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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

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

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

-- + -- = --
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

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

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
```
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