Song of Dividing Gold Among the Five Ranks

The "Five Ranks" refers to the ancient Chinese system of peerage: Gong, Hou, Bo, Zi, and Nan. In mathematical problems of this era, these ranks are frequently used to represent the ratios 5:4:3:2:1.

Duke	Marquis	Earl	Viscount	Baron
Five	Four	Three	Two	One

Suppose there are five cheng A unit of weight; in this problem, 1 cheng is equal to 15 jin. of gold that must be divided according to these rates.

Answer: The Duke receives 1 cheng and 10 jin; the Marquis receives 1 cheng and 5 jin; the Earl receives 1 cheng; the Viscount receives 10 jin; and the Baron receives 5 jin.

The Method says: Set the 5 cheng of gold and multiply by 15 for each cheng to obtain 75, which serves as the dividend original: 實 (shí); the total quantity to be divided. ○ Next, list the ratios for the Duke five, the Marquis four, the Earl three, the Viscount two, and the Baron one. Adding these together yields a sum of 15, which serves as the divisor original: 法 (fǎ).

Divide the dividend [75] by the divisor [15] to obtain 5; this is the amount the Baron receives. Add 5 to this to get 10, which is the amount the Viscount receives. Add 5 again to get 15 1 cheng, which is the amount the Earl receives. Add another 5 to get 20 1 cheng and 5 jin, which is the amount the Marquis receives. Add 5 once more to get 25 1 cheng and 10 jin, which is the amount the Duke receives. These calculations match the requirements of the question.