What is the logic behind this trick in Division?

Question

I was just doing a few sums of Probability in my Statistics subject, and stumbled upon this method of easy division. I don't know if it is already known worldwide, or if there is a Law or Principle about it, but I just found it while doing my assignment.

I got a fraction: 48/120. I had to bring it in the simplest form. What I did in a fluke manner was:

>> Divided 48/120 = 1/2.5

>> Did (48*2)-120= -24

>> Divided both by numerator and denominator by 24 and got 2/5 in an easy manner.

Now, how does this work? I mean what is the logic behind this trick?

Also, I noted that it doesn't work always. For example, if I have 12/45 then the answer I get when I divide it is 3.75. Now if I do (12*3)-45, I won't get the right number to use to divide the both. But if I do (12*4)-45 I get the right nuber to use to divide both numerator and denominator.

I hope you understood what I did and mean to say. I mean that when the division brings the result like 2.5, I can take 2 OR 3 and divide both numer. and denom. by it.

But if I get 2.25 then I HAVE to take 2 to divide both. Same way if I get 2.75 then I HAVE to take 3 to divide both.

Now, what I want to know is how this happens? Sometimes this trick might help in simplifying larger fractions.

Hope you understood my question. Please ask me, and I'll additional info if you have any questions.

Answer 1

Actually,there is no logicality in it.

If we choose 3/9 as an example

divided 3/9 = 1/3 (used your idea we can get 3)

did 3*3-9 = 0  

how can you do next step?

Or if we choose 84/133 as another example

divided 84/133=1/1.58333333(used your idea we can get 1 or 2)

did 84*1-133=-49     84*2-133 =35

we can not divide both numerator and denominator by 49 or 35

and there are many examples like that.......so there is no logicality in your idea.

Answer 2

You seem to be looking for the largest number that can be divided into both top and bottom.  That is called the greatest common divisor of top and bottom, abbreviated gcd.  The systematic method to find it is called the Euclidean algorithm (after Euclid, more than 2000 years ago).  You can find it at this web site:

http://www.math.umn.edu/~garrett/crypto/...

For your first example the computations look like:

120 - (2) * (48) = 24

48 - (2) * (24) = 0

So gcd of 120 and 48 is 24

For the "logic behind it" google "Euclidean algorithm" and poke around a bit.