9514 1404 393
Answer:
5/3
Step-by-step explanation:
In general, you count the number of repeating digits and use that in the following script.
- name the number: x = 1.666...
- multiply by 10 to the power of the number of repeating digits: 10¹x = 16.666...
- subtract the original number. The repeating digits should cancel. 10¹x -x = 9x = 15.000...
- divide by the coefficient of the variable: x = 15/9
- reduce the fraction: x = 5/3 = 1 2/3
The improper fraction is 5/3.
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<em>Additional comment</em>
When the repeat starts at the decimal point, you can create the fraction by dividing the repeating digits by the same number of 9s.
1/7 = 0.142857...(6-digit repeat)
= 142857/999999
When the repeat has a number of digits, reducing the fraction isn't always easy.
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If the repeat doesn't start at the decimal point, you can use the above script, or scale the fraction according to where it starts.
<u>Example</u>: 0.4333...(1-digit repeat) = 0.4 + (1/10)(0.333...) = 0.4 + (1/10)(3/9) = 4/10 +1/30 = 13/30
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Any fraction whose denominator includes factors other than 2 or 5 will be a repeating decimal. It can be helpful to memorize the repeating decimals associated with small-integer fractions:
1/3 = 0.3...(1-digit repeat)
1/6 = 0.16...(1-digit repeat)
1/7 = 0.142857...(6-digit repeat)
1/9 = 0.1...(1-digit repeat) -- of course 2/9, 1/3, etc are multiples of this
1/11 = 0.09...(2-digit repeat) -- as with 1/9, multiples of 1/11 are multiples of this.
1/12 = 0.083...(1-digit repeat)
