jmc

Let $w$ be Yan's walking speed, and let $x$ and $y$ be the distances from Yan to his home and to the stadium, respectively. The time required for Yan to walk to the stadium is $y/w$, and the time required for him to walk home is $x/w$. Because he rides his bicycle at a speed of $7w$, the time required for him to ride his bicycle from his home to the stadium is $(x+y)/(7w)$. Thus $$\frac{y}{w}=\frac{x}{w}+\frac{x+y}{7w}=\frac{8x+y}{7w}.$$As a consequence, $7y=8x+y$, so $8x=6y$. The required ratio is $x/y=6/8=\boxed{\frac{3}{4}}$.