Overview
This lecture explains how to express recurring decimals as fractions using algebraic methods, with examples and step-by-step solutions for different cases.
Converting Recurring Decimals to Fractions
- 0.2 recurring (0.222...) is called x; multiply by 10 to get 2.2 recurring.
- Subtract x from 10x: 10x - x = 9x, so 2.2... - 0.2... = 2, leading to x = 2/9.
- For 0.5 recurring, following the same steps gives x = 5/9.
Recurring Decimals with Two Digits
- 0.45 recurring (0.454545...) is called x; multiply by 100 to get 45.4545...
- 100x - x = 99x; 45.45... - 0.45... = 45, so x = 45/99.
- Simplify 45/99 to 5/11.
Recurring Decimals with More Digits
- 0.15 recurring (0.151515...) becomes x; multiply by 100 to get 15.1515...
- 100x - x = 99x; 15.15... - 0.15... = 15, so x = 15/99 = 5/33.
- For 0.215 recurring (0.215215...), use 1000x to align recurring digits: x = 215/999.
Mixed Recurring and Non-Recurring Decimals
- For 0.215 with only 15 recurring: x = 0.2151515...; use 10x and 1000x to line up recurring parts.
- 1000x - 10x = 990x; 215.15... - 2.15... = 213, so x = 213/990 = 71/330.
- For 0.15 with only 5 recurring: x = 0.155555...; use 10x and 100x for subtraction; x = 14/90 = 7/45.
Additional Examples
- 0.8 recurring: x = 8/9.
- 0.81 recurring (both digits recurring): x = 81/99 = 9/11.
- 0.81 with only 1 recurring: x = 0.811111...; multiply, subtract, get x = 73/90.
Key Terms & Definitions
- Recurring Decimal — A decimal in which one or more digits repeat infinitely.
- Fraction — A way of expressing numbers as the ratio of two integers.
- Algebraic Method — Using variables and equations to manipulate and solve for values.
Action Items / Next Steps
- Practice converting recurring decimals to fractions using the subtraction method.
- Complete assigned questions on expressing recurring decimals as fractions.