converting a repeating decimal to a fraction

The following algorithmMathworldPlanetmath can be used to convert a repeating decimal to a fraction:

  1. 1.

    Set the repeating decimal equal to x.

  2. 2.

    Multiply both sides of the equation by 10n, where n is the number of digits that appear under the bar.

  3. 3.

    If applicable, rewrite the second equation so that its repeating part up with the repeating part in the original equation.

  4. 4.

    Subtract the original equation from the most recently obtained equation. (The repeating part should cancel at this step.)

  5. 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.

  6. 6.

    Divide both sides of the equation by the coefficient of x.

  7. 7.

    Reduce the fraction to lowest terms.

Below, this algorithm is demonstrated for 0.583¯ with the steps indicated on the far .

x=0.583¯ (1)
10x=5.83¯ (2)
10x=5.833¯ (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 0.9¯=1:

x =0.9¯
10x =9.9¯
9x =9
x =1
Title converting a repeating decimal to a fraction
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