( 1)

Suppose the first purchase price of this dried fruit is X yuan per kilogram.

(3000/x)2+300 = 9000/[( 1+20%)x]

Finishing, 1500/x=300.

x=5

The first purchase price of this dried fruit is 5 yuan per kilogram.

(2)

Unit price of the second purchase = 5 (1+20%) = 6 (yuan)

(3000/5) (9-5)+(9000/6 -600) (9-6)+600 (9 0.8-6)

=2400+2700+720

=5820 (yuan)

The supermarket sells this kind of dried fruit with a profit of 5820 yuan.