Convert repeating decimals to fractions practice worksheet.

Let's solve each of these repeating decimals and convert them into fractions. We'll use algebraic methods to do this.

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🔁 General Method for Converting Repeating Decimals to Fractions:

Let’s say we have a repeating decimal like $ 0.\overline{ab} $.
We can use the following steps:

1. Let $ x = $ the repeating decimal.
2. Multiply both sides by $ 10^n $, where $ n $ is the number of digits in the repeating part.
3. Subtract the original equation from this new one to eliminate the repeating part.
4. Solve for $ x $.

Let’s apply this step-by-step.

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✔ 1. $ 0.\overline{3} $

Let $ x = 0.333\ldots $

Multiply both sides by 10 (since one digit repeats):

$$
10x = 3.333\ldots
$$

Subtract original:

$$
10x - x = 3.333\ldots - 0.333\ldots \Rightarrow 9x = 3 \Rightarrow x = \frac{3}{9} = \frac{1}{3}
$$

✔ Answer: $ \boxed{\frac{1}{3}} $

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✔ 2. $ 0.1\overline{6} $

This means $ 0.1666\ldots $

Let $ x = 0.1666\ldots $

Multiply by 10 to shift past the non-repeating part:

$$
10x = 1.666\ldots
$$

Now multiply by 10 again to get repeating part aligned:

$$
100x = 16.666\ldots
$$

Now subtract:

$$
100x - 10x = 16.666\ldots - 1.666\ldots \Rightarrow 90x = 15 \Rightarrow x = \frac{15}{90} = \frac{1}{6}
$$

✔ Answer: $ \boxed{\frac{1}{6}} $

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✔ 3. $ 0.\overline{6} $

Let $ x = 0.666\ldots $

Multiply by 10:

$$
10x = 6.666\ldots
$$

Subtract:

$$
10x - x = 6.666\ldots - 0.666\ldots \Rightarrow 9x = 6 \Rightarrow x = \frac{6}{9} = \frac{2}{3}
$$

✔ Answer: $ \boxed{\frac{2}{3}} $

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✔ 4. $ 0.8\overline{3} $

This is $ 0.8333\ldots $

Let $ x = 0.8333\ldots $

Multiply by 10:

$$
10x = 8.333\ldots
$$

Multiply by 100:

$$
100x = 83.333\ldots
$$

Subtract:

$$
100x - 10x = 83.333\ldots - 8.333\ldots \Rightarrow 90x = 75 \Rightarrow x = \frac{75}{90} = \frac{5}{6}
$$

✔ Answer: $ \boxed{\frac{5}{6}} $

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✔ 5. $ 0.\overline{1} $

Let $ x = 0.111\ldots $

Multiply by 10:

$$
10x = 1.111\ldots
$$

Subtract:

$$
10x - x = 1.111\ldots - 0.111\ldots \Rightarrow 9x = 1 \Rightarrow x = \frac{1}{9}
$$

✔ Answer: $ \boxed{\frac{1}{9}} $

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✔ 6. $ 0.\overline{76} $

Let $ x = 0.767676\ldots $

Multiply by 100 (two digits repeat):

$$
100x = 76.7676\ldots
$$

Subtract:

$$
100x - x = 76.7676\ldots - 0.7676\ldots \Rightarrow 99x = 76 \Rightarrow x = \frac{76}{99}
$$

Check: $ \frac{76}{99} $ is already simplified.

✔ Answer: $ \boxed{\frac{76}{99}} $

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✔ 7. $ 0.\overline{2} $

Let $ x = 0.222\ldots $

Multiply by 10:

$$
10x = 2.222\ldots
$$

Subtract:

$$
10x - x = 2.222\ldots - 0.222\ldots \Rightarrow 9x = 2 \Rightarrow x = \frac{2}{9}
$$

✔ Answer: $ \boxed{\frac{2}{9}} $

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✔ 8. $ 0.\overline{65} $

Let $ x = 0.656565\ldots $

Multiply by 100:

$$
100x = 65.6565\ldots
$$

Subtract:

$$
100x - x = 65.6565\ldots - 0.6565\ldots \Rightarrow 99x = 65 \Rightarrow x = \frac{65}{99}
$$

✔ Answer: $ \boxed{\frac{65}{99}} $

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✔ 9. $ 0.\overline{32} $

Let $ x = 0.323232\ldots $

Multiply by 100:

$$
100x = 32.3232\ldots
$$

Subtract:

$$
100x - x = 32.3232\ldots - 0.3232\ldots \Rightarrow 99x = 32 \Rightarrow x = \frac{32}{99}
$$

✔ Answer: $ \boxed{\frac{32}{99}} $

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✔ 10. $ 0.31\overline{6} $

This is $ 0.31666\ldots $

Let $ x = 0.31666\ldots $

Multiply by 10 to move past non-repeating:

$$
10x = 3.1666\ldots
$$

Multiply by 100:

$$
100x = 31.666\ldots
$$

Now subtract:

$$
100x - 10x = 31.666\ldots - 3.1666\ldots \Rightarrow 90x = 28.5
$$

Wait! Better approach:

Let’s go back.

Let $ x = 0.31666\ldots $

Multiply by 10: $ 10x = 3.1666\ldots $

Now let $ y = 3.1666\ldots $, then:

Multiply $ y $ by 10: $ 10y = 31.666\ldots $

Subtract:

$$
10y - y = 31.666\ldots - 3.1666\ldots = 28.5
\Rightarrow 9y = 28.5 \Rightarrow y = \frac{28.5}{9} = \frac{57}{18} = \frac{19}{6}
$$

But recall $ 10x = y $, so:

$$
x = \frac{y}{10} = \frac{19}{60}
$$

✔ Answer: $ \boxed{\frac{19}{60}} $

(Verify: $ \frac{19}{60} = 0.31666\ldots $ → correct)

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✔ 11. $ 0.\overline{42} $

Let $ x = 0.424242\ldots $

Multiply by 100:

$$
100x = 42.4242\ldots
$$

Subtract:

$$
100x - x = 42.4242\ldots - 0.4242\ldots \Rightarrow 99x = 42 \Rightarrow x = \frac{42}{99} = \frac{14}{33}
$$

✔ Answer: $ \boxed{\frac{14}{33}} $

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✔ 12. $ 0.\overline{524} $

Three digits repeat: $ 0.524524524\ldots $

Let $ x = 0.524524\ldots $

Multiply by 1000:

$$
1000x = 524.524524\ldots
$$

Subtract:

$$
1000x - x = 524.524524\ldots - 0.524524\ldots \Rightarrow 999x = 524 \Rightarrow x = \frac{524}{999}
$$

Check if reducible:

Find GCF of 524 and 999.

- 524 ÷ 2 = 262 → even
- 999 odd → not divisible by 2
- Check 3: 5+2+4=11 → not divisible by 3
- 999 divisible by 3: 9+9+9=27 → yes
- So no common factors? Try dividing 524 and 999.

Use Euclidean algorithm:

- 999 ÷ 524 = 1 rem 475
- 524 ÷ 475 = 1 rem 49
- 475 ÷ 49 = 9 rem 34
- 49 ÷ 34 = 1 rem 15
- 34 ÷ 15 = 2 rem 4
- 15 ÷ 4 = 3 rem 3
- 4 ÷ 3 = 1 rem 1
- 3 ÷ 1 = 3 rem 0

GCF = 1 → fraction is already simplified.

✔ Answer: $ \boxed{\frac{524}{999}} $

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✔ Final Answers:

| Decimal | Fraction |
|--------|----------|
| $ 0.\overline{3} $ | $ \frac{1}{3} $ |
| $ 0.1\overline{6} $ | $ \frac{1}{6} $ |
| $ 0.\overline{6} $ | $ \frac{2}{3} $ |
| $ 0.8\overline{3} $ | $ \frac{5}{6} $ |
| $ 0.\overline{1} $ | $ \frac{1}{9} $ |
| $ 0.\overline{76} $ | $ \frac{76}{99} $ |
| $ 0.\overline{2} $ | $ \frac{2}{9} $ |
| $ 0.\overline{65} $ | $ \frac{65}{99} $ |
| $ 0.\overline{32} $ | $ \frac{32}{99} $ |
| $ 0.31\overline{6} $ | $ \frac{19}{60} $ |
| $ 0.\overline{42} $ | $ \frac{14}{33} $ |
| $ 0.\overline{524} $ | $ \frac{524}{999} $ |

Let me know if you'd like a visual or step-by-step explanation for any of them!