Find Unusual Fractions: 16/64, 10a+b/10b+c = a/c

[dr hannibal]

16/64 is an unusual fraction such that when you cancel 6's from the top and bottom youre left with correct answer of 1/4.
I need to Find a way for finding all other unusal fractions for a fraction n/m such that n>=11 , and m <=99.

I have tried writing it as (10a+b)/(10b+c)=a/c ,but stuck
Thanks for any help

NB:this is not a homework question to the mods

 

[Mentallic]

Here's a start:

[tex]\frac{10a+b}{10b+c}=\frac{a}{c}[/tex]

[tex]10ac+bc=10ab+ac[/tex] ...Multiply through.

[tex]10a(c-b)=c(a-b)[/tex] ...Factorize

[tex]\frac{a}{c}=\frac{a-b}{10(c-b)}[/tex] ...Solve for a/c

[tex]\frac{10a+b}{10b+c}=\frac{a-b}{10(c-b)}[/tex] ...Substitute from the original equation for a/c

[tex]10ab+ac-10b^2-bc=100ac+10bc-100ab-10b^2[/tex] ...multiply through and expand

[tex]10ab=9ac+bc[/tex] ...simplify

Maybe you can work a bit with this equality.