, 2025

There is $1$ dog for every
bowl of food, and so if there were $6$ dogs, then there would be $6$ bowls of food.

There are $2$ dogs for every bowl of
water, and so these same $6$ dogs
would need $\frac{6}{2}=3$ bowls of
water.

There are $3$ dogs for every bowl of
treats, and so the $6$ dogs would
need $\frac{6}{3}=2$ bowls of
treats.

Thus, $6$ dogs require $6$ bowls of food, $3$ bowls of water, and $2$ bowls of treats, or $6+3+2=11$ bowls in total.

There are $11$ bowls for every $6$ dogs, and since there are $77=11\times7$ bowls, then there are $7$ groups of $6$ dogs, or $6\times7=42$ dogs.