Converting Repeating Decimals to Fractions Using Algebraic Method ... - Free Printable

Let's solve each problem step-by-step using the algebraic method to convert repeating decimals into fractions, following the examples provided.

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✔ Section A: 1 Decimal Place

#### Example: $ x = 0.\overline{3} $
- $ 10x = 3.\overline{3} $
- Subtract: $ 10x - x = 3.\overline{3} - 0.\overline{3} \Rightarrow 9x = 3 $
- $ x = \frac{3}{9} = \frac{1}{3} $

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#### 1) $ x = 0.\overline{1} $
- Let $ x = 0.\overline{1} $
- Multiply both sides by 10:
$ 10x = 1.\overline{1} $
- Subtract original equation:
$ 10x - x = 1.\overline{1} - 0.\overline{1} \Rightarrow 9x = 1 $
- $ x = \frac{1}{9} $

✔ Answer: $ x = \frac{1}{9} $

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#### 2) $ x = 0.\overline{7} $
- Let $ x = 0.\overline{7} $
- $ 10x = 7.\overline{7} $
- $ 10x - x = 7.\overline{7} - 0.\overline{7} \Rightarrow 9x = 7 $
- $ x = \frac{7}{9} $

✔ Answer: $ x = \frac{7}{9} $

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#### 3) $ x = 2.\overline{4} $
- Let $ x = 2.\overline{4} $
- $ 10x = 24.\overline{4} $
- $ 10x - x = 24.\overline{4} - 2.\overline{4} \Rightarrow 9x = 22 $
- $ x = \frac{22}{9} $

✔ Answer: $ x = \frac{22}{9} $

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✔ Section B: 2 Decimal Places

#### Example 1: $ x = 0.\overline{43} $
- $ 100x = 43.\overline{43} $
- $ 100x - x = 43.\overline{43} - 0.\overline{43} \Rightarrow 99x = 43 $
- $ x = \frac{43}{99} $

#### Example 2: $ x = 0.4\overline{3} $
- $ 10x = 4.\overline{3} $
- $ 100x = 43.\overline{3} $
- $ 100x - 10x = 43.\overline{3} - 4.\overline{3} \Rightarrow 90x = 39 $
- $ x = \frac{39}{90} = \frac{13}{30} $

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#### 1) $ x = 0.\overline{57} $
- Let $ x = 0.\overline{57} $
- $ 100x = 57.\overline{57} $
- $ 100x - x = 57.\overline{57} - 0.\overline{57} \Rightarrow 99x = 57 $
- $ x = \frac{57}{99} = \frac{19}{33} $ (divide numerator and denominator by 3)

✔ Answer: $ x = \frac{19}{33} $

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#### 2) $ x = 0.\overline{86} $
- Let $ x = 0.\overline{86} $
- $ 100x = 86.\overline{86} $
- $ 100x - x = 86.\overline{86} - 0.\overline{86} \Rightarrow 99x = 86 $
- $ x = \frac{86}{99} $

✔ Answer: $ x = \frac{86}{99} $

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#### 3) $ x = 19.\overline{92} $
- Let $ x = 19.\overline{92} $
- $ 100x = 1992.\overline{92} $
- $ 100x - x = 1992.\overline{92} - 19.\overline{92} \Rightarrow 99x = 1973 $
- $ x = \frac{1973}{99} $

✔ Answer: $ x = \frac{1973}{99} $

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#### 4) $ x = 0.5\overline{7} $
- Let $ x = 0.5\overline{7} $
- This means $ x = 0.57777... $
- Step 1: Multiply by 10 to move decimal past non-repeating part:
$ 10x = 5.7\overline{7} $
- Step 2: Multiply by 10 again to shift one more place:
$ 100x = 57.\overline{7} $
- Now subtract:
$ 100x - 10x = 57.\overline{7} - 5.7\overline{7} \Rightarrow 90x = 52 $
- $ x = \frac{52}{90} = \frac{26}{45} $ (divide numerator and denominator by 2)

✔ Answer: $ x = \frac{26}{45} $

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#### 5) $ x = 0.8\overline{6} $
- Let $ x = 0.8\overline{6} $
- $ 10x = 8.6\overline{6} $
- $ 100x = 86.\overline{6} $
- Subtract: $ 100x - 10x = 86.\overline{6} - 8.6\overline{6} \Rightarrow 90x = 78 $
- $ x = \frac{78}{90} = \frac{13}{15} $ (divide by 6)

✔ Answer: $ x = \frac{13}{15} $

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#### 6) $ x = 3.2\overline{1} $
- Let $ x = 3.2\overline{1} $
- $ 10x = 32.1\overline{1} $
- $ 100x = 321.\overline{1} $
- Subtract: $ 100x - 10x = 321.\overline{1} - 32.1\overline{1} \Rightarrow 90x = 289 $
- $ x = \frac{289}{90} $

✔ Answer: $ x = \frac{289}{90} $

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✔ Final Answers:

#### Section A:
1) $ 0.\overline{1} = \frac{1}{9} $
2) $ 0.\overline{7} = \frac{7}{9} $
3) $ 2.\overline{4} = \frac{22}{9} $

#### Section B:
1) $ 0.\overline{57} = \frac{19}{33} $
2) $ 0.\overline{86} = \frac{86}{99} $
3) $ 19.\overline{92} = \frac{1973}{99} $
4) $ 0.5\overline{7} = \frac{26}{45} $
5) $ 0.8\overline{6} = \frac{13}{15} $
6) $ 3.2\overline{1} = \frac{289}{90} $

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🔍 Summary of Method:

For repeating decimals, use algebra:
- Let $ x = $ the repeating decimal.
- Multiply by a power of 10 to align the repeating parts.
- Subtract the original equation to eliminate the repeating part.
- Solve for $ x $.
- Simplify the fraction.

This works for:
- One-digit repeats → multiply by 10
- Two-digit repeats → multiply by 100
- Mixed (non-repeating + repeating) → adjust multiplication accordingly

Let me know if you'd like this in a printable format or need help simplifying any fraction!