Computing target DPI value from specific dimensions

Question

A rather basic maths problem.

I got an image with a specific width and height in pixels:

WIDTH = 3648 px
HEIGHT = 2736 px

In order to compute the target print size in millimeters, given a specific DPI amount (200) i came up with this:

PRINT-WIDTH = IMAGE-WIDTH-PX / 200 * 2.54 * 10;
PRINT-HEIGHT = IMAGE-HEIGHT-PX / 200 * 2.54 * 10;

This works well. In our example it computes

463 x 347 mm 

as target print size. Perfect.

However, i now must be able to make changes to the widths and heights in millimeters, and based on the fact that we assume 200 DPI for printing, i must compute the new DPI value.

So for instance, when changing 463 x 347 to 400 x 300 i should somehow be able to calculate how that affects the DPI.

The only possible approach that came to my mind was to compute the difference between the old and the new format as a percentage, and then apply that percentage to the DPI. But the results are incorrect.

How can i compute the DPI value from the new width and height, given the original 200 DPI matching the original format?

Answer 1

NewDPI = 200 * 463 / 400

Or without using DPI 200 at all:

NeededDPI = IMAGE-WIDTH-PX(3648) * 25.4 / PRINT-WIDTH(400)