Milo, in your first message to this forum you called yourself an amateur math-historian. Amateurs are known for consulting only secondary literature, whereas professionals go back to the primary sources. If you wish to reach the status of a professional you have to read the Rhind Mathematical Papyrus as written, and if you don't like Peet you have to study the facsimile published by the British Museum. Consulting the 2/n table is not enough, you have to consider the actual working out as performed by Ahmes:
Divide 2 by an odd number with the help of an auxiliary number. How do you proceed? Divide the odd number by the auxiliary number and subtract the resulting unit fraction series from 2. An example:
2 divided by 83 equals '60 plus ???
83 divided by 60 equals 1 '3 '20
2 minus 1 '3 '20 equals '4 '5 '6
result '60 plus '4x83 '5x83 '6x83
2 divided by 83 equals '60 '332 '415 '498
The method used by Ahmes has an algebraic equivalent in
2/q = 1/p + (2 - q/p)/q
while your method, Milo, is based on the identity
2/q = 1/p + (2p - q)/pq
The Egyptians used algorithms, no algebra, so you have to develop your method into a step by step procedure, and then you have to bridge the gap between your method and the one used by Ahmes (should not be so difficult).
In RMP 66 Ahmes divides a yearly portion of 10 hekat or 3200 ro of fat by 365 days and obtains 8 "3 '10 '2190 ro as daily portion. Interestingly, in the contracts of Hapzefa the daily portion is obtained from the yearly by dividing not by 365 but by 360 days, testifying to an ideal early year of 360 days as reconstructed in the hypothetical very early myth going back to the era of the Göbekli Tepe (Sanli-Urfa region, southeast Anatolia, stone pillar temples 11 600 - 9 500 BP, earth mound on top of the limestone hill even earlier).
Who says the Old Kingdom mathematicians knew only the fractions '2 and '4 and '8 and '16 and '32 and '64 ? They built pyramids but had no concept of one third and one fifth and one sixth and one seventh and one nineth and one tenth et cetera ??? I don't buy that.