This is the classic "fix to fix" we all hated in UPT. It can be done in the aircraft without an E6B. I'm going to change the fix just a bit to make things easier (fly from 180/80 to the 090/20)

The easiest way I've been able to describe this process is to look at your RMI or HSI as a smaller "god's eye" view of the actual situation. The middle of the HSI/RMI represents the VOR station.

The tail of the needle at the outer ring of the HSI/RMI card represents the more distant (from the station) fix, in this case the 180/80. In this example it also represents your current location in the "world" of the HSI. You now must "eyeball" a proportionate location for the other fix on the 090 radial on the HSI. Since the outside ring of the HSI or RMI (where the tail of the needle is) represents 80 DME, then the 090/20 (1/4 of 80 DME) would be 1/4 of the distance from the middle of the HSI on the 090 radial.

You can now visualize both fixes on the face of the HSI. Where you are and where you want to go. You turn to make the imaginary course between those two points appear vertically on your HSI. This would be your no wind heading from your current position to the new fix.

I don't know if I helped or just made the situation worse. All I can say is thank goodness for GPS.