All organisms have to consider consequences that vary through time. Theories explaining how animals handle intertemporal choice include delay-discounting models, in which the value of future rewards is discounted by the delay until receipt, and foraging models, which predict that decision-makers maximize rate of reward. We measured the behavior of rats on a 2-option delay-discounting task and a stay/go foraging task that were equivalent for rate of reward and physical demand. Despite the highly shared features of the tasks, rats were willing to wait much longer on the foraging task than on the delay-discounting task. Moreover, choice performance by rats was less optimal in terms of total reward received on the foraging task compared to the delay-discounting task. We applied a suite of intertemporal choice models to the data but found that we needed a novel model incorporating interactions of decision-making systems to successfully explain behavior. Our findings (a) highlight the importance of factors that historically have been seen as irrelevant and (b) indicate the inadequacy of current general theories of intertemporal choice.