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: [tex]x=0.123232323...[/tex].
- 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 [tex]10 ^{2} [/tex], and [tex]10 ^{2} [/tex] is just 100; therefore we get:
[tex]100x=(100)(0.123232323...)[/tex]
[tex]100x=12.32323232...[/tex]
- fourth, subtract our original equation from the second step from the one from above:
[tex] \left \{ {{100x=12.323232...} \atop {-x=1.123232} \right. [/tex]
Now we can cancel the repeated decimals to get:
[tex] \left \{ {{100x=12.3} \atop {-x=0.1}} \right. [/tex]
[tex]99x=12.2[/tex]
[tex]x= \frac{12.2}{99} [/tex]
- Last but not least multiply both numerator and denominator by a power of ten equals to the decimal digits in the numerator:
[tex]x= \frac{12.2(10 ^{1}) }{99(10 ^{1}) } = \frac{122}{990} [/tex]
We now know how to convert a repeating decimal to a fraction.
