Newly Compiled Systematic Guide to Computational Methods, Volume 14

Compiled and edited by Cheng Dawei (styled Rusi, pseudonym Binqu) of Xin’an.
Corrected by great-grandson Su-ting Guangshen, also known as Peizhang.
Reviewed by Yun-zhai Minghong, also known as Shengfu.

Difficult Problems in Proportional Distribution, Section 3

original: "難題衰分" (Nántí Shuāifēn). "Shuaifen" is a classical mathematical method for dividing items or profits according to specific ratios or "weights."

To the Tune of "The Moon over the West River"

Chinese mathematical texts often presented problems in verse to make them easier to memorize. "The Moon over the West River" (Xījiāng Yuè) is a famous poetic meter used here as a mnemonic device.

Pure, picked cotton is carded fine and mixed together in one basket.
9 catties and 12 ounces belong to Zhang Chang; Li De has 5
catties and 4 ounces. Once spun and woven, the cloth measures 108 feet long.
The distribution between the two parties must be clearly shown;
There must be no bias toward either side.

Answer: Zhang Chang receives 7 zhang and 2 inches. Li De receives 3 zhang, 7 feet, and 8 inches.

In this period, 1 zhang = 10 feet (chi), and 1 foot = 10 inches (cun). Zhang Chang’s portion is 70.2 feet; Li De’s is 37.8 feet.

Method: List the respective factors Zhang Chang: 9 catties, 12 ounces; Li De: 5 catties, 4 ounces. Convert each into ounces using the standard conversion: Zhang obtains 156 ounces; Li obtains 84 ounces. Summing these yields 240, which serves as the "divisor" (fa). Take the total cloth of 108 feet and multiply it by each individual's factor, then divide by the divisor to obtain each share.

Catty (斤, jin): A unit of weight. In the Ming Dynasty, 1 catty equaled 16 ounces.

Ounce (兩, liang): A unit of weight. 16 ounces made 1 catty.

Factor (衰, shuai): In proportional distribution, this refers to the ratio or "weight" assigned to a participant based on their initial contribution.

Method (法, fa): In a mathematical context, this word can mean "the rule," "the procedure," or specifically "the divisor" in a division operation.