Question

Complete the table by converting each decimal to a fraction.

Answer 1

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....

$\bf 0.\overline{13}~\hspace{10em}x=0.\overline{13} \\\\[-0.35em] ~\dotfill\\\\ \begin{array}{|lll|ll} \cline{1-3} &&\\ 100\cdot 0.\overline{13}& = & 13.\overline{13}\\ 100\cdot x&& 13 + 0.\overline{13}\\ 100x&&13+x \\&&\\ \cline{1-3} \end{array}\implies \begin{array}{llll} 100x=13+x\implies 99x=13 \\\\ x=\cfrac{13}{99} \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ 0.\overline{1234}~\hspace{10em}x=0.\overline{1234} \\\\[-0.35em] ~\dotfill$

$\bf \begin{array}{|ccc|ll} \cline{1-3} &&\\ 10000\cdot 0.\overline{1234}&=&1234.\overline{1234}\\ 10000\cdot x&=&1234+0.\overline{1234}\\ 10000x&=&1234+x \\&&\\ \cline{1-3} \end{array}\implies \begin{array}{llll} 10000x=1234+x \\\\ 9999x=1234\implies x=\cfrac{1234}{9999} \end{array}$

Answer 2

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!!!!                                                                       : )