Finding fractions with this property

A

apple2357

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So a fraction like 26/65 simplifies to 2/5 which is not very interesting. However, if i cancelled the '6' from the top and bottom it gives a shortcut to the simplification. Clearly this is an illegal operation and not recommended but its fun.
I am trying to generate other fractions like this and the mathematics doesn't feel trivial.
so, i started with (10a+b)/(10b+c) = a/c
Suppose i decide to pick the cancel number as say 3 ( for no reason), this reduces the variables to play with:
(10a+3)/(30+c) =a/c
i then get 10a(3-b) = 2b
What next though? Do i just try whole numbers in a systematic way or is there a smarter way in?
 
(10a+b)/(10b+c) = a/c

First note that a, b and c are digits, c not 0.

Please solve your equation for one of the variables and we will go from there.
 
Jomo said:
(10a+b)/(10b+c) = a/c

First note that a, b and c are digits, c not 0.

Please solve your equation for one of the variables and we will go from there.
Click to expand...

Ok, solving for c, I get c= (10ab)/(9a+b)

Shall i just start running through (a,b) values?
 
I would clear fractions in the equation (without replacing any variable with a specific value); then you'll have a Diophantine equation in three variables, which you can rearrange in factored form. Some use of divisibility considerations and inequalities should help you reduce the number of possibilities to try.
 
Lots of trivial examples like 11/11 or 22/22 etc.
But i found these by searching and thinking about factors, i then verified it using a spreadsheet:

16/64, 19/65, 26/65, 49/98

I think thats all there are...
 
The next thing that i am going to explore is so called printer errors...

2^59^2=2592


The printer has treated the power like a normal number but the answer is still correct..

Why am i doing this? Got to occupy myself in lockdown.



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apple2357 said:
Lots of trivial examples like 11/11 or 22/22 etc.
But i found these by searching and thinking about factors, i then verified it using a spreadsheet:

16/64, 19/65, 26/65, 49/98

I think thats all there are...
Click to expand...
So I thought a bit about this a while this morning. I made it a bit harder by requiring that the resulting fraction have a numerator and denominator that are relatively prime. That would exclude 11/11, 22/22, etc. It also would exclude 49/98 because you get 4/8 rather than 1/2.

And I do not understand 19/65 at all. Did you mean 19/95?

That gives three 16/64, 19/95, and 26/65. More than I would have guessed.
 
apple2357 said:
Lots of trivial examples like 11/11 or 22/22 etc.
But i found these by searching and thinking about factors, i then verified it using a spreadsheet:

16/64, 19/65, 26/65, 49/98

I think thats all there are...
Click to expand...
For a detailed discussion of the problem (and confirmation that there are four, one being 19/95, not 19/65), see here.
 
I just realized that you are dividing just 2=digit numbers.

So if I may write 10a + b as ab and 10b + c as bc, then you want to check all fractions of the form ab/bc. Now there are a finite number of them to check. Keeping b constant for a moment, then there are 9*10 = 90 cases to check. b can only be from 1-9 from in end one needs to check 9*90=900 fractions. I do not like computers too much but I feel that this is a problem for a computer. Just write a program to look at the 900 fractions and give you a list of the fractions you want.

Now, if you not want to use a program, then you either look at the 900 cases or you first figure out (via a theorem/proof) that some pattern will not work or most probably figure out some facts (prove them!) while looking at the initial 900 so that you can rule out some fractions that needs to be looked. This problem is getting to be very interesting to me. Please show us your progress.

By the way, how did you confirm that the fractions that you listed are the only ones?
 
apple2357 said:
Lots of trivial examples like 11/11 or 22/22 etc.
But i found these by searching and thinking about factors, i then verified it using a spreadsheet:

16/64, 19/65, 26/65, 49/98

I think thats all there are...
Click to expand...
Why not include the trivial ones in your list??
 
After having c= (10ab)/(9a+b)

I let a=1 in a column on excel and let b run through 1 to 9, then a=2, b run through 1 to 9.. etc
didn't take long to set up on a spreadsheet and then i typed in the c formula and checked to see when c returned a whole number.
 
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