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:

$$\frac{10a+b}{10b+c}=\frac{a}{c}$$

$$10ac+bc=10ab+ac$$ ...Multiply through.

$$10a(c-b)=c(a-b)$$ ...Factorize

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

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

$$10ab+ac-10b^2-bc=100ac+10bc-100ab-10b^2$$ ...multiply through and expand

$$10ab=9ac+bc$$ ...simplify

Maybe you can work a bit with this equality.