Convert Repeating Decimals to Fractions (examples, solutions ... - Free Printable

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 the 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} $

Two digits repeat: $ 0.767676\ldots $

Let $ x = 0.767676\ldots $

Multiply by 100:

$$
100x = 76.767676\ldots
$$

Subtract:

$$
100x - x = 76.767676\ldots - 0.767676\ldots \Rightarrow 99x = 76 \Rightarrow x = \frac{76}{99}
$$

✔ 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} $

$ 0.656565\ldots $

Let $ x = 0.656565\ldots $

Multiply by 100:

$$
100x = 65.656565\ldots
$$

Subtract:

$$
100x - x = 65.656565\ldots - 0.656565\ldots \Rightarrow 99x = 65 \Rightarrow x = \frac{65}{99}
$$

✔ Answer: $ \boxed{\frac{65}{99}} $

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✔ 9. $ 0.\overline{32} $

$ 0.323232\ldots $

Let $ x = 0.323232\ldots $

Multiply by 100:

$$
100x = 32.323232\ldots
$$

Subtract:

$$
100x - x = 32.323232\ldots - 0.323232\ldots \Rightarrow 99x = 32 \Rightarrow x = \frac{32}{99}
$$

✔ Answer: $ \boxed{\frac{32}{99}} $

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✔ 10. $ 0.3\overline{16} $

This is $ 0.3161616\ldots $ — one non-repeating digit, then two repeating.

Let $ x = 0.3161616\ldots $

Step 1: Multiply by 10 to move past the non-repeating digit:

$$
10x = 3.161616\ldots
$$

Now multiply by 100 to align the repeating part:

$$
1000x = 316.161616\ldots
$$

Now subtract:

$$
1000x - 10x = 316.161616\ldots - 3.161616\ldots \Rightarrow 990x = 313
\Rightarrow x = \frac{313}{990}
$$

Check if reducible: GCD(313, 990) = 1 (313 is prime), so it's already simplified.

✔ Answer: $ \boxed{\frac{313}{990}} $

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✔ 11. $ 0.\overline{42} $

$ 0.424242\ldots $

Let $ x = 0.424242\ldots $

Multiply by 100:

$$
100x = 42.424242\ldots
$$

Subtract:

$$
100x - x = 42.424242\ldots - 0.424242\ldots \Rightarrow 99x = 42 \Rightarrow x = \frac{42}{99} = \frac{14}{33}
$$

✔ Answer: $ \boxed{\frac{14}{33}} $

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✔ 12. $ 0.\overline{524} $

Three digits repeat: $ 0.524524524\ldots $

Let $ x = 0.524524524\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 GCD(524, 999):
524 ÷ 2 = 262 → even
999 odd → not divisible by 2
Try dividing:
GCD(524, 999):
999 ÷ 524 = 1 R 475
524 ÷ 475 = 1 R 49
475 ÷ 49 = 9 R 34
49 ÷ 34 = 1 R 15
34 ÷ 15 = 2 R 4
15 ÷ 4 = 3 R 3
4 ÷ 3 = 1 R 1
3 ÷ 1 = 3 R 0 → GCD = 1

So 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} $ |
| $ 0.3\overline{16} $ | $ \frac{313}{990} $ |
| $ 0.\overline{42} $ | $ \frac{14}{33} $ |
| $ 0.\overline{524} $ | $ \frac{524}{999} $ |

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