The recurring decimal $0.36\overline{6}$ can be converted into a fraction as follows:

Let $x = 0.36\overline{6}$.

Step 1: Multiply to move the decimal point

To eliminate the recurring part, consider $100x$: $100x = 36.66\overline{6}$

Step 2: Subtract to get rid of the repeating part

$100x - x = 36.66\overline{6} - 0.36\overline{6}$ $99x = 36.3$

Step 3: Solve for $x$

$x = \frac{36.3}{99}$

Since $36.3$ can be written as $\frac{363}{10}$, we get: $x = \frac{363}{990}$

Step 4: Simplify the fraction

Divide the numerator and the denominator by their greatest common divisor, which is 9: $x = \frac{363 \div 9}{990 \div 9} = \frac{121}{330}$

This fraction can be further simplified by dividing by 11: $x = \frac{11}{30}$

So, the length of the wire as a fraction is: $\boxed{\frac{11}{30}}$

Would you like further details or explanations?

Here are some related questions:

1. How do you convert a recurring decimal like $0.333\overline{3}$ into a fraction?
2. What is the general process for converting a decimal with more than one repeating digit into a fraction?
3. How can we verify that $\frac{11}{30}$ is equivalent to $0.36\overline{6}$?
4. Why do recurring decimals convert to rational numbers (fractions)?
5. How do you simplify fractions to their lowest terms?

Tip: When converting a recurring decimal to a fraction, remember to write down the repeating part as $x$ and then manipulate it by multiplying by powers of 10.