Question

What is 0.58 repeating as a fraction?

Answer 1

I.
x = 0.(58)=0.58585858...
 
      x = 0.58585858...
·100        ·100                (multiply both sides by100)
100x = 58.585858...
100x = 58 + 0.5858558...
100x = 58 + x
   -x          -x                  (subtract x from both sides)
 99x  =  58
:99        :99                  (divide both sides by 99)
   x  =  $\frac{58}{99}$


II.
x = 0.5(8)=0.588888... = 0.5+0.088888...

      y = 0.08888...
·10        ·10                (multiply both sides by10)
 10y =  0.88888...
·10        ·10                (multiply both sides by10)
100y = 8.88888..
100y = 8 + 0.88888...
100y = 8 + 10y
 -10y     -10y                     (subtract 10y from both sides)
  90y = 8
 :90      :90                  (divide both sides by 90)
   y = $\frac8{90}$

x = 0.5 + y

$x=\frac12+\frac8{90}=\frac{1\cdot45}{2\cdot45}+\frac8{90}=\frac{45}{90}+\frac8{90}=\frac{53}{90}$

Answer 2

-- If it's only the ' 8 ' that repeats forever,
then that decimal is equivalent to                53/90 .

-- If the ' 58 ' repeats forever, then
the decimal is equivalent to                        58/99 .