[Update: Although I discovered this trick independently, I’ve since learned that Thierry Koblentz beat me to it. You can find his 2009 article on A List Apart. Credit where credit is due.]

I know this is an old question, but I encountered a similar problem that I did solve only with CSS. Here is my blog post that discusses the solution. Included in the post is a live example. Code is reposted below.

#container {
  display: inline-block;
  position: relative;
  width: 50%;
}

#dummy {
  margin-top: 75%;
  /* 4:3 aspect ratio */
}

#element {
  position: absolute;
  top: 0;
  bottom: 0;
  left: 0;
  right: 0;
  background-color: silver/* show me! */
}

<div id="container">
  <div id="dummy"></div>
  <div id="element">
    some text
  </div>
</div>

There is a way using CSS!

If you set your width depending on the parent container you can set the height to 0 and set padding-bottom to the percentage which will be calculated depending on the current width:

.some_element {
    position: relative;
    width: 20%;
    height: 0;
    padding-bottom: 20%;
}

This works well in all major browsers.

JSFiddle: https://jsfiddle.net/ayb9nzj3/

It is possible without any Javascript :)

The HTML:

<div class='box'>
    <div class='content'>Aspect ratio of 1:1</div>
</div> 

The CSS:

.box {
    position: relative;
    width:    50%; /* desired width */
}

.box:before {
    content:     "";
    display:     block;
    padding-top: 100%; /* initial ratio of 1:1*/
}

.content {
    position: absolute;
    top:      0;
    left:     0;
    bottom:   0;
    right:    0;
}

/* Other ratios - just apply the desired class to the "box" element */
.ratio2_1:before{
    padding-top: 50%;
}
.ratio1_2:before{
    padding-top: 200%;
}
.ratio4_3:before{
    padding-top: 75%;
}
.ratio16_9:before{
    padding-top: 56.25%;
}