Math worksheet for converting repeating decimals to fractions, ideal for classroom or home practice.

To solve the problem of converting repeating decimals to fractions, we need to carefully analyze each decimal and determine whether it is a terminating decimal or a repeating decimal. Here's how we approach each type:

Terminating Decimals

A terminating decimal can be directly converted to a fraction by expressing it as a numerator over a denominator that corresponds to the place value of the last digit.

Repeating Decimals

For repeating decimals, we use algebraic methods to convert them into fractions. The general steps are:
1. Let \( x \) be the repeating decimal.
2. Multiply \( x \) by a power of 10 such that the repeating part aligns.
3. Subtract the original equation from this new equation to eliminate the repeating part.
4. Solve for \( x \) to get the fraction.

Let's solve each problem step by step.

---

Problem 1: \( 3.33 \)

This is a terminating decimal.
- \( 3.33 = 3 + \frac{33}{100} = \frac{300}{100} + \frac{33}{100} = \frac{333}{100} \)

Answer: \( \frac{333}{100} \)

---

Problem 2: \( 8.999 \)

This is a terminating decimal.
- \( 8.999 = 8 + \frac{999}{1000} = \frac{8000}{1000} + \frac{999}{1000} = \frac{8999}{1000} \)

Answer: \( \frac{8999}{1000} \)

---

Problem 3: \( 1.2525 \)

This is a repeating decimal with a repeating block of "25".
1. Let \( x = 1.252525\ldots \)
2. Multiply by 100 (since the repeating block has 2 digits): \( 100x = 125.252525\ldots \)
3. Subtract the original equation from this new equation:
\[
100x - x = 125.252525\ldots - 1.252525\ldots
\]
\[
99x = 124
\]
4. Solve for \( x \):
\[
x = \frac{124}{99}
\]

Answer: \( \frac{124}{99} \)

---

Problem 4: \( 0.888 \)

This is a repeating decimal with a repeating block of "8".
1. Let \( x = 0.8888\ldots \)
2. Multiply by 10 (since the repeating block has 1 digit): \( 10x = 8.8888\ldots \)
3. Subtract the original equation from this new equation:
\[
10x - x = 8.8888\ldots - 0.8888\ldots
\]
\[
9x = 8
\]
4. Solve for \( x \):
\[
x = \frac{8}{9}
\]

Answer: \( \frac{8}{9} \)

---

Problem 5: \( 0.1717 \)

This is a repeating decimal with a repeating block of "17".
1. Let \( x = 0.171717\ldots \)
2. Multiply by 100 (since the repeating block has 2 digits): \( 100x = 17.171717\ldots \)
3. Subtract the original equation from this new equation:
\[
100x - x = 17.171717\ldots - 0.171717\ldots
\]
\[
99x = 17
\]
4. Solve for \( x \):
\[
x = \frac{17}{99}
\]

Answer: \( \frac{17}{99} \)

---

Problem 6: \( 0.9999 \)

This is a terminating decimal.
- \( 0.9999 = \frac{9999}{10000} \)

Answer: \( \frac{9999}{10000} \)

---

Problem 7: \( 10.9898 \)

This is a repeating decimal with a repeating block of "98".
1. Let \( x = 10.989898\ldots \)
2. Multiply by 100 (since the repeating block has 2 digits): \( 100x = 1098.989898\ldots \)
3. Subtract the original equation from this new equation:
\[
100x - x = 1098.989898\ldots - 10.989898\ldots
\]
\[
99x = 1088
\]
4. Solve for \( x \):
\[
x = \frac{1088}{99}
\]

Answer: \( \frac{1088}{99} \)

---

Problem 8: \( 1.4444 \)

This is a repeating decimal with a repeating block of "4".
1. Let \( x = 1.4444\ldots \)
2. Multiply by 10 (since the repeating block has 1 digit): \( 10x = 14.4444\ldots \)
3. Subtract the original equation from this new equation:
\[
10x - x = 14.4444\ldots - 1.4444\ldots
\]
\[
9x = 13
\]
4. Solve for \( x \):
\[
x = \frac{13}{9}
\]

Answer: \( \frac{13}{9} \)

---

Problem 9: \( 11.111 \)

This is a repeating decimal with a repeating block of "1".
1. Let \( x = 11.1111\ldots \)
2. Multiply by 10 (since the repeating block has 1 digit): \( 10x = 111.1111\ldots \)
3. Subtract the original equation from this new equation:
\[
10x - x = 111.1111\ldots - 11.1111\ldots
\]
\[
9x = 100
\]
4. Solve for \( x \):
\[
x = \frac{100}{9}
\]

Answer: \( \frac{100}{9} \)

---

Problem 10: \( 0.8787 \)

This is a repeating decimal with a repeating block of "87".
1. Let \( x = 0.878787\ldots \)
2. Multiply by 100 (since the repeating block has 2 digits): \( 100x = 87.878787\ldots \)
3. Subtract the original equation from this new equation:
\[
100x - x = 87.878787\ldots - 0.878787\ldots
\]
\[
99x = 87
\]
4. Solve for \( x \):
\[
x = \frac{87}{99} = \frac{29}{33} \quad (\text{after simplifying})
\]

Answer: \( \frac{29}{33} \)

---

Problem 11: \( 0.1222 \)

This is a repeating decimal with a repeating block of "2".
1. Let \( x = 0.12222\ldots \)
2. Multiply by 10 (to shift the decimal point past the non-repeating part): \( 10x = 1.2222\ldots \)
3. Multiply by 100 (to align the repeating parts): \( 100x = 12.2222\ldots \)
4. Subtract the second equation from the third equation:
\[
100x - 10x = 12.2222\ldots - 1.2222\ldots
\]
\[
90x = 11
\]
5. Solve for \( x \):
\[
x = \frac{11}{90}
\]

Answer: \( \frac{11}{90} \)

---

Problem 12: \( 0.444 \)

This is a repeating decimal with a repeating block of "4".
1. Let \( x = 0.4444\ldots \)
2. Multiply by 10 (since the repeating block has 1 digit): \( 10x = 4.4444\ldots \)
3. Subtract the original equation from this new equation:
\[
10x - x = 4.4444\ldots - 0.4444\ldots
\]
\[
9x = 4
\]
4. Solve for \( x \):
\[
x = \frac{4}{9}
\]

Answer: \( \frac{4}{9} \)

---

Problem 13: \( 125.777 \)

This is a repeating decimal with a repeating block of "7".
1. Let \( x = 125.7777\ldots \)
2. Multiply by 10 (since the repeating block has 1 digit): \( 10x = 1257.7777\ldots \)
3. Subtract the original equation from this new equation:
\[
10x - x = 1257.7777\ldots - 125.7777\ldots
\]
\[
9x = 1132
\]
4. Solve for \( x \):
\[
x = \frac{1132}{9}
\]

Answer: \( \frac{1132}{9} \)

---

Problem 14: \( 99.2323 \)

This is a repeating decimal with a repeating block of "23".
1. Let \( x = 99.232323\ldots \)
2. Multiply by 100 (since the repeating block has 2 digits): \( 100x = 9923.232323\ldots \)
3. Subtract the original equation from this new equation:
\[
100x - x = 9923.232323\ldots - 99.232323\ldots
\]
\[
99x = 9824
\]
4. Solve for \( x \):
\[
x = \frac{9824}{99}
\]

Answer: \( \frac{9824}{99} \)

---

Problem 15: \( 0.1111 \)

This is a repeating decimal with a repeating block of "1".
1. Let \( x = 0.1111\ldots \)
2. Multiply by 10 (since the repeating block has 1 digit): \( 10x = 1.1111\ldots \)
3. Subtract the original equation from this new equation:
\[
10x - x = 1.1111\ldots - 0.1111\ldots
\]
\[
9x = 1
\]
4. Solve for \( x \):
\[
x = \frac{1}{9}
\]

Answer: \( \frac{1}{9} \)

---

Problem 16: \( 0.555 \)

This is a repeating decimal with a repeating block of "5".
1. Let \( x = 0.5555\ldots \)
2. Multiply by 10 (since the repeating block has 1 digit): \( 10x = 5.5555\ldots \)
3. Subtract the original equation from this new equation:
\[
10x - x = 5.5555\ldots - 0.5555\ldots
\]
\[
9x = 5
\]
4. Solve for \( x \):
\[
x = \frac{5}{9}
\]

Answer: \( \frac{5}{9} \)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & \frac{333}{100} \\
2. & \frac{8999}{1000} \\
3. & \frac{124}{99} \\
4. & \frac{8}{9} \\
5. & \frac{17}{99} \\
6. & \frac{9999}{10000} \\
7. & \frac{1088}{99} \\
8. & \frac{13}{9} \\
9. & \frac{100}{9} \\
10. & \frac{29}{33} \\
11. & \frac{11}{90} \\
12. & \frac{4}{9} \\
13. & \frac{1132}{9} \\
14. & \frac{9824}{99} \\
15. & \frac{1}{9} \\
16. & \frac{5}{9} \\
\end{array}
}
\]