Rational and Irrational Numbers worksheet: Convert non-terminating recurring decimals to fractions.

Let's solve each of these non-terminating recurring decimals and express them in the form $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \ne 0 $. We'll use algebraic methods to convert repeating decimals into fractions.

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General Method:

For a repeating decimal:
- Let $ x = $ the decimal.
- Multiply $ x $ by a power of 10 so that the repeating part aligns.
- Subtract the original $ x $ to eliminate the repeating part.
- Solve for $ x $.

We’ll go step-by-step.

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1) $ 0.\overline{3} $

Let $ x = 0.333\ldots $

Multiply both sides by 10:
$ 10x = 3.333\ldots $

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

✔ Answer: $ \frac{1}{3} $

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2) $ 1.\overline{4} $

Let $ x = 1.444\ldots $

Multiply by 10:
$ 10x = 14.444\ldots $

Subtract:
$ 10x - x = 14.444\ldots - 1.444\ldots $
$ 9x = 13 $
$ x = \frac{13}{9} $

✔ Answer: $ \frac{13}{9} $

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3) $ 0.\overline{57} $

Let $ x = 0.575757\ldots $

Multiply by 100 (since two digits repeat):
$ 100x = 57.5757\ldots $

Subtract:
$ 100x - x = 57.5757\ldots - 0.5757\ldots $
$ 99x = 57 $
$ x = \frac{57}{99} = \frac{19}{33} $ (divide numerator and denominator by 3)

✔ Answer: $ \frac{19}{33} $

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4) $ 0.\overline{001} $

Let $ x = 0.001001001\ldots $

Multiply by 1000 (three digits repeat):
$ 1000x = 1.001001\ldots $

Subtract:
$ 1000x - x = 1.001001\ldots - 0.001001\ldots $
$ 999x = 1 $
$ x = \frac{1}{999} $

✔ Answer: $ \frac{1}{999} $

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5) $ 0.\overline{53} $

Let $ x = 0.535353\ldots $

Multiply by 100:
$ 100x = 53.5353\ldots $

Subtract:
$ 100x - x = 53.5353\ldots - 0.5353\ldots $
$ 99x = 53 $
$ x = \frac{53}{99} $

✔ Answer: $ \frac{53}{99} $

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6) $ 2.\overline{93} $

Let $ x = 2.939393\ldots $

Multiply by 100:
$ 100x = 293.9393\ldots $

Subtract:
$ 100x - x = 293.9393\ldots - 2.9393\ldots $
$ 99x = 291 $
$ x = \frac{291}{99} = \frac{97}{33} $ (divide by 3)

✔ Answer: $ \frac{97}{33} $

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7) $ 18.\overline{48} $

Let $ x = 18.484848\ldots $

Multiply by 100:
$ 100x = 1848.4848\ldots $

Subtract:
$ 100x - x = 1848.4848\ldots - 18.4848\ldots $
$ 99x = 1830 $
$ x = \frac{1830}{99} = \frac{610}{33} $ (divide numerator and denominator by 3)

✔ Answer: $ \frac{610}{33} $

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8) $ 0.\overline{235} $

Let $ x = 0.235235235\ldots $

Multiply by 1000:
$ 1000x = 235.235235\ldots $

Subtract:
$ 1000x - x = 235.235235\ldots - 0.235235\ldots $
$ 999x = 235 $
$ x = \frac{235}{999} $

Check if reducible: GCD(235, 999).
235 = 5 × 47
999 = 3³ × 37 → no common factors

✔ Answer: $ \frac{235}{999} $

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9) $ 0.00\overline{32} $

This is $ 0.00323232\ldots $

Let $ x = 0.00323232\ldots $

The repeating part is "32", starting after two decimal places.

Multiply by 100 to shift past the non-repeating part:
$ 100x = 0.323232\ldots $

Now multiply by 100 again to get alignment:
$ 10000x = 32.323232\ldots $

Subtract:
$ 10000x - 100x = 32.3232\ldots - 0.3232\ldots $
$ 9900x = 32 $
$ x = \frac{32}{9900} = \frac{8}{2475} $ (divide numerator and denominator by 4)

✔ Answer: $ \frac{8}{2475} $

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10) $ 0.\overline{9} $

Let $ x = 0.999\ldots $

Multiply by 10:
$ 10x = 9.999\ldots $

Subtract:
$ 10x - x = 9.999\ldots - 0.999\ldots $
$ 9x = 9 $
$ x = 1 $

So $ 0.\overline{9} = 1 $

✔ Answer: $ \frac{1}{1} $ or simply $ 1 $

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11) $ 2.4\overline{178} $

Let $ x = 2.4178178178\ldots $

Repeating part is "178" (3 digits), but it starts after one digit.

Let’s move the decimal:

Multiply by 10 to handle the non-repeating digit:
$ 10x = 24.178178178\ldots $

Now multiply by 1000 to align the repeating part:
$ 10000x = 24178.178178\ldots $

Subtract:
$ 10000x - 10x = 24178.178178\ldots - 24.178178\ldots $
$ 9990x = 24154 $

So $ x = \frac{24154}{9990} $

Simplify:

Divide numerator and denominator by 2:
$ \frac{12077}{4995} $

Check if reducible:
GCD(12077, 4995) — let's test.

Use Euclidean algorithm:
- 12077 ÷ 4995 ≈ 2 → 12077 - 2×4995 = 12077 - 9990 = 2087
- 4995 ÷ 2087 ≈ 2 → 4995 - 2×2087 = 4995 - 4174 = 821
- 2087 ÷ 821 ≈ 2 → 2087 - 1642 = 445
- 821 ÷ 445 ≈ 1 → 821 - 445 = 376
- 445 ÷ 376 ≈ 1 → 69
- 376 ÷ 69 ≈ 5 → 376 - 345 = 31
- 69 ÷ 31 ≈ 2 → 69 - 62 = 7
- 31 ÷ 7 ≈ 4 → 31 - 28 = 3
- 7 ÷ 3 = 2 → 1
- 3 ÷ 1 = 3 → remainder 0

GCD is 1 → already simplified.

✔ Answer: $ \frac{12077}{4995} $

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12) $ 2.\overline{36} $

Let $ x = 2.363636\ldots $

Multiply by 100:
$ 100x = 236.3636\ldots $

Subtract:
$ 100x - x = 236.3636\ldots - 2.3636\ldots $
$ 99x = 234 $
$ x = \frac{234}{99} = \frac{78}{33} = \frac{26}{11} $

✔ Answer: $ \frac{26}{11} $

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13) $ 1.\overline{323} $

Let $ x = 1.323323323\ldots $

Multiply by 1000:
$ 1000x = 1323.323323\ldots $

Subtract:
$ 1000x - x = 1323.323323\ldots - 1.323323\ldots $
$ 999x = 1322 $
$ x = \frac{1322}{999} $

Check simplification: GCD(1322, 999)

- 1322 ÷ 999 = 1 → remainder 323
- 999 ÷ 323 = 3 → 999 - 969 = 30
- 323 ÷ 30 = 10 → 323 - 300 = 23
- 30 ÷ 23 = 1 → 7
- 23 ÷ 7 = 3 → 2
- 7 ÷ 2 = 3 → 1
- 2 ÷ 1 = 2 → 0

GCD = 1 → irreducible

✔ Answer: $ \frac{1322}{999} $

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14) $ 0.3\overline{178} $

Let $ x = 0.3178178178\ldots $

Repeating part "178" starts after first digit.

Multiply by 10:
$ 10x = 3.178178178\ldots $

Multiply by 1000:
$ 10000x = 3178.178178\ldots $

Subtract:
$ 10000x - 10x = 3178.178178\ldots - 3.178178\ldots $
$ 9990x = 3175 $

$ x = \frac{3175}{9990} $

Simplify: divide numerator and denominator by 5:

$ \frac{635}{1998} $

Check GCD(635, 1998):

- 635 = 5 × 127
- 1998 = 2 × 3³ × 37 → no common factors

✔ Answer: $ \frac{635}{1998} $

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15) $ 0.4\overline{07} $

Let $ x = 0.4070707\ldots $

Multiply by 10:
$ 10x = 4.070707\ldots $

Multiply by 100:
$ 1000x = 407.070707\ldots $

Subtract:
$ 1000x - 10x = 407.0707\ldots - 4.0707\ldots $
$ 990x = 403 $

$ x = \frac{403}{990} $

Check GCD(403, 990):

- 403 = 13 × 31
- 990 = 2 × 3² × 5 × 11 → no common factors

✔ Answer: $ \frac{403}{990} $

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16) $ 32.1\overline{235} $

Let $ x = 32.1235235235\ldots $

Repeating part "235" starts after one decimal place.

Multiply by 10:
$ 10x = 321.235235235\ldots $

Multiply by 1000:
$ 10000x = 321235.235235\ldots $

Subtract:
$ 10000x - 10x = 321235.235235\ldots - 321.235235\ldots $
$ 9990x = 320914 $

$ x = \frac{320914}{9990} $

Simplify:

Divide numerator and denominator by 2:

$ \frac{160457}{4995} $

Check if reducible: GCD(160457, 4995)

Use Euclidean algorithm:

- 160457 ÷ 4995 ≈ 32 → 4995 × 32 = 159840
- 160457 - 159840 = 617
- 4995 ÷ 617 ≈ 8 → 617 × 8 = 4936
- 4995 - 4936 = 59
- 617 ÷ 59 ≈ 10 → 590 → remainder 27
- 59 ÷ 27 = 2 → 54 → rem 5
- 27 ÷ 5 = 5 → 25 → rem 2
- 5 ÷ 2 = 2 → rem 1
- 2 ÷ 1 = 2 → rem 0 → GCD = 1

So fraction is simplified.

✔ Answer: $ \frac{160457}{4995} $

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) $ 0.\overline{3} $ | $ \frac{1}{3} $ |
| 2) $ 1.\overline{4} $ | $ \frac{13}{9} $ |
| 3) $ 0.\overline{57} $ | $ \frac{19}{33} $ |
| 4) $ 0.\overline{001} $ | $ \frac{1}{999} $ |
| 5) $ 0.\overline{53} $ | $ \frac{53}{99} $ |
| 6) $ 2.\overline{93} $ | $ \frac{97}{33} $ |
| 7) $ 18.\overline{48} $ | $ \frac{610}{33} $ |
| 8) $ 0.\overline{235} $ | $ \frac{235}{999} $ |
| 9) $ 0.00\overline{32} $ | $ \frac{8}{2475} $ |
| 10) $ 0.\overline{9} $ | $ 1 $ |
| 11) $ 2.4\overline{178} $ | $ \frac{12077}{4995} $ |
| 12) $ 2.\overline{36} $ | $ \frac{26}{11} $ |
| 13) $ 1.\overline{323} $ | $ \frac{1322}{999} $ |
| 14) $ 0.3\overline{178} $ | $ \frac{635}{1998} $ |
| 15) $ 0.4\overline{07} $ | $ \frac{403}{990} $ |
| 16) $ 32.1\overline{235} $ | $ \frac{160457}{4995} $ |

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