We report a study on students' approaches to quantitative synthesis problems with varying mathematical complexities. Synthesis problems involve multiple concepts typically taught in different chapters. In this study, mathematical complexity is determined by the number and the type of equations that must be simultaneously solved. Students from a second year calculus-based physics course were randomly assigned to solve one of three synthesis problems varying in mathematical complexity: simple, medium, or complex. Results from extended written and interview responses revealed four major problem-solving approaches used by the students: trial-and-error, flawed reasoning, knowledgeable, and expert-like approach. Students solving the simple problem used all the four approaches, whereas those solving the other two mainly used the "trial-and-error" or "flawed reasoning" approaches. A common phenomenon is that many students could identify the appropriate concepts but failed to correctly apply them. Additionally, the students made similar mistakes on all the three problems.