Steps to Convert Repeating Decimals to Fractions

Example: Convert 0.3‾ to a Fraction

Set Up an Equation:

  • Let x=0.3‾
  • This means x=0.3333

Multiply by 10 (or a power of 10) to Shift the Decimal Point:

  • Multiply both sides of the equation by 10 to shift the decimal point one place to the right.
  • 10x = 3.3333..

Subtract the Original Equation from the New Equation:

  • Subtract the original x from 10x to eliminate the repeating part.
  • 10x−x=3.3333… − 0.3333…
  • 9x=3.

Solve for x:

  • x= 3⁄9
  • Simplify the fraction.
  • x= 1⁄3
  • So, 0.3‾= 1⁄3

Example: Convert 0.8̅1 to a Fraction

Set Up an Equation:

  • Let x=0.8̅1
  • This means x=0.818181…

Multiply by 100 (to Shift the Decimal Two Places):

  • Multiply both sides of the equation by 100 to shift the decimal two places to the right.
  • 100x = 81.818181…

Subtract the Original Equation from the New Equation:

  • Subtract the original x from 100x to eliminate the repeating part.
  • 100x−x = 81.818181…−0.818181…
  • 99x=81.

Solve for x:

  • x = 81⁄99
  • Simplify the fraction.
  • x = 9⁄11

so, 0.8̅1 = 9⁄11

General Formula

For a repeating decimal 0.a̅

  • x= 0.a̅
  • Multiply by 10ⁿ where n is the number of repeating digits.
  • Subtract the original equation from the new equation.
  • Solve for x and simplify.

For a repeating decimal with a non-repeating part (e.g., 0.1̅6):

Set Up an Equation:

  • Let x = 0.1̅6

Multiply by a Power of 10 to Shift the Decimal Past the Non-Repeating Part:

  • Multiply both sides by 10: 10x =1.6̅
  • Multiply both sides by 100 to shift past the repeating part: 100x=16.6‾

Subtract the Two Equations to Isolate the Repeating Part:

  • 100x−10x=16.6‾−1.6‾.
  • 90x=15.

Solve for x:

  • x = 15⁄90
  • Simplify: x=15⁄90
  • x = 1⁄6

So, 0.16‾=1⁄6

Let’s Practice!

Pages: 1 2