Recurring Decimals to Fractions - Algebraic Method worksheet with step-by-step examples and practice problems for converting recurring decimals into fractions.

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

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✔ Section A: 1 Decimal Place

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

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1) $ 0.\dot{1} $

Let $ x = 0.\dot{1} $

- $ 10x = 1.\dot{1} $
- Subtract: $ 10x - x = 1.\dot{1} - 0.\dot{1} \Rightarrow 9x = 1 $
- $ x = \frac{1}{9} $

✔ Answer: $ \boxed{\frac{1}{9}} $

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2) $ 0.\dot{7} $

Let $ x = 0.\dot{7} $

- $ 10x = 7.\dot{7} $
- Subtract: $ 10x - x = 7.\dot{7} - 0.\dot{7} \Rightarrow 9x = 7 $
- $ x = \frac{7}{9} $

✔ Answer: $ \boxed{\frac{7}{9}} $

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3) $ 2.\dot{4} $

Let $ x = 2.\dot{4} $

- $ 10x = 24.\dot{4} $
- Subtract: $ 10x - x = 24.\dot{4} - 2.\dot{4} \Rightarrow 9x = 22 $
- $ x = \frac{22}{9} $

✔ Answer: $ \boxed{\frac{22}{9}} $

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✔ Section B: 2 Decimal Places

#### Example 1: $ x = 0.\dot{4}\dot{3} $
- $ 100x = 43.\dot{4}\dot{3} $
- $ 100x - x = 43.\dot{4}\dot{3} - 0.\dot{4}\dot{3} \Rightarrow 99x = 43 $
- $ x = \frac{43}{99} $

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

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1) $ 0.\dot{5}\dot{7} $

Let $ x = 0.\dot{5}\dot{7} $

- $ 100x = 57.\dot{5}\dot{7} $
- $ 100x - x = 57.\dot{5}\dot{7} - 0.\dot{5}\dot{7} \Rightarrow 99x = 57 $
- $ x = \frac{57}{99} = \frac{19}{33} $ (divide numerator and denominator by 3)

✔ Answer: $ \boxed{\frac{19}{33}} $

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2) $ 0.\dot{8}\dot{6} $

Let $ x = 0.\dot{8}\dot{6} $

- $ 100x = 86.\dot{8}\dot{6} $
- $ 100x - x = 86.\dot{8}\dot{6} - 0.\dot{8}\dot{6} \Rightarrow 99x = 86 $
- $ x = \frac{86}{99} $ → already in simplest form

✔ Answer: $ \boxed{\frac{86}{99}} $

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3) $ 19.\dot{9}\dot{2} $

Let $ x = 19.\dot{9}\dot{2} $

- $ 100x = 1992.\dot{9}\dot{2} $
- $ 100x - x = 1992.\dot{9}\dot{2} - 19.\dot{9}\dot{2} \Rightarrow 99x = 1973 $
- $ x = \frac{1973}{99} $

Check if reducible:
- GCD of 1973 and 99?
99 = 9×11; 1973 ÷ 11 = 179.36… not divisible
Try dividing: 1973 ÷ 99 ≈ 19.929 → not a whole number
So $ \frac{1973}{99} $ is already simplified.

✔ Answer: $ \boxed{\frac{1973}{99}} $

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4) $ 0.5\dot{7} $

Let $ x = 0.5\dot{7} $

This means $ x = 0.57777... $

We need to eliminate the repeating part.

- Let $ x = 0.5\dot{7} $
- Multiply by 10 to shift decimal: $ 10x = 5.7\dot{7} $
- Multiply by 100: $ 100x = 57.7\dot{7} $
- Now subtract: $ 100x - 10x = 57.7\dot{7} - 5.7\dot{7} \Rightarrow 90x = 52 $
- $ x = \frac{52}{90} = \frac{26}{45} $

✔ Answer: $ \boxed{\frac{26}{45}} $

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5) $ 0.8\dot{6} $

Let $ x = 0.8\dot{6} $ → $ x = 0.86666... $

- $ 10x = 8.6\dot{6} $
- $ 100x = 86.6\dot{6} $
- Subtract: $ 100x - 10x = 86.6\dot{6} - 8.6\dot{6} \Rightarrow 90x = 78 $
- $ x = \frac{78}{90} = \frac{13}{15} $

✔ Answer: $ \boxed{\frac{13}{15}} $

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6) $ 3.2\dot{1} $

Let $ x = 3.2\dot{1} $ → $ x = 3.21111... $

- $ 10x = 32.1\dot{1} $
- $ 100x = 321.1\dot{1} $
- Subtract: $ 100x - 10x = 321.1\dot{1} - 32.1\dot{1} \Rightarrow 90x = 289 $
- $ x = \frac{289}{90} $

Check if reducible:
GCD of 289 and 90?
289 = 17², 90 = 2×3²×5 → no common factors

✔ Answer: $ \boxed{\frac{289}{90}} $

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## ✔ Final Answers:

Section A:

1) $ 0.\dot{1} = \frac{1}{9} $
2) $ 0.\dot{7} = \frac{7}{9} $
3) $ 2.\dot{4} = \frac{22}{9} $

Section B:

1) $ 0.\dot{5}\dot{7} = \frac{19}{33} $
2) $ 0.\dot{8}\dot{6} = \frac{86}{99} $
3) $ 19.\dot{9}\dot{2} = \frac{1973}{99} $
4) $ 0.5\dot{7} = \frac{26}{45} $
5) $ 0.8\dot{6} = \frac{13}{15} $
6) $ 3.2\dot{1} = \frac{289}{90} $

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