Practice converting recurring decimals into their simplest fractional forms using this step-by-step algebra worksheet.

Absolutely! Let's solve each problem step-by-step using the algebraic method for converting repeating decimals to fractions, as shown in the examples.

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## 🔹 SECTION A: 1 Decimal Place (Single digit repeating)

📌 Example Given:

> $ x = 0.\overline{3} $
> $ 10x = 3.\overline{3} $
> Subtract: $ 10x - x = 3.\overline{3} - 0.\overline{3} \Rightarrow 9x = 3 $
> So, $ x = \frac{3}{9} = \frac{1}{3} $

This is our model. We multiply by 10 to shift the decimal so that the repeating part aligns, then subtract.

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✔ Problem 1: $ 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} $
→ $ 9x = 1 $
→ $ x = \frac{1}{9} $

✔ Answer: $ \boxed{\frac{1}{9}} $

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✔ Problem 2: $ 0.\overline{7} $

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

Multiply by 10:
→ $ 10x = 7.\overline{7} $

Subtract:
→ $ 10x - x = 7.\overline{7} - 0.\overline{7} $
→ $ 9x = 7 $
→ $ x = \frac{7}{9} $

✔ Answer: $ \boxed{\frac{7}{9}} $

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✔ Problem 3: $ 2.\overline{4} $

This has a whole number part. Let’s handle it similarly.

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

Multiply by 10:
→ $ 10x = 24.\overline{4} $

Subtract original:
→ $ 10x - x = 24.\overline{4} - 2.\overline{4} $
→ $ 9x = 22 $
→ $ x = \frac{22}{9} $

We can leave it as an improper fraction or convert to mixed number:
→ $ \frac{22}{9} = 2\frac{4}{9} $

✔ Answer: $ \boxed{\frac{22}{9}} $ or $ \boxed{2\frac{4}{9}} $

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## 🔹 SECTION B: 2 Decimal Places (Two digits repeating)

📌 Example 1: $ 0.\overline{43} $

Let $ x = 0.\overline{43} $

Multiply by 100 (since 2 digits repeat):
→ $ 100x = 43.\overline{43} $

Subtract:
→ $ 100x - x = 43.\overline{43} - 0.\overline{43} $
→ $ 99x = 43 $
→ $ x = \frac{43}{99} $

✔ This is already in simplest form (43 is prime, doesn’t divide 99).

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✔ Problem 1: $ 0.\overline{57} $

Let $ x = 0.\overline{57} $

Multiply by 100:
→ $ 100x = 57.\overline{57} $

Subtract:
→ $ 100x - x = 57.\overline{57} - 0.\overline{57} $
→ $ 99x = 57 $
→ $ x = \frac{57}{99} $

Simplify: Divide numerator and denominator by 3:
→ $ \frac{57 ÷ 3}{99 ÷ 3} = \frac{19}{33} $

✔ Answer: $ \boxed{\frac{19}{33}} $

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✔ Problem 2: $ 0.\overline{86} $

Let $ x = 0.\overline{86} $

Multiply by 100:
→ $ 100x = 86.\overline{86} $

Subtract:
→ $ 100x - x = 86.\overline{86} - 0.\overline{86} $
→ $ 99x = 86 $
→ $ x = \frac{86}{99} $

Check if simplifiable: GCF of 86 and 99 is 1 → already simplified.

✔ Answer: $ \boxed{\frac{86}{99}} $

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✔ Problem 3: $ 19.\overline{92} $

Let $ x = 19.\overline{92} $

Multiply by 100:
→ $ 100x = 1992.\overline{92} $

Subtract:
→ $ 100x - x = 1992.\overline{92} - 19.\overline{92} $
→ $ 99x = 1973 $
→ $ x = \frac{1973}{99} $

Check if reducible: 1973 ÷ 99? Let’s see — 99 × 19 = 1881, 1973 - 1881 = 92 → not divisible.
GCF(1973,99): 99 = 9×11; 1973 ÷ 11 = 179.36… not integer → no common factors.

✔ Answer: $ \boxed{\frac{1973}{99}} $ or mixed number $ \boxed{19\frac{92}{99}} $

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## 🔹 SECTION C: Mixed Repeating Decimals (One digit non-repeating, one digit repeating)

📌 Example: $ 0.4\overline{3} $

Let $ x = 0.4\overline{3} $

Multiply by 10 to move the non-repeating digit:
→ $ 10x = 4.\overline{3} $

Now multiply by 10 again to align repeating parts:
→ $ 100x = 43.\overline{3} $

Subtract:
→ $ 100x - 10x = 43.\overline{3} - 4.\overline{3} $
→ $ 90x = 39 $
→ $ x = \frac{39}{90} = \frac{13}{30} $

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✔ Problem 4: $ 0.5\overline{7} $

Let $ x = 0.5\overline{7} $

Multiply by 10:
→ $ 10x = 5.\overline{7} $

Multiply by 100:
→ $ 100x = 57.\overline{7} $

Subtract:
→ $ 100x - 10x = 57.\overline{7} - 5.\overline{7} $
→ $ 90x = 52 $
→ $ x = \frac{52}{90} $

Simplify: divide numerator and denominator by 2 → $ \frac{26}{45} $

✔ Answer: $ \boxed{\frac{26}{45}} $

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✔ Problem 5: $ 0.8\overline{6} $

Let $ x = 0.8\overline{6} $

Multiply by 10:
→ $ 10x = 8.\overline{6} $

Multiply by 100:
→ $ 100x = 86.\overline{6} $

Subtract:
→ $ 100x - 10x = 86.\overline{6} - 8.\overline{6} $
→ $ 90x = 78 $
→ $ x = \frac{78}{90} $

Simplify: divide by 6 → $ \frac{13}{15} $

✔ Answer: $ \boxed{\frac{13}{15}} $

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✔ Problem 6: $ 3.2\overline{1} $

Let $ x = 3.2\overline{1} $

Multiply by 10:
→ $ 10x = 32.\overline{1} $

Multiply by 100:
→ $ 100x = 321.\overline{1} $

Subtract:
→ $ 100x - 10x = 321.\overline{1} - 32.\overline{1} $
→ $ 90x = 289 $
→ $ x = \frac{289}{90} $

Check simplification: 289 = 17², 90 = 9×10 → no common factors.

✔ Answer: $ \boxed{\frac{289}{90}} $ or mixed number $ \boxed{3\frac{19}{90}} $

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## ✔ Final Answers Summary:

Section A:

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

Section B:

1) $ 0.\overline{57} = \boxed{\frac{19}{33}} $
2) $ 0.\overline{86} = \boxed{\frac{86}{99}} $
3) $ 19.\overline{92} = \boxed{\frac{1973}{99}} $

Section C:

4) $ 0.5\overline{7} = \boxed{\frac{26}{45}} $
5) $ 0.8\overline{6} = \boxed{\frac{13}{15}} $
6) $ 3.2\overline{1} = \boxed{\frac{289}{90}} $

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Let me know if you’d like these written out with all steps filled in for your worksheet! 😊