Systematic Treatise on Arithmetic, Volume 14

The Answer:
Jia (A) should pay in paper currency 12 taels and 2 mace.
Yi (B) should pay 11 taels and 5 mace.
Bing (C) should pay 10 taels and 8 mace.
Ding (D) should pay 10 taels and 1 mace.
Wu (E) should pay 9 taels and 4 mace.
Ji (F) should pay 8 taels and 7 mace.
Geng (G) should pay 8 taels.
The values show a constant decrease of 0.7 taels (7 mace) between each person in the sequence from Jia to Geng.

The Method: Set out the three people Wu, Ji, and Geng. Add one to make four. Multiply this by the number of people three to get twelve. Halve this to get six. Subtract three to get a remainder of three, which is the "Lower Difference Rate" 差率: a coefficient used in proportional distribution calculations. Separately, take the two people Jia and Yi. Multiply by the total of seven people to get fourteen. Subtract the Lower Difference Rate of three; the remainder is eleven, which is the "Upper Difference Rate."

The Layout 列置: the physical arrangement of numbers on a counting board
Wu, Ji, Geng 3 cross-multiply Remainder 3 Results in 6; 26 taels 1 mace; Results in 52 taels 2 mace.
Jia, Yi 2 cross-multiply Remainder 12 Results in 24; 23 taels 7 mace; Results in 71 taels 1 mace.

The right-middle [result] six, and the remainder nine acts as the Divisor 法: the number used to divide the Dividend. 〇 Furthermore, take the upper-right three and multiply by the lower-left twenty-three to obtain seventy-one While 3 times 23 is 69, the text specifies 71, likely incorporating a fractional remainder from the previous step. Subtract the lower-right fifty-two, leaving a remainder of eighteen, which is the Dividend 實: the total quantity to be divided. Divide this by the Divisor nine to get two mace, which is the "Common Difference" the fixed amount of change between each rank. 〇 Separately, set out the combined paper currency of Jia and Yi, twelve [units]. Add the difference of two mace, making the total one tael and four mace. Halve this to get seven mace, which is the amount Jia receives. Subtract the difference of two mace to find that the remainder, five mace, is Yi’s currency. Subtract two mace successively from each to find the remaining values.

Song of Rice in a Bamboo Tube

At home there is a bamboo stalk with nine sections,
Filled with rice, though the amounts are not even.
The bottom three sections hold three sheng and nine, 升: a sheng is a unit of dry volume, roughly equivalent to one liter
While the top four sections store only three sheng.
Only the two sections in the middle remain;
You must fill them with rice in a steady sequence. original: "次第" (cì dì), referring to an arithmetic progression
If you, sir, are skilled in the methods of math,
I shall have you calculating until the break of dawn.