A cat takes 5 leaps for every 4 leaps of a dog, but 3 leaps of the dog are equal to 4 leaps of the cat. What is the ratio of the speed of the cat to that of the dog?

Step 1: Let's denote the distance covered in one leap by the cat as C and by the dog as D.

Step 2: From the first part of the problem, we know that the cat takes 5 leaps for every 4 leaps of the dog. This can be written as 5C = 4D.

Step 3: The second part of the problem tells us that 3 leaps of the dog are equal to 4 leaps of the cat. This can be written as 3D = 4C.

Step 4: Now we have a system of two equations, and we can solve it. From the first equation (5C = 4D), we can express D as D = 5C/4.

Step 5: Substitute D = 5C/4 into the second equation (3D = 4C) to get 3*(5C/4) = 4C. Simplify this to get 15C/4 = 4C.

Step 6: Solve for C to get C = 16/15.

Step 7: Substitute C = 16/15 into the first equation (5C = 4D) to get 5*(16/15) = 4D. Simplify this to get D = 20/15.

Step 8: The ratio of the speed of the cat to that of the dog is the ratio of the distances they cover in one leap, which is C/D = (16/15) / (20/15) = 16/20 = 4/5.

So, the ratio of the speed of the cat to that of the dog is 4:5.