I really mostly think algebraicly, for better or for worse :)

For every 1 dog there are 2 cats results in the ratio 1:2 for dogs to cats: D/C=D:C=1:2=1/2

The equivalent equation is 2D=1C, so now the 2 is with the dogs and the 1 is with the cats. This is what confused you. But 2 times the smaller quantity equals 1 times the larger quantity, which makes sense!

Check:

2D=1C

2D/C=1

D/C=1/2

In general: If X/Y=a/b, then bX=aY

(Not aX=By)