Combining the Image Size Dialog with Further Pixel Math

If you start with a picture that is 3" square at 300 pixels per inch and change the size to 6" square WITHOUT checking the resample button in the Image > Image Size dialog box, what is the resolution of the resized image?

NOTE: The illustration below shows concept, but not the true number of pixels in this problem.

Solution

Remember the when Resample Images is unchecked, the number of pixels in the picture won't change, but you will change the dimensions (inches) of it. The first step is to calculate the number of pixels. Multiply the picture size (inches) by resolution to find that number. Because the picture is square, height = width, and the answer is 3 x 300 = 900 pixels in each dimension (H and W).

However the question asks how much resolution is left after the resize, not how many pixels there are (which we know already). Resolution will change because  the dimensions of the picture increased while keeping the number of pixels the same.

Resolution is defined as pixels/inches, so the answer is 900 pixels/6 inches = 150 pixels per inch after the resize operation.

Another way to approach this specific kind of problem (keeping the same pixels, but stretching or shrinking the picture dimensions) is that if you double the picture size, the resolution is halved.