https://r2.erweima.ai/mathbot/image/1723494611929euyqqavx.jpg

obten la fraccion generatriz de las siguientes expresiones decimales periodicas puras

To solve these problems, we'll need to convert each repeating decimal into its equivalent fraction, known as the "fracción generatriz."

### Steps to Convert a Pure Repeating Decimal to a Fraction:
1. **Let the repeating decimal be represented by \( x \).**
2. **Multiply \( x \) by a power of 10 such that the decimal part aligns.**
3. **Subtract the original equation from the new equation to eliminate the repeating part.**
4. **Solve for \( x \), giving the fraction.**
5. **Simplify the fraction if possible.**

Let's go through each one from the image:

---

**a) \( 0.\overline{36} \)**

Let \( x = 0.\overline{36} \).

Multiply by 100 (since there are 2 repeating digits):

\[
100x = 36.\overline{36}
\]

Subtract the original equation:

\[
100x - x = 36.\overline{36} - 0.\overline{36}
\]

\[
99x = 36
\]

\[
x = \frac{36}{99} = \frac{4}{11}
\]

So, \( 0.\overline{36} = \frac{4}{11} \).

---

**b) \( 0.\overline{202} \)**

Let \( x = 0.\overline{202} \).

Multiply by 1000:

\[
1000x = 202.\overline{202}
\]

Subtract the original equation:

\[
1000x - x = 202.\overline{202} - 0.\overline{202}
\]

\[
999x = 202
\]

\[
x = \frac{202}{999}
\]

So, \( 0.\overline{202} = \frac{202}{999} \).

---

**c) \( 21.\overline{025} \)**

Let \( x = 21.\overline{025} \).

Multiply by 1000:

\[
1000x = 21025.\overline{025}
\]

Subtract the original equation:

\[
1000x - x = 21025.\overline{025} - 21.\overline{025}
\]

\[
999x = 21004
\]

\[
x = \frac{21004}{999} = \frac{5251}{249.75}
\]

Simplify further if needed.

---

**d) \( 0.\overline{048} \)**

Let \( x = 0.\overline{048} \).

Multiply by 1000:

\[
1000x = 48.\overline{048}
\]

Subtract the original equation:

\[
1000x - x = 48.\overline{048} - 0.\overline{048}
\]

\[
999x = 48
\]

\[
x = \frac{48}{999}
\]

Simplify the fraction:

\[
\frac{48}{999} = \frac{16}{333}
\]

So, \( 0.\overline{048} = \frac{16}{333} \).

---

**e) \( 13.\overline{342} \)**

Let \( x = 13.\overline{342} \).

Multiply by 1000:

\[
1000x = 13342.\overline{342}
\]

Subtract the original equation:

\[
1000x - x = 13342.\overline{342} - 13.\overline{342}
\]

\[
999x = 13329
\]

\[
x = \frac{13329}{999} = \frac{1481}{111}
\]

So, \( 13.\overline{342} = \frac{1481}{111} \).

---

**f) \( 3.\overline{14} \)**

Let \( x = 3.\overline{14} \).

Multiply by 100:

\[
100x = 314.\overline{14}
\]

Subtract the original equation:

\[
100x - x = 314.\overline{14} - 3.\overline{14}
\]

\[
99x = 311
\]

\[
x = \frac{311}{99}
\]

So, \( 3.\overline{14} = \frac{311}{99} \).

---

Would you like more details on any of these steps or further clarification? 

### Related Questions:
1. How do you convert a repeating decimal that begins after some non-repeating digits to a fraction?
2. What is the difference between a pure repeating decimal and a mixed repeating decimal?
3. How do you identify if a fraction will result in a repeating decimal or a terminating decimal?
4. How do you simplify fractions obtained from repeating decimals?
5. Can all repeating decimals be converted into fractions?
6. How do you find the repeating part of a decimal manually?
7. What role does the denominator play in determining the length of the repeating block?
8. How can you use algebra to convert a repeating decimal into a fraction?

**Tip:** When converting a repeating decimal to a fraction, the number of repeating digits will determine the power of 10 you multiply by. This ensures the decimals line up correctly for subtraction.

Learn how to convert pure repeating decimals to fractions with detailed step-by-step examples. Understand the concept of fracción generatriz and apply it to solve decimal to fraction conversions. Suitable for high school students and beyond.

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