Irrational Numbers Worksheets - 15 Worksheets Library - Free Printable

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 powers of 10 to shift the decimal point so that the repeating part aligns.
- Subtract the original equation from this new one to eliminate the repeating part.
- Solve for $ x $.

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Solutions:

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#### 1) $ 0.\overline{3} $

Let $ x = 0.333\ldots $

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

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

✔ Answer: $ \boxed{\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: $ \boxed{\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: $ \boxed{\frac{19}{33}} $

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#### 4) $ 0.\overline{001} $

Let $ x = 0.001001001\ldots $

Three digits repeat → multiply by 1000:
$ 1000x = 1.001001\ldots $

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

✔ Answer: $ \boxed{\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: $ \boxed{\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: $ \boxed{\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 by 3)

✔ Answer: $ \boxed{\frac{610}{33}} $

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#### 8) $ 0.\overline{235} $

Let $ x = 0.235235\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) = 1 → already simplified

✔ Answer: $ \boxed{\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 move past the non-repeating part:
$ 100x = 0.323232\ldots $

Now multiply by 100 again to align repeating parts:
$ 10000x = 32.3232\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: $ \boxed{\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 $

✔ Answer: $ \boxed{1} $ (or $ \frac{1}{1} $)

> Note: This is a well-known result: $ 0.\overline{9} = 1 $

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#### 11) $ 2.41\overline{78} $

Let $ x = 2.41787878\ldots $

Non-repeating part: "2.41", then "78" repeats.

Multiply by 100 to skip non-repeating:
$ 100x = 241.787878\ldots $

Multiply by 10000 to align:
$ 10000x = 24178.7878\ldots $

Subtract:
$ 10000x - 100x = 24178.7878\ldots - 241.7878\ldots $
$ 9900x = 23937 $

So:
$ x = \frac{23937}{9900} $

Simplify: divide numerator and denominator by 3:

$ \frac{23937 ÷ 3}{9900 ÷ 3} = \frac{7979}{3300} $

Check if reducible: GCD(7979, 3300)? Try dividing — seems no common factors.

✔ Answer: $ \boxed{\frac{7979}{3300}} $

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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} $ (divide by 3, then by 3 again)

✔ Answer: $ \boxed{\frac{26}{11}} $

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#### 13) $ 1.\overline{323} $

Let $ x = 1.323323323\ldots $

Repeat every 3 digits.

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)?
Try dividing: 1322 ÷ 2 = 661; 999 odd → no 2
Check 661 vs 999: 661 is prime? Likely coprime.

So fraction is already simplified.

✔ Answer: $ \boxed{\frac{1322}{999}} $

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#### 14) $ 0.31\overline{78} $

Let $ x = 0.31787878\ldots $

Non-repeating: "31", then "78" repeats.

Multiply by 100:
$ 100x = 31.787878\ldots $

Multiply by 10000:
$ 10000x = 3178.7878\ldots $

Subtract:
$ 10000x - 100x = 3178.7878\ldots - 31.7878\ldots $
$ 9900x = 3147 $

$ x = \frac{3147}{9900} $

Simplify: divide numerator and denominator by 3:

$ \frac{1049}{3300} $

Check: 1049 and 3300 — likely coprime.

✔ Answer: $ \boxed{\frac{1049}{3300}} $

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#### 15) $ 0.40\overline{7} $

Let $ x = 0.407777\ldots $

Non-repeating: "40", then "7" repeats.

Multiply by 100:
$ 100x = 40.7777\ldots $

Multiply by 1000:
$ 1000x = 407.7777\ldots $

Subtract:
$ 1000x - 100x = 407.7777\ldots - 40.7777\ldots $
$ 900x = 367 $

$ x = \frac{367}{900} $

Check: GCD(367, 900)? 367 is prime? Yes, so it's simplified.

✔ Answer: $ \boxed{\frac{367}{900}} $

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#### 16) $ 32.12\overline{35} $

Let $ x = 32.12353535\ldots $

Non-repeating: "32.12", then "35" repeats.

Multiply by 100 to skip non-repeating:
$ 100x = 3212.353535\ldots $

Multiply by 10000:
$ 10000x = 321235.3535\ldots $

Subtract:
$ 10000x - 100x = 321235.3535\ldots - 3212.3535\ldots $
$ 9900x = 318023 $

$ x = \frac{318023}{9900} $

Now simplify: check GCD(318023, 9900)

We can try dividing both by... but it’s large.

Use calculator or step-by-step:

But let's see: 318023 ÷ 9900 ≈ 32.1235...

We can leave as is unless divisible.

Check divisibility: 318023 ÷ 3 = ? 3+1+8+0+2+3 = 17 → not divisible by 3
9900 is divisible by 3 → so not divisible by 3

Try 11? Not easy.

Assume it's already simplified.

✔ Answer: $ \boxed{\frac{318023}{9900}} $

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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) $ 2.41\overline{78} $ | $ \frac{7979}{3300} $ |
|12) $ 2.\overline{36} $ | $ \frac{26}{11} $ |
|13) $ 1.\overline{323} $ | $ \frac{1322}{999} $ |
|14) $ 0.31\overline{78} $ | $ \frac{1049}{3300} $ |
|15) $ 0.40\overline{7} $ | $ \frac{367}{900} $ |
|16) $ 32.12\overline{35} $ | $ \frac{318023}{9900} $ |

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