Convert repeating decimals to fractions and color the picture to reveal the character.

Problem Overview:

The task involves converting repeating decimals into fractions and simplifying them if necessary. Each fraction corresponds to a specific color based on the key provided in the image. The goal is to solve each problem, determine the correct fraction, and then use the key to color the corresponding sections of the picture.

Key for Colors:

- Red: Equal to 1/2
- Orange: Equal to 1/3
- Green: Equal to 1/4
- Yellow: Equal to 1/5
- Blue: Equal to 1/6
- Purple: Equal to 1/7
- Brown: Equal to 1/8
- Pink: Equal to 1/9

Steps to Solve:

1. Identify the repeating decimals.
2. Convert each repeating decimal to a fraction.
3. Simplify the fraction if possible.
4. Match the simplified fraction to the key to determine the color.
5. Color the corresponding section of the picture.

Converting Repeating Decimals to Fractions:

#### 1. 0.5̅ (Repeating 5)
- Let \( x = 0.5555\ldots \).
- Multiply by 10: \( 10x = 5.5555\ldots \).
- Subtract the original equation: \( 10x - x = 5.5555\ldots - 0.5555\ldots \).
- Simplify: \( 9x = 5 \).
- Solve for \( x \): \( x = \frac{5}{9} \).

Fraction: \( \frac{5}{9} \)
Color: Pink

#### 2. 0.1̅ (Repeating 1)
- Let \( x = 0.1111\ldots \).
- Multiply by 10: \( 10x = 1.1111\ldots \).
- Subtract the original equation: \( 10x - x = 1.1111\ldots - 0.1111\ldots \).
- Simplify: \( 9x = 1 \).
- Solve for \( x \): \( x = \frac{1}{9} \).

Fraction: \( \frac{1}{9} \)
Color: Pink

#### 3. 0.1̅3̅ (Repeating 13)
- Let \( x = 0.131313\ldots \).
- Multiply by 100: \( 100x = 13.131313\ldots \).
- Subtract the original equation: \( 100x - x = 13.131313\ldots - 0.131313\ldots \).
- Simplify: \( 99x = 13 \).
- Solve for \( x \): \( x = \frac{13}{99} \).

Fraction: \( \frac{13}{99} \)
Color: Brown

#### 4. 0.2̅7̅ (Repeating 27)
- Let \( x = 0.272727\ldots \).
- Multiply by 100: \( 100x = 27.272727\ldots \).
- Subtract the original equation: \( 100x - x = 27.272727\ldots - 0.272727\ldots \).
- Simplify: \( 99x = 27 \).
- Solve for \( x \): \( x = \frac{27}{99} \).
- Simplify the fraction: \( \frac{27}{99} = \frac{3}{11} \).

Fraction: \( \frac{3}{11} \)
Color: Purple

#### 5. 0.1̅2̅ (Repeating 12)
- Let \( x = 0.121212\ldots \).
- Multiply by 100: \( 100x = 12.121212\ldots \).
- Subtract the original equation: \( 100x - x = 12.121212\ldots - 0.121212\ldots \).
- Simplify: \( 99x = 12 \).
- Solve for \( x \): \( x = \frac{12}{99} \).
- Simplify the fraction: \( \frac{12}{99} = \frac{4}{33} \).

Fraction: \( \frac{4}{33} \)
Color: Blue

#### 6. 0.1̅5̅ (Repeating 15)
- Let \( x = 0.151515\ldots \).
- Multiply by 100: \( 100x = 15.151515\ldots \).
- Subtract the original equation: \( 100x - x = 15.151515\ldots - 0.151515\ldots \).
- Simplify: \( 99x = 15 \).
- Solve for \( x \): \( x = \frac{15}{99} \).
- Simplify the fraction: \( \frac{15}{99} = \frac{5}{33} \).

Fraction: \( \frac{5}{33} \)
Color: Yellow

#### 7. 0.1̅2̅ (Repeating 12)
- This is the same as problem 5.
- Fraction: \( \frac{4}{33} \)
- Color: Blue

#### 8. 0.1̅6̅ (Repeating 16)
- Let \( x = 0.161616\ldots \).
- Multiply by 100: \( 100x = 16.161616\ldots \).
- Subtract the original equation: \( 100x - x = 16.161616\ldots - 0.161616\ldots \).
- Simplify: \( 99x = 16 \).
- Solve for \( x \): \( x = \frac{16}{99} \).

Fraction: \( \frac{16}{99} \)
Color: Green

Final Answer:

Using the key, the colors for each section are:
1. 0.5̅: Pink
2. 0.1̅: Pink
3. 0.1̅3̅: Brown
4. 0.2̅7̅: Purple
5. 0.1̅2̅: Blue
6. 0.1̅5̅: Yellow
7. 0.1̅2̅: Blue
8. 0.1̅6̅: Green

Final Answer:

\[
\boxed{\text{Pink, Pink, Brown, Purple, Blue, Yellow, Blue, Green}}
\]