Resolution is a relationship, not a property
The single most useful idea here is that an image does not have a DPI on its own. It has a pixel count. DPI only exists once you decide how big to print it, because it is pixels divided by inches. A 1600 pixel wide photograph is 400 DPI at four inches wide, 200 DPI at eight inches and 100 DPI at sixteen. Same file, three answers, and only the printed size changed.
This is why the number stored in a file's metadata is close to meaningless and why changing it in an editor achieves nothing. Setting a file to 300 DPI without resampling relabels it and moves no pixels at all. Resampling upward is worse, because it invents pixels by averaging the ones already there, producing a bigger soft image rather than a sharper one. There is no software that recovers detail a camera never captured.
Why the calculator reports the lower of two figures
An image placed into a space is scaled to fit that space, and it scales by the same factor in both directions. So whichever axis runs out of pixels first determines the quality of the entire picture. A photograph might be comfortably above 300 DPI across its width and well below it down its height, and if you looked only at the width you would send it to press believing it was fine.
The working table shows both figures and marks which one binds. If they are far apart, the usual explanation is that the image is a different shape from the space it is going into, and it is being cropped or stretched. That is worth noticing on its own, because a crop decided by software is rarely the crop you would have chosen.
What newsprint actually needs
Coated magazine stock holds fine detail, which is why 300 DPI became the standard answer. Newsprint does not. It is absorbent and uncoated, ink spreads slightly into the fibres, and detail beyond a certain point is simply not resolvable no matter how many pixels you send. That is why Newspaper Club accepts a range of 150 to 300 rather than insisting on a single figure, while MakeMyNewspaper asks for 300.
The practical reading is that 300 is comfortable, the printer's stated minimum is the floor, and below that floor you are gambling. The calculator marks the three bands rather than giving a bare pass or fail, because the honest answer for a 200 DPI image on a Newspaper Club page is not that it fails, it is that it is acceptable and slightly soft.
The number worth writing down
The most useful output on this page is the maximum size line. Rather than telling you a photograph is too small, it tells you the largest you can place it while holding the target resolution. That converts a dead end into a layout decision: an image that cannot be a half page spread might be perfectly good at a quarter, and knowing the exact boundary means you can design around it instead of hunting for a replacement.
One last habit worth forming. Photographs from phone galleries, messaging apps and social media downloads are usually recompressed and much smaller than the originals. Before concluding that an image is unusable, find the file that came off the camera. The difference is often the whole problem.