xbox to monitor question

I'm looking to hook my xbox 360 up to a monitor and I don't know too much about this stuff. My monitor has vga and dvi inputs only. What I would like to do is hook my xbox to my monitor using the xbox's hdmi output, and do it as cheaply as possible.

I've read that dvi is the same as hdmi, but without the audio, so I'm guessing I would prefer to use this type of input on my monitor. I've also read that the vga input would potentially not work since hdmi is digital and vga is analog. Also that vga would potentially have lower quality. Feel free to correct that if it's wrong.

I've looked around and found a cable that is an hdmi male to dvi male, but this wouldn't give me sound. I've also found an hdmi male to vga male + rca cables, which would give me the sound, but again, I would be dealing with vga. What I would like to know, is if I could buy the hdmi to vga + rca and just put a dvi adapter on the vga end. Is there going to be any kind of loss in quality from going from hdmi to vga to dvi? I know that it's not actually changing the signal in this process at all but I have no idea how the wiring works and if it's essentially the same as hdmi to dvi or not.

If it works, I could get the two things I need from ebay for about 7 dollars total. Thanks in advance for any insight anyone can offer.