How do I find the ratio of 5&3/5 to 2&1/10?

Switch both mixed fractions to improper fractions to get

(28/5) : (21/10)
= (28/5) ÷ (21/10)
= 28/5 * 10/21
= 8/3 or the ratio 8:3

Since both of your original fractions yield exact decimals, an alternate way would be

..
..
= 5.6/2.1
= 56/21
= 8/3