To express the decimal \( 3.63636 \) as a fraction, we start by analyzing the repeating part of the decimal. The decimal can be written as \( 3.63636 \ldots \), where the "36" repeats indefinitely.
Let's define the repeating decimal:
\[ x = 3.63636 \ldots \]
Next, we can separate the whole number part from the fractional part:
\[ x = 3 + 0.63636 \ldots \]
Now, let's focus on the fractional part \( 0.63636 \ldots \). We can set this part as \( y \):
\[ y = 0.63636 \ldots \]
To eliminate the repeating decimal, multiply \( y \) by \( 100 \) (since the repeating part has two digits):
\[ 100y = 63.63636 \ldots \]
Now we have two equations:
- \( y = 0.63636 \ldots \)
- \( 100y = 63.63636 \ldots \)
Next, we subtract the first equation from the second:
\[ 100y - y = 63.63636 \ldots - 0.63636 \ldots \]
This simplifies to:
\[ 99y = 63 \]
Now, we can solve for \( y \):
\[ y = \frac{63}{99} \]
We can simplify this fraction by finding the greatest common divisor (GCD) of 63 and 99, which is 9:
\[ y = \frac{63 \div 9}{99 \div 9} = \frac{7}{11} \]
Now we can substitute \( y \) back into the equation for \( x \):
\[ x = 3 + y = 3 + \frac{7}{11} \]
To combine into a single fraction, convert \( 3 \) to a fraction with a denominator of \( 11 \):
\[ 3 = \frac{33}{11} \]
Thus,
\[ x = \frac{33}{11} + \frac{7}{11} = \frac{33 + 7}{11} = \frac{40}{11} \]
Therefore, the real fraction representation of \( 3.63636 \ldots \) is:
\[ \boxed{\frac{40}{11}} \]