Your way is much simpler than mine, Ray - it uses the awkward 54T as an idler, so it takes no part in determining the overall ratio.
But you have transposed things: Dave's leadscrew is 6tpi, not 8 tpi. It is his desired thread which is 8tpi.
So using your method:
If 34T was on the leadscrew, as well as the spindle, and they were connected by a simple train of idlers, Dave would cut a thread equal to the leadscrew: 6 tpi.
So he has a constant: desired tpi x 34/6 (or 17/3) = leadscrew gear teeth.
8tpi required = 8x17/3 = 45.3333 teeth.
Can't get a gear with 45.3333 teeth, so one three times the size, being 136T, on the leadscrew and a 1:3 compound (say 60 and 20) in the train would do it.
To check, apply those figures to the formula I quoted, which was
Product of all the Drivens = leadscrew tpi
Product of all the Drivens desired tpi
Then
34x60 = 6
136x20 8
Both sides of that equation work out to 0.75, so that’s OK.
The problem is that a 136T gear for the leadscrew will be hard to find, and even if one were bought or made, there might not be enough distance between Dave’s spindle and leadscrew for a train which included 54T as an idler and 136T on the leadscrew. If there is room, the position of Dave’s 54T stud may be fixed, so he can’t adjust it get the 54T into mesh with the 136T, rendering another idler necessary to bridge the gap.
Andy