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Write 0.611111111111 as a fraction PLEASE EXPLAIN STEP BY STEP!

User XMythicx
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We have to write N=0.6111... as a fraction.

This is a periodic number.

We start by transforming the number as:


10\cdot N=10\cdot0.6111\ldots=6.111\ldots=6+0.111\ldots

Now we take the periodic part we have (x=0.111...) and express it like this:


10x=10\cdot0.111\ldots=1.111\ldots=1+0.111\ldots=1+x

Then, we have:


\begin{gathered} 10x=1+x \\ 10x-x=1 \\ 9x=1 \\ x=(1)/(9) \end{gathered}

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:


\begin{gathered} 10N=6+0.111\ldots=6+(1)/(9) \\ N=(1)/(10)(6+(1)/(9))=(1)/(10)((6\cdot9)/(9)+(1)/(9))=(1)/(10)\cdot(54+1)/(9)=(1)/(10)\cdot(55)/(9)=(55)/(90) \end{gathered}

Then, 0.6111... as a fraction is 55/90.

User Dmestrovic
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