The decimal representation of any number is a linear combination of powers of 10. In other words, given a number like 123.456, we can expand it as
[tex]1\cdot10^2+2\cdot10^1+3\cdot10^0+4\cdot10^{-1}+5\cdot10^{-2}+6\cdot10^{-3}[/tex]
[tex]10^{-n}=\dfrac1{10^n}[/tex] for any [tex]n[/tex], so the above is the same as
[tex]100+20+3+\dfrac4{10}+\dfrac5{100}+\dfrac6{1000}=\dfrac{100000+20000+3000+400+50+6}{1000}=\dfrac{123456}{1000}[/tex]
Similarly, we can write
[tex]0.768=\dfrac{768}{1000}[/tex]
Now it's a question of reducing the fraction as much as possible. We have [tex]\mathrm{gcd}(768,1000)=8[/tex] so
[tex]\dfrac{768}{1000}=\dfrac{96\cdot8}{125\cdot8}=\dfrac{96}{125}[/tex]
