DPI versus pixel dimensions

A raster file contains a grid of pixels and a header. The grid is the image; the DPI figure in the header is a note to the printer saying how densely to lay those pixels down. Changing that number without resampling changes nothing about the image data and everything about how large it prints, which is why a file can be described as 72 DPI and still print beautifully at A4 if it has enough pixels.

The arithmetic is one division. Pixels divided by DPI gives inches, so a 3000 pixel wide image at 300 DPI prints ten inches wide, and the same file at 150 DPI prints twenty inches wide with visibly less detail per inch. Going the other way, a required print size multiplied by the target DPI gives the pixel count you need to supply.

On screen the number is ignored completely. A browser lays out an image by its pixel dimensions and the page's own CSS, so the DPI in the metadata has no effect at all on a web page. This is the source of the persistent belief that web images should be 72 DPI, which was true of a particular class of monitor decades ago and has meant nothing since.

Three hundred DPI is a convention rather than a rule. It matches roughly the resolving limit of the eye at normal reading distance, so it is the right target for something held in the hand. A poster read from three metres away is routinely produced at half that, and doubling the resolution beyond it adds file size and nothing a viewer can see.