Systematic Treatise on Arithmetic, Volume 14

The Answer:
Section 1: 1 sheng 4 ge of non-glutinous rice 秔: non-glutinous or "jing" rice.
Section 2: 1 sheng 3 ge of non-glutinous rice.
Section 3: 1 sheng 2 ge of non-glutinous rice.
Section 4: 1 sheng 1 ge of non-glutinous rice.
Section 5: 1 sheng of non-glutinous rice.
Section 6: 9 ge of non-glutinous rice.
Section 7: 8 ge of non-glutinous rice.
Section 8: 7 ge of non-glutinous rice.
Section 9: 6 ge of non-glutinous rice.
The amounts follow a clear arithmetic sequence, decreasing by exactly 1 ge (one-tenth of a sheng) per section, starting from 1.4 sheng and ending at 0.6 sheng.

The Method: Set out the number of the upper sections and add one 4 plus 1 to make 5. Multiply this by the number of sections 4 to get 20. Halve 折半: a standard term for dividing by two this to get 10. Subtract the number of sections 4 to get a remainder of 6, which is the Lower Difference Rate 差率: a coefficient used to determine the share of a total based on a progression. ○ Separately, take the lower 3 sections and multiply by the total of 9 sections to get 27. Subtract the Lower Difference Rate of 6; the remainder is 21, which is the Upper Difference Rate.

The Layout 列置: the arrangement of numbers on the counting board for calculation
Set out 4 ✕ 4
Cross Remainder 6, results in 18; 3 sheng; results in 9 parts.
Remainder 21, results in 84; 3 sheng 9 ge; results in 15 parts 6 li 厘: a decimal unit, here representing 0.06 of a part.

First, take the top-left 3 and cross-multiply 互乘: diagonal multiplication used in proportional calculations with the middle-right 6 to get 18. Next, take the top-left 4 and cross-multiply with the left...