Share of dogs in an endless line of pets

Along an endless pavement of cats and dogs, $6$ out of every $7$ cats are immediately followed by a dog, and $1$ out of every $4$ dogs is immediately followed by a cat. What fraction of the animals are dogs?

$24/31$. Treat the next animal as a two-state Markov chain. In steady state the flows balance: $c\cdot\tfrac67=d\cdot\tfrac14$, so $d/c=\tfrac{24}{7}$ and $d=\tfrac{24}{31}$.