# fraction paradox that is probably wrong...

Duck Duck Goose
Posers and Puzzles 25 May '05 01:09
1. 25 May '05 01:091 edit
let's say that we have 2y+10 over x + 3. We need to add that fraction to 2y+20 over 1/2x + 1.5 . the logical thing to do would be to multiply the second fraction by 2 to get common denominators, right? so we should multiply the top of the second fraction by 2 over the bottom half times 2. but if you multiply something by 2 over 2, isn't that kind of just like multiplying it by one? 2/2= 1...

2y+ 20 x 2
-------- --
1/2x+1.5 2... 2/2 is one, so why do we get a different fraction when multiplying by one?

what about simplifying? how can you simplify something like 20/4 by dividing it by 2 on each side? 2/2 is one
2. orfeo
Paralysed analyst
25 May '05 03:291 edit
Originally posted by Duck Duck Goose
let's say that we have 2y+10 over x + 3. We need to add that fraction to 2y+20 over 1/2x + 1.5 . the logical thing to do would be to multiply the second fraction by 2 to get common denominators, right? so we should multiply the ...[text shortened]... y something like 20/4 by dividing it by 2 on each side? 2/2 is one
The whole POINT of multiplying by, say, 2/2 is that you're not changing the value of something, just the way of writing it. What's paradoxical about that?

20/4 = 5, and 10/2 = 5, and 5/1 = 5. The whole aim is to write something in the simplest way possible.

There are no 'sides' to your 20/4 example, by the way, an EQUATION has sides.
3. 25 May '05 10:511 edit
but isn't 2/2 one? that way, when you multiply the fraction by 1, it doesn't change anything
4. orfeo
Paralysed analyst
25 May '05 11:021 edit
Originally posted by Duck Duck Goose
but isn't 2/2 one? that way, when you multiply the fraction by 1, it doesn't change anything
EXACTLY! That's what I'm saying. What is your problem with that idea?

You said in your first post, "you get a different fraction". It's a differently written fraction WITH THE SAME VALUE.

Try putting any value of x in both of your fractions, you will get exactly the same answer.
5. Hegemon
A Lost Bobby
25 May '05 13:22

2y+20 2
--------- x -
.5x+1.5 2

= 4y+40
-------
x+3

you are doublying both the numerator and the denominator.

what's the issue..
6. Bowmann
Non-Subscriber
26 May '05 00:35
What a plonker!
7. 30 May '05 07:21
Haha, at least he asks the question, I spose
8. genius
Wayward Soul
31 May '05 19:11
yeah-if you don't understand something, ask! not everyone's quite so rude as bowmann. in general, we're a friendly bunch. at times. every now and then. if you're very nice. and bring cookies...but you've gotta leave your cat at the door, okay?
9. TheMaster37
Kupikupopo!
01 Jun '05 08:37
Yeah, we're friendly...

You'll agree that your second fraction is 'ugly'. To write is in a simpler form you have to do something to it, without changing the value of the fraction in all points x.

EG instead of writing 1/(1/2) you simply write 2. What we did here is multiply 1/(1/2) with 1=2/2. The same trick can be done to other fractions as well, like yours for example.

Multiplying with 1 doesn't do anything, you're right about that. But i CAN change the way we write something. There are many way to write one single fraction. We resort to tricks like this to write them in the most convenient form.
10. genius
Wayward Soul
01 Jun '05 09:25
Originally posted by TheMaster37
Yeah, we're friendly...

You'll agree that your second fraction is 'ugly'. To write is in a simpler form you have to do something to it, without changing the value of the fraction in all points x.

EG instead of writing 1/(1/2) you simply write 2. What we did here is multiply 1/(1/2) with 1=2/2. The same trick can be done to other fractions as we ...[text shortened]... te one single fraction. We resort to tricks like this to write them in the most convenient form.
it can be very useful, for instance 2/(2^0.5), i.e. two divided by root two, is actually root two! miltiply by (2^0.5)/(2^0.5),

2 2^0.5
------- x -------
2^0.5 2^0.5

=2 x 2^0.5
--------------
. 2
the two's cancel, so your left with 2^0.5. i still find that beautiful...
11. 02 Jun '05 16:59
Originally posted by genius
it can be very useful, for instance 2/(2^0.5), i.e. two divided by root two, is actually root two! miltiply by (2^0.5)/(2^0.5),

2 2^0.5
------- x -------
2^0.5 2^0.5

=2 x 2^0.5
--------------
. 2
the two's cancel, so your left with 2^0.5. i still find that beautiful...
actualy this example isn't even as complex as that.
x/(x^.5)=(x^.5)^2*(x^.5)^-1=(x^.5)^(2-1)=(x^.5)^1=x^.5