Respuesta :
first off, we'll move the non-repeating part in the decimal to the left-side, by doing a division by a power of 10.
then we'll equate the value to some variable, and move the repeating part over to the left as well.
anyhow, the idea being, we can just use that variable, say "x" for the repeating bit, let's proceed,
[tex]\bf 0.580\overline{80}\implies \boxed{\cfrac{5.80\overline{80}}{10}}\qquad \textit{now, let's say }x= 5.80\overline{80}\\\\ -------------------------------[/tex]
[tex]\bf thus\qquad \begin{array}{llll} 100\cdot x&=&580.80\overline{80}\\ &&575+5.80\overline{80}\\ &&575+x \end{array}\qquad \implies 100x=575+x \\\\\\ 99x=575\implies x=\cfrac{575}{99}\qquad therefore\qquad \boxed{\cfrac{5.80\overline{80}}{10}}\implies \cfrac{\quad \frac{575}{99}\quad }{10} \\\\\\ \cfrac{\quad \frac{575}{99}\quad }{\frac{10}{1}}\implies \cfrac{575}{99}\cdot \cfrac{1}{10}\implies \cfrac{575}{990}\implies \stackrel{simplified}{\cfrac{115}{198}}[/tex]
and you can check that in your calculator.
then we'll equate the value to some variable, and move the repeating part over to the left as well.
anyhow, the idea being, we can just use that variable, say "x" for the repeating bit, let's proceed,
[tex]\bf 0.580\overline{80}\implies \boxed{\cfrac{5.80\overline{80}}{10}}\qquad \textit{now, let's say }x= 5.80\overline{80}\\\\ -------------------------------[/tex]
[tex]\bf thus\qquad \begin{array}{llll} 100\cdot x&=&580.80\overline{80}\\ &&575+5.80\overline{80}\\ &&575+x \end{array}\qquad \implies 100x=575+x \\\\\\ 99x=575\implies x=\cfrac{575}{99}\qquad therefore\qquad \boxed{\cfrac{5.80\overline{80}}{10}}\implies \cfrac{\quad \frac{575}{99}\quad }{10} \\\\\\ \cfrac{\quad \frac{575}{99}\quad }{\frac{10}{1}}\implies \cfrac{575}{99}\cdot \cfrac{1}{10}\implies \cfrac{575}{990}\implies \stackrel{simplified}{\cfrac{115}{198}}[/tex]
and you can check that in your calculator.
Repeating decimals are defined as decimal numbers that have a number or a group of numbers that is occurring over and over again.
0.58080 as a repeating fraction is [tex]\frac{363}{625}[/tex]
To make 0.58080 a repeating fraction:
- Step 1: We multiply 0.58080 by 10 raised to the power of 5
0.58080 x [tex]10^{5}[/tex]
= 58080
- Step 2: We divide 58080 by [tex]10^{5}[/tex] until we get the simplest fraction.
[tex]\frac{58080}{100000}[/tex]
The greatest common factor of 58080 and 100000 is 160, so we divide the numerator and denominator by 160
= [tex]\frac{363}{625}[/tex]
Therefore, 0.58080 as a repeating fraction is [tex]\frac{363}{625}[/tex]
To learn more, visit the link below:
https://brainly.com/question/21846566
