ImageMagick change aspect ratio without scaling the image

Just create a wrapper <div> with a percentage value for padding-bottom, like this:

.demoWrapper {
  padding: 10px;
  background: white;
  box-sizing: border-box;
  resize: horizontal;
  border: 1px dashed;
  overflow: auto;
  max-width: 100%;
  height: calc(100vh - 16px);
}

div {
  width: 100%;
  padding-bottom: 75%;
  background: gold; /** <-- For the demo **/
}

<div class="demoWrapper">
  <div></div>
</div>

It will result in a <div> with height equal to 75% of the width of its container (a 4:3 aspect ratio).

This relies on the fact that for padding :

The percentage is calculated with respect to the width of the generated box's containing block [...] (source: w3.org, emphasis mine)

Padding-bottom values for other aspect ratios and 100% width :

aspect ratio  | padding-bottom value
--------------|----------------------
    16:9      |       56.25%
    4:3       |       75%
    3:2       |       66.66%
    8:5       |       62.5%

Placing content in the div :

In order to keep the aspect ratio of the div and prevent its content from stretching it, you need to add an absolutely positioned child and stretch it to the edges of the wrapper with:

div.stretchy-wrapper {
  position: relative;
}

div.stretchy-wrapper > div {
  position: absolute;
  top: 0; bottom: 0; left: 0; right: 0;
}

Here's a demo and another more in depth demo

If I'm interpreting your spec correctly, you want a result that is no larger than the one the end-user laid out originally; you want one of the two dimensions to shrink, and the other to stay the same. In other words, the new size should fill the designer space in one direction while shortening the size in the other direction to retain the original aspect ratio.

original_ratio = original_width / original_height
designer_ratio = designer_width / designer_height
if original_ratio > designer_ratio
    designer_height = designer_width / original_ratio
else
    designer_width = designer_height * original_ratio

Often you'll be working with integer coordinates, but the divisions to produce the ratios above need to be floating point. Here's a rearrangement of the formula to be more integer friendly. Make sure your integers have the range to handle the maximum width*height.

if original_width * designer_height > designer_width * original_height
    designer_height = (designer_width * original_height) / original_width
else
    designer_width = (designer_height * original_width) / original_height