A repeating decimal is a decimal number where one or more of the digits repeats indefinitely. For example, when you convert the fraction 1/3 to a decimal, you get 0.333333 . . . where the 3s repeat indefinitely. That makes the decimal representation of 1/3 a repeating decimal. It can be written with a line above the number or numbers that are repeated, like this:
0.3 = 0.333333 . . .
Repeated numbers can be more than a single digit.
For example, if you divide 5 by 7 you get 0.714285 714285 714285 . . . , which can be represented as
Note: All repeating decimals are the result of dividing one whole number by another.
In other words, by converting a fraction to a decimal.
It is possible, since all repeating decimals are fractions, to reconstruct the fraction, given the repeating decimal. This is how you do it:
Now here are some problems for you (use your calculator):
- Multiply your number by a power of 10 in order to get the repeating part of the decimal to the LEFT of the decimal point.
For example, in the repeating decimal 0.55555555 . . . multiply by 10 to get:
5.5555555 . . .
- Calling the original repeating decimal X, this new value is 10X
- Subtract the smaller value from the larger value:
10X = 5.55555555555 . . .|
- X = 0.55555555555 . . .
9X = 5
X = 5⁄9
This last step has gotten all the repeating decimals to cancel each other, except for the first one(s).|
Now, it's easy! 10X - X = 5, so 9X = 5 and the fraction is 5⁄9
(divide 5 by 9 on your calculator and see if this is true!)
Convert the following fractions to decimals.
They will result in repeating decimals. First find the decimal representation of the fraction and then write them with the line above the repeating part
(and only the repeating part!):
1/12 = _____________ = ______
7/9   = _____________ = ______
11/13 = ____________ = ______
Convert 0.4 back into a fraction:
- Expand it to a repeating decimal:
0.4 = _____________ . . .<= this is X
- Multiply it by 10:
__________________ . . .<= this is 10X
- Subtract the original expanded decimal from it:
10X: ___________ . . .
- X: ___________ . . .
9X: ____________,    so X = ______