Rewrite 0.580 80 repeating decimal

Step-by-step explanation:

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,

\begin{gathered}\bf 0.580\overline{80}\implies \boxed{\cfrac{5.80\overline{80}}{10}}\qquad \textit{now, let's say }x= 5.80\overline{80} \end{gathered}

0.580

80

⟹

10

5.80

80

now, let’s say x=5.80

80

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

\begin{gathered}\bf thus\qquad \begin{array}{} 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}}\end{gathered}

thus

100⋅x

=

580.80

80

575+5.80

80

575+x

⟹100x=575+x

99x=575⟹x=

99

575

therefore

10

5.80

80

⟹

10

99

575

1

10

99

575

⟹

99

575

⋅

10

1

⟹

990

575

⟹

198

115

simplified

Because 80 is repeat and simply .80 = .8

So, answer is 0.5(8)

Good luck.