**Question 4**    –    4 marks

*You should be able to answer this question after studying Unit 10.*

Find a fraction equivalent to the recurring decimal

0*.*513 251 325 132 *. . . .*                                                                                            [*4*]
 

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?

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**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.

Discover how to find a fraction equivalent to the recurring decimal 0.513251325132 through a systematic step-by-step solution. This problem involves converting a recurring decimal to a simplified fraction using the concept of the greatest common divisor (GCD). Suitable for high school students studying advanced fractions and decimals.

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