In the event of the filing with the county clerk of a petition signed by ten percent of the qualified electors in the county, the board of county supervisors shall adopt the provisions of the county road unit system by resolution at the next regular meeting of the board. The resolution shall be published once a week for three consecutive weeks in a legal newspaper published in the county or, if none is published in the county, in a legal newspaper of general circulation in the county. The adoption of the county unit plan shall take effect ninety days after the date of the first publication of the resolution providing for such adoption or any later date designated by the resolution, but not later than one year after the date of the first publication of the resolution unless before the effective date of adoption there is filed with the county clerk a petition signed by ten percent of the qualified electors in the county protesting such adoption. In that event, the board of county supervisors shall submit the question of a county road unit system to the electors of the respective counties.