converting a repeating decimal to a fraction
The following algorithm can be used to convert a repeating decimal to a fraction:
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1.
Set the repeating decimal equal to x.
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2.
Multiply both sides of the equation by 10n, where n is the number of digits that appear under the bar.
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3.
If applicable, rewrite the second equation so that its repeating part up with the repeating part in the original equation.
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4.
Subtract the original equation from the most recently obtained equation. (The repeating part should cancel at this step.)
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5.
If applicable, multiply both sides by a large enough power of 10 so that the equation is of the form ax=b, where a and b are integers.
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6.
Divide both sides of the equation by the coefficient of x.
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7.
Reduce the fraction to lowest terms.
Below, this algorithm is demonstrated for 0.58ˉ3 with the steps indicated on the far .
x=0.58ˉ3 | (1) |
10x=5.8ˉ3 | (2) |
10x=5.83ˉ3 | (3) |
9x=5.25 | (4) |
900x=525 | (5) |
x=525900 | (6) |
x=712 | (7) |
An important application of this algorithm is that it supplies a proof for the fact that 0.ˉ9=1:
x | =0.ˉ9 | ||
10x | =9.ˉ9 | ||
9x | =9 | ||
x | =1 |
Title | converting a repeating decimal to a fraction |
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Canonical name | ConvertingARepeatingDecimalToAFraction |
Date of creation | 2013-03-22 16:55:22 |
Last modified on | 2013-03-22 16:55:22 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 10 |
Author | Wkbj79 (1863) |
Entry type | Algorithm |
Classification | msc 11A99 |
Classification | msc 11-00 |