https://file.aiquickdraw.com/mathbot/file/172711070607945ces8x0.jpg

The problem in the image asks to show that the number 0.747474... (a repeating decimal) is rational by writing it as a ratio of two integers.

### Solution:

To express \( 0.747474\ldots \) (repeating) as a fraction:

1. Let \( x = 0.74747474\ldots \).

2. Multiply both sides by 100 to shift the decimal two places to the right:
   \[
   100x = 74.74747474\ldots
   \]

3. Now subtract the original equation \( x = 0.747474\ldots \) from this new equation:
   \[
   100x - x = 74.747474\ldots - 0.747474\ldots
   \]
   This simplifies to:
   \[
   99x = 74
   \]

4. Solve for \( x \):
   \[
   x = \frac{74}{99}
   \]

Thus, \( 0.747474\ldots = \frac{74}{99} \), showing that it is a rational number.

This matches the answer shown in the image, which is \( \frac{74}{99} \).

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Do you have any questions or would you like more details on this solution? 

Here are 5 related questions for further exploration:

1. How can you convert a non-repeating decimal to a fraction?
2. What is the general method for converting repeating decimals into fractions?
3. Can every repeating decimal be expressed as a fraction?
4. What is the difference between a rational and an irrational number?
5. How can you simplify fractions derived from repeating decimals?

**Tip:** When converting repeating decimals to fractions, always try multiplying by powers of 10 to eliminate the repeating part during subtraction.

Show that the following number is rational by writing it as a ratio of two integers. 0.747474... (repeating decimal)

Learn how to convert the repeating decimal 0.747474... into a fraction using algebraic methods. This detailed step-by-step guide shows how the repeating decimal is expressed as the ratio 74/99. Suitable for students in Grades 7-9.

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