To convert the repeating decimal \( 0.78\overline{78} \) into a fraction, we can follow these steps:
Let \( x = 0.787878...\)
Now, to eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part consists of two digits):
\[ 100x = 78.787878... \]
Next, we can set up the equation:
\[ 100x = 78.78\overline{78} \] \[ x = 0.78\overline{78} \]
Now, we subtract the first equation from the second:
\[ 100x - x = 78.787878... - 0.787878... \]
This simplifies to:
\[ 99x = 78 \]
Now, solve for \( x \):
\[ x = \frac{78}{99} \]
Next, we need to simplify the fraction \( \frac{78}{99} \). To do this, we can divide both the numerator and the denominator by their greatest common divisor (GCD).
The GCD of 78 and 99 is 3. Dividing both by 3 gives us:
\[ x = \frac{78 \div 3}{99 \div 3} = \frac{26}{33} \]
Thus, the repeating decimal \( 0.78\overline{78} \) as a simplified fraction is:
\[ \frac{26}{33} \]