Converting repeating decimals to fractions worksheet with six problems requiring students to show their work.

To convert repeating decimals to fractions, we use algebraic methods. Let's solve each problem step by step.

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Problem 1: Convert \( \overline{0.88} \) to a fraction

#### Step 1: Let \( x = 0.\overline{88} \)
\[ x = 0.8888\ldots \]

#### Step 2: Multiply both sides by 100 (since the repeating block has 2 digits)
\[ 100x = 88.8888\ldots \]

#### Step 3: Subtract the original equation from this new equation
\[ 100x - x = 88.8888\ldots - 0.8888\ldots \]
\[ 99x = 88 \]

#### Step 4: Solve for \( x \)
\[ x = \frac{88}{99} \]

#### Step 5: Simplify the fraction
The greatest common divisor (GCD) of 88 and 99 is 11.
\[ x = \frac{88 \div 11}{99 \div 11} = \frac{8}{9} \]

#### Final Answer:
\[ \boxed{\frac{8}{9}} \]

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Problem 2: Convert \( \overline{0.66} \) to a fraction

#### Step 1: Let \( x = 0.\overline{66} \)
\[ x = 0.6666\ldots \]

#### Step 2: Multiply both sides by 100 (since the repeating block has 2 digits)
\[ 100x = 66.6666\ldots \]

#### Step 3: Subtract the original equation from this new equation
\[ 100x - x = 66.6666\ldots - 0.6666\ldots \]
\[ 99x = 66 \]

#### Step 4: Solve for \( x \)
\[ x = \frac{66}{99} \]

#### Step 5: Simplify the fraction
The GCD of 66 and 99 is 33.
\[ x = \frac{66 \div 33}{99 \div 33} = \frac{2}{3} \]

#### Final Answer:
\[ \boxed{\frac{2}{3}} \]

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Problem 3: Convert \( \overline{0.42} \) to a fraction

#### Step 1: Let \( x = 0.\overline{42} \)
\[ x = 0.424242\ldots \]

#### Step 2: Multiply both sides by 100 (since the repeating block has 2 digits)
\[ 100x = 42.424242\ldots \]

#### Step 3: Subtract the original equation from this new equation
\[ 100x - x = 42.424242\ldots - 0.424242\ldots \]
\[ 99x = 42 \]

#### Step 4: Solve for \( x \)
\[ x = \frac{42}{99} \]

#### Step 5: Simplify the fraction
The GCD of 42 and 99 is 3.
\[ x = \frac{42 \div 3}{99 \div 3} = \frac{14}{33} \]

#### Final Answer:
\[ \boxed{\frac{14}{33}} \]

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Problem 4: Convert \( \overline{0.36} \) to a fraction

#### Step 1: Let \( x = 0.\overline{36} \)
\[ x = 0.363636\ldots \]

#### Step 2: Multiply both sides by 100 (since the repeating block has 2 digits)
\[ 100x = 36.363636\ldots \]

#### Step 3: Subtract the original equation from this new equation
\[ 100x - x = 36.363636\ldots - 0.363636\ldots \]
\[ 99x = 36 \]

#### Step 4: Solve for \( x \)
\[ x = \frac{36}{99} \]

#### Step 5: Simplify the fraction
The GCD of 36 and 99 is 9.
\[ x = \frac{36 \div 9}{99 \div 9} = \frac{4}{11} \]

#### Final Answer:
\[ \boxed{\frac{4}{11}} \]

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Problem 5: Convert \( \overline{0.16} \) to a fraction

#### Step 1: Let \( x = 0.\overline{16} \)
\[ x = 0.161616\ldots \]

#### Step 2: Multiply both sides by 100 (since the repeating block has 2 digits)
\[ 100x = 16.161616\ldots \]

#### Step 3: Subtract the original equation from this new equation
\[ 100x - x = 16.161616\ldots - 0.161616\ldots \]
\[ 99x = 16 \]

#### Step 4: Solve for \( x \)
\[ x = \frac{16}{99} \]

#### Final Answer:
\[ \boxed{\frac{16}{99}} \]

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Problem 6: Convert \( \overline{0.369} \) to a fraction

#### Step 1: Let \( x = 0.\overline{369} \)
\[ x = 0.369369369\ldots \]

#### Step 2: Multiply both sides by 1000 (since the repeating block has 3 digits)
\[ 1000x = 369.369369369\ldots \]

#### Step 3: Subtract the original equation from this new equation
\[ 1000x - x = 369.369369369\ldots - 0.369369369\ldots \]
\[ 999x = 369 \]

#### Step 4: Solve for \( x \)
\[ x = \frac{369}{999} \]

#### Step 5: Simplify the fraction
The GCD of 369 and 999 is 9.
\[ x = \frac{369 \div 9}{999 \div 9} = \frac{41}{111} \]

#### Final Answer:
\[ \boxed{\frac{41}{111}} \]

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Final Answers:

1. \( \overline{0.88} = \boxed{\frac{8}{9}} \)
2. \( \overline{0.66} = \boxed{\frac{2}{3}} \)
3. \( \overline{0.42} = \boxed{\frac{14}{33}} \)
4. \( \overline{0.36} = \boxed{\frac{4}{11}} \)
5. \( \overline{0.16} = \boxed{\frac{16}{99}} \)
6. \( \overline{0.369} = \boxed{\frac{41}{111}} \)