In a signal detection theory (SDT) approach to multiple choice exams, examinees are viewed as choosing, for each item, the alternative that is perceived as being the most plausible, with perceived plausibility depending in part on whether or not an item is known. The SDT model is a process model and provides measures of item difficulty, item discrimination, and the relative plausibility of each alternative. It is shown how to incorporate information from response times into the model, which has potential benefits for estimation and also offers a way to study underlying processes. The SDT model is joined with a lognormal response time (RT) model in a manner similar to that used in hierarchical models. In addition, a mixture extension of the RT model is joined in a novel way with the SDT choice model, using the idea that the probability of "knowing" in the SDT model might be related to the probability of working in one of two speed states in the mixture RT model. A semiparametric version of the mixture RT model is also used to assess robustness. The fused SDT/RT models are examined with science items from the 2015 Program for International Student Assessment.