The ratio of dogs to cats is $5: 3$ The ratio of fish to dogs is $6: 1$ Find the ratio of cats to fish. Give your answer in its simplest form.

Step 1 ：Let the number of dogs be 5x and the number of cats be 3x.

Step 2 ：Let the number of fish be 6y and the number of dogs be y.

Step 3 ：Since the number of dogs is the same in both ratios, we can equate them: 5x = y.

Step 4 ：Now we can find the ratio of cats to fish: \(\frac{3x}{6y}\).

Step 5 ：Substitute the value of y from step 3: \(\frac{3x}{6(5x)}\).

Step 6 ：Simplify the expression: \(\frac{3x}{30x}\).

Step 7 ：Cancel out the x terms: \(\frac{3}{30}\).

Step 8 ：Simplify the fraction: \(\frac{1}{10}\).

Step 9 ：\(\boxed{\frac{1}{10}}\) is the ratio of cats to fish.