Whenever a petition in conformity to Kan. Stat. Ann. § 24-464, and amendments thereto, signed by not less than 2/5 of the total number of taxpayers residing within both the original district and the territory named in the petition or by all of the directors of any drainage district incorporated pursuant to Kan. Stat. Ann. § 24-401 et seq., and amendments thereto, is presented to the board of county commissioners of the county designated in Kan. Stat. Ann. § 24-463, and amendments thereto, praying that the limits of the drainage district be extended to include within its boundaries the territory described in the petition it shall be the duty forthwith of such board of county commissioners to fix a time for the hearing of such petition and to cause the county clerk to give notice thereof by one publication in a newspaper of general circulation in the county at least 10 days before the day fixed for the hearing. If such petition is signed by any number in excess of 2/5 of the total number of taxpayers residing within the territory over which the extension is to be made, it shall not be necessary to furnish a petition signed by the taxpayers residing within the original drainage district.