Let \( x = 0.7\overline{7} \).
This means that \( x = 0.7777777\ldots \)
To convert this repeating decimal into a fraction, we can use the method of multiplying by a power of 10 to shift the decimal point. Since the repeating part (7) consists of only one digit, we will multiply by 10.
So, we have:
\[ 10x = 7.7777777\ldots \]
Now we can set up an equation by subtracting the original \( x \) from this new equation:
\[ 10x - x = 7.7777777\ldots - 0.7777777\ldots \]
This simplifies to:
\[ 9x = 7 \]
Now, we can solve for \( x \) by dividing both sides by 9:
\[ x = \frac{7}{9} \]
Thus, the fraction equivalent of \( 0.7\overline{7} \) is
\[ \boxed{\frac{7}{9}}. \]