To convert the recurring decimal $5.272727\overline{27}$ to a fraction, follow these steps:

Step 1: Let $x = 5.272727\overline{27}$.

We denote the repeating part of the decimal as $27$. This repeating part has two digits.

Step 2: Multiply both sides by 100.

Since the repeating block is two digits long, we multiply both sides by $10^2 = 100$:

$100x = 527.272727\overline{27}$

Step 3: Subtract the original equation from the new equation.

Subtract $x = 5.272727\overline{27}$ from $100x = 527.272727\overline{27}$:

$100x - x = 527.272727\overline{27} - 5.272727\overline{27}$ $99x = 522$

Step 4: Solve for $x$.

Solve for $x$ by dividing both sides of the equation by 99:

$x = \frac{522}{99}$

Step 5: Simplify the fraction.

Now, simplify the fraction $\frac{522}{99}$. The greatest common divisor (GCD) of 522 and 99 is 9, so we divide both the numerator and the denominator by 9:

$\frac{522}{99} = \frac{522 \div 9}{99 \div 9} = \frac{58}{11}$

Thus, the fraction equivalent of the recurring decimal $5.272727\overline{27}$ is:

$5.272727\overline{27} = \frac{58}{11}$

Final Answer:

The recurring decimal $5.272727\overline{27}$ is equal to the fraction $\frac{58}{11}$.

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Related Questions:

1. How do you convert a recurring decimal with more complex repeating patterns?
2. What is the method to convert a non-repeating decimal to a fraction?
3. Can all repeating decimals be converted to fractions?
4. What are terminating decimals and how are they converted to fractions?
5. How does the length of the repeating block affect the conversion process?

Tip:

When simplifying fractions, always check for the greatest common divisor (GCD) to ensure the fraction is in its simplest form.