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To express the repeating decimal \( 0.\overline{56} \) as a ratio of integers, follow these steps:

1. Let \( x = 0.\overline{56} \), so \( x = 0.56565656\ldots \).
2. Multiply both sides by 100 to shift the decimal two places to the right (since the repeating part is two digits long):
   \[
   100x = 56.565656\ldots
   \]
3. Now, we have:
   \[
   100x = 56.565656\ldots \quad \text{and} \quad x = 0.565656\ldots
   \]
4. Subtract the second equation from the first:
   \[
   100x - x = 56.565656\ldots - 0.565656\ldots
   \]
   This simplifies to:
   \[
   99x = 56
   \]
5. Solve for \( x \):
   \[
   x = \frac{56}{99}
   \]

Thus, \( 0.\overline{56} = \frac{56}{99} \).

Would you like more details or have any questions?

Here are some related questions to explore further:
1. How do you convert other repeating decimals, like \(0.\overline{3}\), into fractions?
2. Why does the method of multiplying by 100 work for two-digit repeating patterns?
3. How can you determine the fraction for decimals with non-repeating and repeating parts, such as \(0.123\overline{4}\)?
4. Can repeating decimals always be converted into fractions?
5. What is the difference between terminating and repeating decimals in terms of fraction representation?

**Tip:** When converting repeating decimals to fractions, remember that the number of repeating digits determines the power of 10 used for multiplication.

Express the repeating decimal 0.56 (with repeating 56) as a ratio of integers.

Learn how to convert the repeating decimal 0.56 (0.565656...) into a fraction. This step-by-step guide explains the algebraic method to express 0.56̅ as the fraction 56/99, suitable for students in Grades 7-9.

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