Convert repeating decimals to fractions worksheet with ten problems and answer spaces.

To solve the problem of converting repeating decimals to fractions, we will use a systematic approach. Let's go through each repeating decimal step by step.

General Method:

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 \( x \) from this new equation to eliminate the repeating part.
4. Solve for \( x \) to get the fraction.

Now, let's apply this method to each given repeating decimal.

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1. \( 0.\overline{4354} \)

#### Step 1: Let \( x = 0.\overline{4354} \).
\[ x = 0.435443544354\ldots \]

#### Step 2: Multiply by \( 10^4 = 10000 \) (since the repeating block has 4 digits).
\[ 10000x = 4354.435443544354\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 10000x - x = 4354.435443544354\ldots - 0.435443544354\ldots \]
\[ 9999x = 4354 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{4354}{9999} \]

Thus, the fraction is:
\[ \boxed{\frac{4354}{9999}} \]

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2. \( 0.\overline{588} \)

#### Step 1: Let \( x = 0.\overline{588} \).
\[ x = 0.588588588\ldots \]

#### Step 2: Multiply by \( 10^3 = 1000 \) (since the repeating block has 3 digits).
\[ 1000x = 588.588588588\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 1000x - x = 588.588588588\ldots - 0.588588588\ldots \]
\[ 999x = 588 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{588}{999} \]

Simplify the fraction:
\[ \frac{588}{999} = \frac{196}{333} \]

Thus, the fraction is:
\[ \boxed{\frac{196}{333}} \]

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3. \( 0.\overline{680} \)

#### Step 1: Let \( x = 0.\overline{680} \).
\[ x = 0.680680680\ldots \]

#### Step 2: Multiply by \( 10^3 = 1000 \) (since the repeating block has 3 digits).
\[ 1000x = 680.680680680\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 1000x - x = 680.680680680\ldots - 0.680680680\ldots \]
\[ 999x = 680 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{680}{999} \]

Thus, the fraction is:
\[ \boxed{\frac{680}{999}} \]

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4. \( 5.\overline{484} \)

#### Step 1: Let \( x = 5.\overline{484} \).
\[ x = 5.484484484\ldots \]

#### Step 2: Multiply by \( 10^3 = 1000 \) (since the repeating block has 3 digits).
\[ 1000x = 5484.484484484\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 1000x - x = 5484.484484484\ldots - 5.484484484\ldots \]
\[ 999x = 5479 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{5479}{999} \]

Thus, the fraction is:
\[ \boxed{\frac{5479}{999}} \]

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5. \( 8.\overline{180} \)

#### Step 1: Let \( x = 8.\overline{180} \).
\[ x = 8.180180180\ldots \]

#### Step 2: Multiply by \( 10^3 = 1000 \) (since the repeating block has 3 digits).
\[ 1000x = 8180.180180180\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 1000x - x = 8180.180180180\ldots - 8.180180180\ldots \]
\[ 999x = 8172 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{8172}{999} \]

Simplify the fraction:
\[ \frac{8172}{999} = \frac{2724}{333} = \frac{908}{111} \]

Thus, the fraction is:
\[ \boxed{\frac{908}{111}} \]

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6. \( 0.\overline{81668} \)

#### Step 1: Let \( x = 0.\overline{81668} \).
\[ x = 0.816688166881668\ldots \]

#### Step 2: Multiply by \( 10^5 = 100000 \) (since the repeating block has 5 digits).
\[ 100000x = 81668.8166881668\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 100000x - x = 81668.8166881668\ldots - 0.8166881668\ldots \]
\[ 99999x = 81668 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{81668}{99999} \]

Thus, the fraction is:
\[ \boxed{\frac{81668}{99999}} \]

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7. \( 5.\overline{24600} \)

#### Step 1: Let \( x = 5.\overline{24600} \).
\[ x = 5.246002460024600\ldots \]

#### Step 2: Multiply by \( 10^5 = 100000 \) (since the repeating block has 5 digits).
\[ 100000x = 524600.2460024600\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 100000x - x = 524600.2460024600\ldots - 5.2460024600\ldots \]
\[ 99999x = 524595 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{524595}{99999} \]

Simplify the fraction:
\[ \frac{524595}{99999} = \frac{174865}{33333} \]

Thus, the fraction is:
\[ \boxed{\frac{174865}{33333}} \]

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8. \( 9.\overline{2331} \)

#### Step 1: Let \( x = 9.\overline{2331} \).
\[ x = 9.233123312331\ldots \]

#### Step 2: Multiply by \( 10^4 = 10000 \) (since the repeating block has 4 digits).
\[ 10000x = 92331.233123312331\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 10000x - x = 92331.233123312331\ldots - 9.233123312331\ldots \]
\[ 9999x = 92322 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{92322}{9999} \]

Simplify the fraction:
\[ \frac{92322}{9999} = \frac{30774}{3333} = \frac{10258}{1111} \]

Thus, the fraction is:
\[ \boxed{\frac{10258}{1111}} \]

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9. \( 27.\overline{1} \)

#### Step 1: Let \( x = 27.\overline{1} \).
\[ x = 27.111111\ldots \]

#### Step 2: Multiply by \( 10 \) (since the repeating block has 1 digit).
\[ 10x = 271.111111\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 10x - x = 271.111111\ldots - 27.111111\ldots \]
\[ 9x = 244 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{244}{9} \]

Thus, the fraction is:
\[ \boxed{\frac{244}{9}} \]

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10. \( 4.\overline{5935} \)

#### Step 1: Let \( x = 4.\overline{5935} \).
\[ x = 4.593559355935\ldots \]

#### Step 2: Multiply by \( 10^4 = 10000 \) (since the repeating block has 4 digits).
\[ 10000x = 45935.593559355935\ldots \]

#### Step 3: Subtract the original \( x \) from this equation.
\[ 10000x - x = 45935.593559355935\ldots - 4.593559355935\ldots \]
\[ 9999x = 45931 \]

#### Step 4: Solve for \( x \).
\[ x = \frac{45931}{9999} \]

Thus, the fraction is:
\[ \boxed{\frac{45931}{9999}} \]

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Final Answers:

1. \( \boxed{\frac{4354}{9999}} \)
2. \( \boxed{\frac{196}{333}} \)
3. \( \boxed{\frac{680}{999}} \)
4. \( \boxed{\frac{5479}{999}} \)
5. \( \boxed{\frac{908}{111}} \)
6. \( \boxed{\frac{81668}{99999}} \)
7. \( \boxed{\frac{174865}{33333}} \)
8. \( \boxed{\frac{10258}{1111}} \)
9. \( \boxed{\frac{244}{9}} \)
10. \( \boxed{\frac{45931}{9999}} \)