Skylar multiplied the price of the video game by 1.0775 to include the tax in her total. This can be interpreted as:
\[ \text{Total with tax} = \text{Price of video game} \times (1 + \text{Tax rate}) \]
From this equation, we can set the following:
\[ 1 + \text{Tax rate} = 1.0775 \]
To find the tax rate, we subtract 1 from both sides:
\[ \text{Tax rate} = 1.0775 - 1 = 0.0775 \]
To express the tax rate as a percentage, we multiply by 100:
\[ \text{Tax rate percentage} = 0.0775 \times 100 = 7.75% \]
Thus, the percent tax that Skylar paid is:
\[ \boxed{7.75%} \]