To find a fraction equivalent to the recurring decimal $0.\overline{513251325132}$, let's go through the steps systematically.

Step 1: Express the recurring decimal

Let $x = 0.\overline{513251325132}$. This means that $x$ is a decimal that repeats every 12 digits.

Step 2: Multiply by a power of 10

To eliminate the repeating part, we multiply $x$ by $10^{12}$ because there are 12 digits in the repeating sequence.

$10^{12}x = 513251325132.\overline{513251325132}$

Step 3: Subtract the original equation from the new equation

Now, subtract the original equation $x = 0.\overline{513251325132}$ from the new equation:

$10^{12}x - x = 513251325132.\overline{513251325132} - 0.\overline{513251325132}$

This simplifies to:

$(10^{12} - 1)x = 513251325132$

$999999999999x = 513251325132$

Step 4: Solve for $x$

$x = \frac{513251325132}{999999999999}$

Step 5: Simplify the fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator.

Let's calculate the GCD and simplify the fraction:

$\text{GCD}(513251325132, 999999999999) = 3$

Dividing the numerator and the denominator by the GCD:

$x = \frac{513251325132 \div 3}{999999999999 \div 3} = \frac{171083775044}{333333333333}$

Thus, the fraction equivalent to the recurring decimal $0.\overline{513251325132}$ is:

$\boxed{\frac{171083775044}{333333333333}}$

Would you like to see the detailed process of finding the GCD, or do you have any other questions?

---

Related Questions:

1. How do you convert a recurring decimal with a different number of repeating digits to a fraction?
2. What is the decimal representation of a fraction like $\frac{1}{7}$?
3. How can you determine if a fraction will result in a terminating or recurring decimal?
4. What is the procedure for converting a mixed recurring decimal to a fraction?
5. Can all repeating decimals be represented as fractions?

Tip: Understanding how to convert recurring decimals to fractions is helpful when you need to represent infinite decimals in a finite form.