It’ll be pretty hot in august, and quite busy, but its still nice out there.
The absolute cheapest way will almost certainly be some sort of package, but you might not like where you end up. if you dont care about the accomodation as long as there is a beach/pool then this will almost certainly be the cheapest way. But if all you want is sun and sea then there are cheaper destinations than majorca, I think greece/turkey is supposed to be very cheap.
If you want a higher standard of accomodation then booking seperately will probably be cheaper than booking as a package, especially if you book the accomodation from a private owner rather than through a company.
Majorca is a very in demand place and the prices are not necesarily as low as you might think they should be.
If you book seperately, flights will be in the region of £200 each maybe more by the time you have added bags and charges on.