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}

Read more about this topic:  Repeating Decimal

Famous quotes containing the words converting and/or repeating:

    A way of certifying experience, taking photographs is also a way of refusing it—by limiting experience to a search for the photogenic, by converting experience into an image, a souvenir. Travel becomes a strategy for accumulating photographs.
    Susan Sontag (b. 1933)

    Alone with our madness and favorite flower
    We see that there really is nothing left to write about.
    Or rather, it is necessary to write about the same old things
    In the same way, repeating the same things over and over
    For love to continue and be gradually different.
    John Ashbery (b. 1927)