1. A storage system, comprising: a virtual volume that is constituted by a plurality of virtual areas; a logical volume; a pool that has a plurality of pages which are storage areas that can be assigned to the virtual areas of the virtual volume; mapping information that indicates which storage areas correspond with which virtual areas; an I/O processing unit that receives a write request from a computer, assigns a page in the pool to a virtual area of a virtual volume that is designated by the write request, writes data fragments corresponding to the write request to the assigned page, and associates information representing the page with information representing the virtual area to which the page is assigned, in the mapping information; and a backup processing unit that copies a plurality of data fragments that are stored in each of a plurality of pages that are assigned to the virtual volume to the logical volume, updates information representing the respective copy source pages in the mapping information to information that represents respective copy destination storage areas for the data fragments stored in the respective copy source pages, and copies the updated mapping information to the logical volume.