Class Notes: Converting Fractions to Repeating Decimals

Objective

Learn how to convert fractions (especially those with denominators 3, 6, 9, or their multiples) into repeating decimals using long division and observe common patterns.

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Steps to Convert a Fraction to a Repeating Decimal

1. Set Up Long Division

   • The denominator is the divisor (outside).
   • The numerator is the dividend (inside).

2. Perform Division

   • If the numerator is smaller, add a decimal point and zeros.
   • Divide as normal, track remainders.

3. Detect Repeating Patterns

   • If the same remainder repeats, the decimal will start repeating at that point.
   • Use a bar over repeating digits to indicate repetition.

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Examples

1. ( \frac{1}{3} )

• Set up: 1 divided by 3.
• 3 goes into 10 three times (3 × 3 = 9).
• Remainder is 1, repeats the process.
• Decimal: ( 0.\overline{3} )

2. ( \frac{2}{3} )

• ( 2 \times \frac{1}{3} = 2 \times 0.\overline{3} = 0.\overline{6} )

3. ( \frac{1}{6} )

• 1 divided by 6.
• 6 goes into 10 once, remainder 4.
• Bring down a zero: 6 goes into 40 six times.
• Pattern: ( 0.1666... ) (only 6 repeats)
• Decimal: ( 0.1\overline{6} )

4. ( \frac{1}{9} )

• 1 divided by 9.
• 9 goes into 10 once, remainder 1.
• Pattern repeats.
• Decimal: ( 0.\overline{1} )

5. ( \frac{2}{9} )

• ( 2 \times 0.\overline{1} = 0.\overline{2} )

6. ( \frac{4}{9} )

• ( 4 \times 0.\overline{1} = 0.\overline{4} )

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Pattern with Denominators of 9, 99, 999, etc.

• ( \frac{17}{99} = 0.\overline{17} )
• ( \frac{25}{99} = 0.\overline{25} )
• ( \frac{247}{999} = 0.\overline{247} )

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Notes on Repeating Decimals

• Dividing by 3, 6, 9, or their multiples often results in repeating decimals.
• Not all divisions by multiples of three produce repeating decimals if the fraction simplifies (e.g., ( \frac{3}{6} = \frac{1}{2} = 0.5 ), not repeating).
• For simplified fractions with denominators that are multiples of three, check for repeating decimals.

Other Examples

• ( \frac{1}{12} = 0.083\overline{3} )
• ( \frac{1}{15} = 0.06\overline{6} )
• ( \frac{1}{18} = 0.05\overline{5} )

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Tips

• If you keep getting the same remainder during long division, the decimal part will repeat.
• Use a bar notation ((\overline{ })) to show repeating parts in the decimal.
• Quickly recognize patterns for denominators 9, 99, 999, etc.: numerator repeats as the decimal.

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Summary

• Long division helps convert fractions to decimals.
• Recognize repeating decimals, especially with denominators of 3, 6, 9, their multiples, 9, 99, 999, etc.
• Use bar notation for repeating decimals.

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