https://file.aiquickdraw.com/mathbot/file/1726508562792xkpfrvdy.jpg

Let’s break down and solve the exercise step by step based on the image you provided. 

The task is to:
1. Identify if each number is *exact*, *repeating (periodic)*, or *semi-repeating (semiperiodic)*.
2. Convert each number to its equivalent fraction.

The list of numbers to work with is:
1. \( 0.6 \)
2. \( 1.454545\ldots \)
3. \( 3.025 \)
4. \( 2.5 \)
5. \( 0.8333\ldots \)
6. \( 4.181818\ldots \)
7. \( 0.090909\ldots \)
8. \( 5.0125 \)

### Solutions:

1. **\( 0.6 \)**:
   - **Type**: Exact decimal (terminates).
   - **Fraction**: \( 0.6 = \frac{6}{10} = \frac{3}{5} \).

2. **\( 1.454545\ldots \)**:
   - **Type**: Repeating decimal.
   - **Fraction**: \( 1.454545\ldots = 1 + 0.454545\ldots \). To convert the repeating part \( 0.454545\ldots \), let:
     \[
     x = 0.454545\ldots
     \]
     Multiply by 100 to shift the decimal:
     \[
     100x = 45.454545\ldots
     \]
     Subtract the original equation:
     \[
     100x - x = 45.454545\ldots - 0.454545\ldots
     \]
     This gives:
     \[
     99x = 45 \implies x = \frac{45}{99} = \frac{5}{11}
     \]
     Therefore:
     \[
     1.454545\ldots = 1 + \frac{5}{11} = \frac{16}{11}
     \]

3. **\( 3.025 \)**:
   - **Type**: Exact decimal (terminates).
   - **Fraction**: \( 3.025 = \frac{3025}{1000} = \frac{605}{200} = \frac{121}{40} \).

4. **\( 2.5 \)**:
   - **Type**: Exact decimal (terminates).
   - **Fraction**: \( 2.5 = \frac{25}{10} = \frac{5}{2} \).

5. **\( 0.8333\ldots \)**:
   - **Type**: Repeating decimal.
   - **Fraction**: \( 0.8333\ldots = \frac{5}{6} \) (this is a known repeating decimal).

6. **\( 4.181818\ldots \)**:
   - **Type**: Repeating decimal.
   - **Fraction**: \( 4.181818\ldots = 4 + 0.181818\ldots \).
     Let:
     \[
     y = 0.181818\ldots
     \]
     Multiply by 100:
     \[
     100y = 18.181818\ldots
     \]
     Subtracting the original equation:
     \[
     100y - y = 18.181818\ldots - 0.181818\ldots
     \]
     This gives:
     \[
     99y = 18 \implies y = \frac{18}{99} = \frac{2}{11}
     \]
     Therefore:
     \[
     4.181818\ldots = 4 + \frac{2}{11} = \frac{46}{11}
     \]

7. **\( 0.090909\ldots \)**:
   - **Type**: Repeating decimal.
   - **Fraction**: \( 0.090909\ldots = \frac{1}{11} \).

8. **\( 5.0125 \)**:
   - **Type**: Exact decimal (terminates).
   - **Fraction**: \( 5.0125 = \frac{50125}{10000} = \frac{802}{160} = \frac{401}{80} \).

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Let me know if you'd like more details on any of the steps. Here are five related questions to deepen your understanding:

1. How do you convert a repeating decimal into a fraction?
2. What is the difference between exact and repeating decimals?
3. Can a semiperiodic decimal always be converted to a simple fraction?
4. What methods exist to simplify complex fractions?
5. How does the position of the repeating part in a decimal affect the conversion process?

**Tip**: Always check if a fraction can be simplified after converting a decimal. This ensures that the fraction is in its lowest terms.

Exercici 7: Identificació i transformació de decimals a fracció generatriu. Llegeix atentament els següents nombres decimals i determina si són decimals exactes, periòdics o semiperiodics. Transforma'ls a la seva fracció generatriu corresponent. Nombres a treballar: 0.6, 1.454545…, 3.025, 2.5, 0.8333…, 4.181818…, 0.090909…, 5.0125.

Learn how to identify and transform decimals to their corresponding fractions. This exercise covers exact decimals, repeating decimals, and semi-repeating decimals with step-by-step solutions for numbers like 0.6, 1.454545…, 3.025, and more. Suitable for students in Grades 7-9.

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