Math worksheet for converting repeating 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.

---

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.

---

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} $

---

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} $

---

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} $

---

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} $

---

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} $

---

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} $

---

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} $

---

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} $

---

9) $ 0.00\overline{32} $

This is $ 0.00323232\ldots $

Let $ x = 0.00323232\ldots $

First, shift decimal to make repeating part start:

Multiply by 100:
$ 100x = 0.323232\ldots $

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

Now 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} $

---

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 $

---

11) $ 2.41\overline{78} $

Let $ x = 2.41787878\ldots $

The repeating part is "78", which starts after two decimal places.

Multiply by 100 to move decimal past non-repeating part:
$ 100x = 241.787878\ldots $

Now multiply by 100 again:
$ 10000x = 24178.787878\ldots $

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

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

Simplify: divide numerator and denominator by 3:

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

Check if further simplification possible:
GCD(7979, 3300)? Try dividing:

7979 ÷ 11 = 725.36… not divisible
Try small primes: seems irreducible.

✔ Answer: $ \frac{7979}{3300} $

---

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} $

---

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 if reducible: GCD(1322, 999)

Use Euclidean algorithm:
1322 ÷ 999 = 1 R 323
999 ÷ 323 = 3 R 30
323 ÷ 30 = 10 R 23
30 ÷ 23 = 1 R 7
23 ÷ 7 = 3 R 2
7 ÷ 2 = 3 R 1
2 ÷ 1 = 2 R 0 → GCD = 1

So fraction is already simplified.

✔ Answer: $ \frac{1322}{999} $

---

14) $ 0.31\overline{78} $

Let $ x = 0.31787878\ldots $

Multiply by 100 to shift past non-repeating part:
$ 100x = 31.787878\ldots $

Multiply by 100 again:
$ 10000x = 3178.787878\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 if reducible: GCD(1049, 3300) — 1049 is prime? Check divisibility:

1049 ÷ 7 ≈ 149.8 → not divisible
1049 ÷ 11 = 95.36 → no
13: 1049 ÷ 13 ≈ 80.6 → no
17: ~61.7 → no
19: ~55.2 → no
23: ~45.6 → no
29: ~36.1 → no
31: ~33.8 → no
So likely irreducible.

✔ Answer: $ \frac{1049}{3300} $

---

15) $ 0.40\overline{7} $

Let $ x = 0.407777\ldots $

Multiply by 100 to get to non-repeating part:
$ 100x = 40.7777\ldots $

Now multiply by 10:
$ 1000x = 407.7777\ldots $

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

Check if reducible: GCD(367, 900)

367 is prime? Yes, it’s a known prime number.

So irreducible.

✔ Answer: $ \frac{367}{900} $

---

16) $ 32.12\overline{35} $

Let $ x = 32.12353535\ldots $

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

Multiply by 100 again:
$ 10000x = 321235.353535\ldots $

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

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

Check if reducible: GCD(318023, 9900)

Try dividing: 318023 ÷ 3 = 106007.666... → not divisible
9900 = 99 × 100 = 9×11×100 → check divisibility by 11?

Alternating sum: 3-1+8-0+2-3 = 3-1=2; +8=10; -0=10; +2=12; -3=9 → not divisible by 11

Likely irreducible.

✔ Answer: $ \frac{318023}{9900} $

---

✔ 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} $ |

Let me know if you'd like a PDF version or explanations for any specific one!