Simply put, and this sounds silly, but the answer is because that was the only number they could agree upon. Remember, the slaves were being counted for both representation in congress and for tax purposes since the national government based its tax revenue on the population of each state. My guess is that 3/5 is less than 2/3, but more than 1/2, but that is purely a guess.

It was a compromise between the northern states, who didn't think slaves should be counted towards congressional apportionment at all, and the southern states, who did because huge fractions of their populations were made up of slaves. So they settled on 3/5.

It was called (drumroll) The Three-fifths Compromise.