Converting Recurring Decimals to Fractions

Introduction

• Recurring Decimals: Decimals where digits repeat indefinitely, e.g., 0.473, 473, 473...
• Representation: Use dots over the first and last repeating digits (e.g., 0.473 with dots over 4 and 3).
• Converting recurring decimals to fractions is a multi-step process.

Conversion Process

Steps for Recurring Decimals Starting Right After the Decimal Point

1. Assign a Variable: Let r represent the recurring decimal, e.g., r = 0.473 recurring.
2. Multiply by 10: Continue multiplying until one full set of repeating digits is before the decimal point.
   • Example:
     • 10r = 4.734 recurring
     • 100r = 47.347 recurring
     • 1000r = 473.473 recurring
3. Subtract: Subtract the smallest equation (r) from the largest (1000r):
   • 1000r - r = 999r = 473
4. Solve for r: Divide by 999 to isolate r:
   • r = 473 / 999

Steps for Recurring Decimals Where Repetition Doesn't Start Immediately

1. Assign and Multiply: Same initial step, but requires two different multiplications:
   • First, ensure full sets of repeating digits are left of the decimal.
   • Example:
     • 10r = 2.3 recurring
     • 100r = 23.3 recurring
2. Separate Non-Recurring: Multiply enough to move only non-recurring digits:
   • 10r = 2.3 recurring
3. Subtract: Subtract the smaller equation from the larger:
   • 100r - 10r = 90r = 21
4. Solve for r: Simplify the fraction:
   • r = 21/90
   • Simplified: 7/30

Conclusion

• Complex Topic: Understand that converting recurring decimals can be challenging.
• Practice: More practice helps solidify the concept.
• Simplification: Always check if the resulting fraction can be simplified.

Remember, this method is applicable to any recurring decimal, and practice will aid understanding.