Solution: If the number of toothbrush handles produced by manufacturer B is X, the number of toothbrush heads is 20X.

(8-4)X+(2- 1)×20X = 3200×(5-2)×2

4x+20x= 19200

24x= 19200

x=800

20x=20×800= 16000

Answer: Manufacturer B produced 800 toothbrush handles and 16000 toothbrush heads during this period.