Six specific modes of reasoning about ratio and proportion have been delineated as a means of operationalizing expert practice. These modes stem from consideration of how physicists reason in context, are informed by prior work in physics and mathematics education, and have grain size matched to the steps in reasoning needed to solve problems commonly used in physics instruction. A suite of assessment questions has been developed and validated to probe student facility with the reasoning modes. Responses to open-ended and multiple-choice versions of the assessment questions have been collected from more than 3000 students at Western Washington University, Rutgers University, and New Mexico State University. Results have been used to identify specific reasoning difficulties, to document differences in performance between student populations, and to explore the effect of question context on student reasoning. We find that students enrolled in university physics courses have difficulty interpreting and applying ratios in context, and in many cases lack facility with the reasoning underlying basic arithmetic operations of division and multiplication.