Lecture Notes: Converting Repeating Decimals to Fractions

Introduction

• Subject: Converting repeating decimals into fractions.
• Example:
  • Repeating decimal 0.7
  • Notation: 0.7 with a bar over 7 or 0.777...

Methodology

1. Setting the Decimal Equal to a Variable

   • Assign variable: Let x = 0.7 (repeating).

2. Multiplication by 10

   • Multiply both sides by 10: 10x = 7.777... (7.7 repeating).
   • The purpose is to shift the decimal point.

3. Subtraction to Eliminate Repetition

   • Subtract x from 10x:
     • 10x - x = 9x
     • Result: 9x equals 7
   • Explanation: The repeating decimals cancel out.

4. Solve for x

   • Divide both sides by 9:
   • Result: x = 7/9

Further Example

• Decimal with 1 Before Repetition: 1.2 repeating
  • Equivalent to 1.222...
  • Assign variable: x = 1.2 (repeating).

1. Multiply by 10

   • 10x = 12.222... (12.2 repeating).

2. Subtraction

   • Subtract x from 10x:
     • 10x - x = 9x
     • Result: 12.222... - 1.222... = 11
   • Explanation: The repeating parts cancel out.

3. Solve for x

   • Divide both sides by 9:
   • Result: x = 11/9

Conclusion

• Method involves:
  • Assigning a variable to the repeating decimal.
  • Multiplying by powers of 10 to shift repetition.
  • Subtracting to eliminate repetitions.
  • Solving simple algebra to find the fraction form.