Barbs are generally speaking far more active swimmers. This means they need more linear space that would for example comparable-sized rasbora that are more sedate (species vary, I am being very general to get points across). Please, always use the scientific name so others know exactly which species; common names can vary from country to country, even among aquarists in the same place, and this makes looking up data more difficult.
The purple headed barb I assume it what is often called black ruby barb or ruby barb (see?) which is
Pethia nigrofasciata. This species needs an aquarium at minimum 3 feet/90cm length. I had a group of 11 or 12 in such a tank (a 33 gallon), but they were better by far in my 4-foot 90g tank.
The gold barb I assume is
Barbodes semifasciolatus. This one too needs a 3-foot minimum tank.
The 8 banded barb is presumably
Eirmotus octoxona. At a size of 3-4 cm this is doable in your size tank. This is in fact very well suited here. It is eaily out-competed by bold fish but the sort of fish being considered like the pygmy cories, rasbora species, peaceful small characins are ideal tankmates. This fish comes from quiet waters, so that is also suitable here. So far as I know, it is not a super-active fish, madly racing everywhere (which can easily scare quiet fish) like many barbs do, so again well suited here. You must have 10-12, it will not be settled with too few, and it will display its lovely colouration and behaviours in a good group.
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