Systematic Treatise on Arithmetic, Volume 14

Six

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.

First, take the 2 in the upper left and cross-multiply the 3 in the middle right to get 6. Then take the 2 in the upper left and multiply the 12 in the lower right referring to the numerical grid described above to get 24. Next, take the 3 in the upper right and multiply by the 12 in the middle left to get 36. Subtract 24 from this; the remainder is 12, which is the "Divisor" (Fa). Next, take the 3 in the upper right and multiply by the 23 taels and 7 mace in the lower left to get 71 taels and 1 mace. Then take the 2 in the upper left and multiply by the 26 taels and 1 mace in the lower right to get 52 taels and 2 mace. Subtract this...

original terminology: 差分 (Proportional Distribution), 差率 (Difference Rate), 列置 (Calculational Layout), 互乘 (Cross-multiplication), 算法統宗 (Systematic Treatise on Arithmetic), 籌算 (Rod Calculus)