Repeating Decimal - Converting Repeating Decimals To Fractions

Converting Repeating Decimals To Fractions

Given a repeating decimal, it is possible to calculate the fraction that produced it. For example:

\begin{alignat}2 x &= 0.333333\ldots\\ 10x &= 3.333333\ldots&\quad&\text{(multiplying each side of the above line by 10)}\\ 9x &= 3 &&\text{(subtracting the 1st line from the 2nd)}\\ x &= 3/9 = 1/3 &&\text{(reducing to lowest terms)}\\ \end{alignat}

Another example:

\begin{align} x &= 0.836363636\ldots\\ 10x &= 8.3636363636\ldots\text{(multiplying by a power of 10 to move decimal to start of repetition)}\\ 1000x &= 836.36363636\ldots\text{(multiplying by a power of 100 to move decimal to end of first repeating decimal)}\\ 990x &= 836.36363636\ldots - 8.36363636\ldots = 828 \text{ (subtracting to clear decimals)}\\ x &= \frac{828}{990} = \frac{18 \times 46}{18 \times 55} = \frac{46}{55}. \end{align}

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