*13

          13*13 = 13*13

Is there a pattern here that we can exploit? 

Suppose we took a different square, like 17*17, and broke 17 into, 
say, 5 + 12:

    17*(5 + 12) = 17*17

    17*5 + 17*12 = 17*17

     a d    c  b    a  c

This would give us the equation

       17   17   17   17
       -- + -- = -- * --
       12    5   12    5

  17*5 + 17*12    17*17
  ------------ = -------
      12*5        12*5

  17*(5 + 12)     17*17
  ------------ = -------
      12*5        12*5


Do you think this would work for 

  17   17   17   17
  -- + -- = -- * --  ?
   1   16    1   16

How about 

  17   17   17   17
  -- + -- = -- * --  ?
   2   15    2   15

How about 

  53   53   53   53
  -- + -- = -- * --  ?
  26   27   26   27

Can you generalize this, to come up with a formula for cranking out 
more pairs? 
  
Note that there are also trivial pairs, like

  0   0   0   0
  - + - = - * -
  1   2   1   2

As you've stated the problem, these are sneaky, but not illegal.  

I hope this helps. This was an interesting question! Thanks for asking 
it.  Let me know if you'd like to talk more about this, or anything 
else. 

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

Date: 03/01/2002 at 13:24:30
From: Ryan
Subject: Obtaining the same answer when multiplying and adding 2 
fractions

Wow! That was interesting, and you did a great job figuring that out 
and explaining it. I appreciate your quick response.

Ryan