When it shall be necessary to build, construct, or repair any road in any township, which would be an unreasonable burden to the same, the cost of which will be more than can be raised in one year by the levy provided in section 23-259, the township board shall present a petition to the county board of the county in which such township is situated, praying for an appropriation from the county treasury to aid in the building, constructing, or repairing of such road, and such county board may, by a majority vote of all the elected members thereof, make an appropriation of so much for that purpose as in its judgment the nature of the case requires and the funds of the county will justify; such appropriation to be expended under the supervision of an authorized agent or agents of the county, if the county board shall so order. In such case, where the county grants aid as provided in this section, the contract shall be let by the township board, under the provisions of law relating to letting contracts by county boards.