if you've read those links already, you'd know what we're doing here.
we'll move the repeating part to the left-side of the dot, by multiplying by "1" and as many zeros as needed, or 10 at some power pretty much.
on 0.13 we need 100 to get 13.13.... and on 0.1234, we need 10000 to get 1234.1234....
[tex] \bf 0.\overline{13}~\hspace{10em}x=0.\overline{13} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{|lll|ll} \cline{1-3} &&\\ 100\cdot 0.\overline{13}& = & 13.\overline{13}\\ 100\cdot x&& 13 + 0.\overline{13}\\ 100x&&13+x \\&&\\ \cline{1-3} \end{array}\implies \begin{array}{llll} 100x=13+x\implies 99x=13 \\\\ x=\cfrac{13}{99} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 0.\overline{1234}~\hspace{10em}x=0.\overline{1234} \\\\[-0.35em] ~\dotfill [/tex]
[tex] \bf \begin{array}{|ccc|ll} \cline{1-3} &&\\ 10000\cdot 0.\overline{1234}&=&1234.\overline{1234}\\ 10000\cdot x&=&1234+0.\overline{1234}\\ 10000x&=&1234+x \\&&\\ \cline{1-3} \end{array}\implies \begin{array}{llll} 10000x=1234+x \\\\ 9999x=1234\implies x=\cfrac{1234}{9999} \end{array} [/tex]
Question 1.). 0.8 (8 Repeating) ======> 8/9
Showing the work:
Let X equal the decimal number: 0.8 (8 Repeating)
Equation 1:
X = 0.8 (8 Repeating)
With 1 digit in the repeating decimal group, create a second equation by multiplying both sides by 10^1 = 10:
Equation 2:
10X = 8. 8 (8 Repeating).
Subtract equation (1) from equation (2)
10X 8.8....
X = 0.8...
9X = 8
We get ======> 9X = 8
Solve For X:
X = 8 / 9 ======> In Conclusion, 0. 8 (8 Repeating) ======> Answer 8 / 9
Question 2.). 0.13 ====> 13 / 99
Showing the work
Let X equal the decimal number: 0.13 (13 Repeating)
Equation 1:
With 2 digits in the repeating decimal group, create a second equation by multiplying both sides by 10^2 = 100:
Equation 2:
100X = 0.13 (13 Repeating).
Subtract equation (1) from equation (2)
100X = 13.13....
X = 0.13...
99X = 13
We get ======> 99X = 13
Solve For X:
X = 13 / 99 ======> In Conclusion, 0. 13 (13Repeating) ====> Answer 13 / 99
Question 3.). 0.175 ===> 175 / 9990.175 ===> 175 / 999
Showing the work
Let X equal the decimal number: 0.175 (175 Repeating)
Equation 1:
With 3 digits in the repeating decimal group, create a second equation by multiplying both sides by 10^3 = 1000:
Equation 2:
1000X = 0.175 (175 Repeating).
Subtract equation (1) from equation (2)
1000X = 175.175 (175 Repeating) ….
X = 175. 175..
X = 0.175
We get ======> 999X = 175
Solve For X:
X = 175 / 999 => In Conclusion, 0. 175 (175 Repeating) ==> Answer 175 / 999
Question 4.). 0.1234 =====> 1234 / 9999
0.1234 ===> 1234 / 9999
Showing the work
Let X equal the decimal number: 0.1234 (1234 Repeating)
Equation 1:
With 4 digits in the repeating decimal group, create a second equation by multiplying both sides by 10^4 = 10000:
Equation 2:
10000X = 1234. 1234 (1234 Repeating).
Subtract equation (1) from equation (2)
10000X = 1234.1234 (1234 Repeating) ….
X = 1234. 1234..
X = 0.1234
We get ======> 9999X = 1234
Solve For X:
X = 1234 / 9999 ===> In Conclusion, 0. 1234 (1234 Repeating) ==> Answer 1234 / 9999
Hope that helps!!!! : )
