Unit rates are a great concept to learn.
The ratio of distance to time is correct. You just get it by writing distance over time.
Then, when multiplying to create the unit rate you make the denominator 1. In this case you would multiply by 3 to transform the 1/3 to 1. I think you meant denominator but you did it to the numerator.
Then the unit rate is just 2/5 * 3 = 6/5 like you said.
Equivalently, to find the unit rate you can just do the division 2/5 over 1/3 rather than making the denominator 1. Or in other words all you need to do is simplify. The problem is being a little laborious here cause the unit rate is already calculated in step 1.