Answer:
- Count the number of repeating digits. Call that number n.
- Multiply the number by 10^n
- Subtract the original number. Repeating digits should cancel.
- Divide that result by (10^n)-1. This is your fraction. Simplify as required.
Step-by-step explanation:
Following the above steps for the repeating decimal 0.083333... we have
1. n=1. There is one repeating digit: 3.
2. 10^1 times the number is 0.833333...
3. Subtracting the original number, we get ...
0.8333... -0.0833... = 0.7500...
4. Dividing this by 10^1 -1 = 9, we have ...
0.75/9 = 75/900 = 1/12
For this example, 0.0833333... = 1/12.
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Comment on the process
These steps always work, but they are not always easy. 1/17 is a repeating fraction with a 16-digit repeat. That means the denominator in step 4 is a string of 9s that is 16 digits long. Finding the common divisor between numerator and denominator so you can reduce the fraction may be beyond the capability of your calculator.
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Alternate solution
You can also convert the number to a "continued fraction". For each step of this stage, you write down the integer part of the number, then find the reciprocal of the fractional part and repeat.
If the integer parts are n1, n2, n3, ... then the equivalent fraction is ...
n1 +1/(n2 +1/(n3 +1/( ...)))
This process is expected to terminate after a few iterations. If your calculation is carried to full accuracy throughout, then the desired fraction (or mixed number) will result from simplifying this continued fraction. Otherwise, you may find that one of the nx values is substantially larger than the others. This generally means the fraction can be truncated at the previous result. Calculator internal rounding or truncation can come into play after several iterations, corrupting the result.
Example:
Consider the 16-digit repeat ...
[tex]0.\overline{2941176470588235}[/tex]
Using the first method above, we would be faced with reducing the fraction ...
[tex]\dfrac{2941176470588235}{9999999999999999}[/tex]
Using the second method, we get the integer sequence ...
0/3/2/2 ⇒ 0 +1/(3 +1/(2 +1/2)) = 1/(3 +2/5) = 5/17
(Note that the fact of a 16-digit repeat suggests the denominator is the prime number 16+1 = 17 or a multiple of it.)
