In this work we address the challenging case of answering count queries in web search, such as number of songs by John Lennon. Prior methods merely answer these with a single, and sometimes puzzling number or return a ranked list of text snippets with different numbers. This paper proposes a methodology for answering count queries with inference, contextualization and explanatory evidence. Unlike previous systems, our method infers final answers from multiple observations, supports semantic qualifiers for the counts, and provides evidence by enumerating representative instances. Experiments with a wide variety of queries, including existing benchmark show the benefits of our method, and the influence of specific parameter settings. Our code, data and an interactive system demonstration are publicly available at https://github.com/ghoshs/CoQEx and https://nlcounqer.mpi-inf.mpg.de/.