Respuesta :
We have to write N=0.6111... as a fraction.
This is a periodic number.
We start by transforming the number as:
[tex]10\cdot N=10\cdot0.6111\ldots=6.111\ldots=6+0.111\ldots[/tex]Now we take the periodic part we have (x=0.111...) and express it like this:
[tex]10x=10\cdot0.111\ldots=1.111\ldots=1+0.111\ldots=1+x[/tex]Then, we have:
[tex]\begin{gathered} 10x=1+x \\ 10x-x=1 \\ 9x=1 \\ x=\frac{1}{9} \end{gathered}[/tex]We use 10 to have the non-periodic part as an integer and the periodic part as a decimal.
Now we know that our periodic part of the number is equal to 1/9.
So we come back to N and complete:
[tex]\begin{gathered} 10N=6+0.111\ldots=6+\frac{1}{9} \\ N=\frac{1}{10}(6+\frac{1}{9})=\frac{1}{10}(\frac{6\cdot9}{9}+\frac{1}{9})=\frac{1}{10}\cdot\frac{54+1}{9}=\frac{1}{10}\cdot\frac{55}{9}=\frac{55}{90} \end{gathered}[/tex]Then, 0.6111... as a fraction is 55/90.
