All a business needs to do to be profitable is to make at least 1 cent more in revenues than they had in expenses for whatever time period you re examining. Any less than that would be unprofitable. By the way, your example sentence ( a business need to earn between 15% and 35% of profits in order to be profitable. ) makes absolutely no sense at all.
the percentage profit depends on the nature of your business as percentage profit (profit margin) from sales is also known as contributioncontribution - fixed costs and overheads = profit (pre tax if applicable)as long as contribution (selling price minus cost price) take away fixed costs and overheads is above 0 then your company is seen as profitable