In electronics industry, aircraft manufacturing, distributed computer systems and supply chains, it is common that many jobs are divisible and can be considered as a batch of potentially infinitely small and independent items. Subcontracting divisible jobs means that a job can be partially processed by an in-house machine and the remaining of it can be processed by a subcontractor's machine. Considering three subcontracting pricing strategies: non-increasing, non-decreasing and constant over time, this paper studies a scheduling problem with divisible jobs and subcontracting option in which both the manufacturer and the subcontractor are in single-machine environment. The objective is to minimize the sum of total weighted tardiness and total subcontracting costs. A mixed integer programming (MIP) model is formulated. Then a Lagrangian-based Benders Dual Decomposition (denoted by LB-BDD) method is developed based on the MIP formulation. Extensive computational experiments are conducted on five groups of randomly generated problem instances, and the results show that the proposed LB-BDD method outperforms the basic Benders Decomposition (denoted by BD) method and solving the MIP model directly in Gurobi solver.