Which of the following fractions is exactly equal to the recurring decimal 0.565656... (where the digits 56 repeat indefinitely)?

Introduction / Context:
This question asks you to convert a recurring decimal with a two digit repeating block into an exact fraction. Understanding how to convert repeating decimals to fractions is important in number system topics and helps improve comfort with non terminating representations of rational numbers.

Given Data / Assumptions:
The decimal is 0.565656..., where the pair of digits 56 repeats forever.We must find a fraction that is exactly equal to this recurring decimal.All fraction options are in simplest form or easily simplifiable.

Concept / Approach:
Let x represent the repeating decimal. Because the repeating block has two digits, multiplying x by 100 shifts the decimal point two places to the right, aligning the repeating parts. Subtracting the original x from 100x eliminates the repeating part and leaves a simple linear equation. Solving for x gives the required fraction. This is a standard method for handling repeating decimals.

Step-by-Step Solution:
Step 1: Let x = 0.565656..., where 56 repeats indefinitely.Step 2: Multiply both sides by 100 to shift two digits: 100x = 56.565656...Step 3: Notice that the decimal parts of x and 100x are identical (both .565656...).Step 4: Subtract the original equation from this new one: 100x − x = 56.565656... − 0.565656...Step 5: The repeating parts cancel out, leaving 99x = 56.Step 6: Solve for x: x = 56 / 99.Step 7: Therefore, 0.565656... is exactly equal to 56 / 99.

Verification / Alternative check:
To verify, divide 56 by 99 using long division. The quotient starts with 0.56 and the pattern 56 continues repeating: 0.565656..., matching the original decimal. This confirms that the fraction 56 / 99 is correct. Also, the fraction is already in simplest form because 56 and 99 do not share any common factor greater than 1.

Why Other Options Are Wrong:
The fraction 56 / 100 is equal to 0.56, which terminates and does not repeat. The fraction 56 / 1000 equals 0.056, which is much smaller. The fraction 560 / 90 simplifies to 56 / 9, which equals 6.222..., not close to 0.5656.... None of these alternative fractions produce a repeating decimal with the repeating block 56 starting immediately after the decimal point.

Common Pitfalls:
Some learners confuse the repeating decimal 0.565656... with the terminating decimal 0.56 and pick 56 / 100. Others incorrectly move the decimal only one place instead of two, which does not align the repeating block. Remember that the multiplier should be 10 raised to the power of the number of repeating digits, which is 100 in this case.

Final Answer:
The recurring decimal 0.565656... is exactly equal to the fraction 56/99.