Question

Explain step by step how to make a repeating decimal become into a fraction form?

Answer 1

Suppose we have the repeat decimal 0.123232323.... or 0.123 with a line over the repeated part (check the diagram)
- The first thing we need to do is identify the part of the decimal that repeats, 23 in  our case.
- Second, we are going to assign a variable to our original decimal:      $x=0.123232323...$.
- Third, we are going to multiply both sides by a power of ten whose denominator will be the number of repeating digits. We know that we have 2 repeating digits (23), so we are going to multiply both sides by $10 ^{2}$, and $10 ^{2}$ is just 100; therefore we get:
 $100x=(100)(0.123232323...)$
$100x=12.32323232...$
- fourth, subtract our original equation from the second step from the one from above:
$\left \{ {{100x=12.323232...} \atop {-x=1.123232} \right.$
Now we can cancel the repeated decimals to get:
$\left \{ {{100x=12.3} \atop {-x=0.1}} \right.$
$99x=12.2$
$x= \frac{12.2}{99}$
- Last but not least multiply both numerator and denominator by a power of ten equals to the decimal digits in the numerator:
$x= \frac{12.2(10 ^{1}) }{99(10 ^{1}) } = \frac{122}{990}$

We now know how to convert a repeating decimal to a fraction.