TCM Arithmetic 3: Rates and Unit Conversions

By: Tao Steven Zheng (郑涛)

Exchange Rates and Unit Conversions

Figure 1. The jin you method or "rule of three"

The concept of "rate" or "proportion" (率 lü) was highly important in ancient Chinese mathematics. We will later discover that the notion of rates is vital in explaining the reasoning behind traditional Chinese problems on arithmetic, linear equations, and geometry. One important method discussed in Chapter 2, titled "Grains" (粟米 su mi), of the Jiuzhang Suanshu is the jinyou method (今有术 jin you shu). The jinyou method most likely arose from commercial transactions of antiquity, for every problem in this chapter dealt with the exchange of different grains that followed a defined market rate. The market rates are used to calculate an unknown quantity of grain. In the Jiuzhang Suanshu, at the beginning of Chapter 2 (粟米 su mi), there is a table of market rates of various grains provided for the reader, as well as a detailed description of the jinyou method.

粟米之法： 粟率五十；粝米三十 粺米二十七；糳米二十四                御米二十一；小䵂十三半    大䵂五十四；粝饭七十五                粺饭五十四；糳饭四十八                御饭四十二；菽、荅、麻、麦各四十五 稻六十；豉六十三    飧九十；熟菽一百三半                    蘖一百七十五 今有术曰：以所有数乘所求率为实，以所有率为法，实如法而一。

The regulated [rates of exchange] for grains: Unhusked millet 50; Hulled millet 30 Milled millet 27; Refined millet 24 Imperial millet 21; Refined wheat 13 ½ Coarse wheat 54; Cooked coarse wheat 75 Cooked milled millet 54; Cooked refined millet 48 Cooked imperial millet 42; Soy beans, Small beans, Sesame seed, Wheat 45 Paddy rice 60; Fermented soy beans 63 Porridge 90; Cooked beans 103 ½ Fermented grain 175 Jinyou method: Take the given amount multiplied by the sought rate as the dividend. The given rate is the divisor. Divide the dividend by the divisor.

What the jinyou method describes is that given two rates, the given rate $r_1$ and sought rate $r_2$, and the given amount $y$, one can determine the sought amount $x$ as follows:

$x = \frac{r_2 \times y}{r_1}$

The jinyou method went by many names over the centuries and across different cultures. In ancient India, the astronomer-mathematician Aryabhata (476 - 550 AD) formulated a method for calculating rates equivalent to the jinyou method. Aryabhata called it the Trairāśika, which literally translates to the "rule of three quantities". Knowledge of the jinyou method travelled along the Eurasian trade routes (often called the Silk Road), making its way to the Middle East and Europe through India. For centuries, the mathematicians and merchants of Muslim empires and Christian Europe still refer it as the "rule of three".

In actuality, the jinyou method is an abstract arithmetic principle related to rates and proportion. Knowing how the jinyou method can be used for calculating rates, one can discover that it can naturally be applied to the conversion of standardized units. In ancient China, there are three fundamental measures: linear measure (度 du), capacity measure (量 liang), and weight measure (衡 heng). The three sets of measures are collectively called 度量衡 (du liang heng, "weights and measures").

Table 1. Linear Measure (度 du)

10 毫 = 1 厘 10 厘 = 1 分 10 分 = 1 寸 10 寸 = 1 尺 10尺 = 1 丈 6 尺 = 1 步 40 尺 = 1 匹（疋） 50 尺 = 1 端 300 步 = 1 里

10 hao = 1 li 10 li = 1 fen 10 fen = 1 cun 10 cun = 1 chi 10 chi = 1 zhang 6 chi = 1 bu 40 chi = 1 pi 50 chi = 1 duan 300 bu = 1 li

Table 2. Capacity Measure (量 liang)

10 圭 = 1 撮 10 撮 = 1 抄 10 抄 = 1 勺 10 勺 = 1 合 10 合 = 1 升 10 升 = 1 斗 10 斗 = 1 斛

10 gui = 1 cuo 10 cuo = 1 chao 10 chao = 1 shao 10 shao = 1 ge 10 ge = 1 sheng 10 sheng = 1 dou 10 dou = 1 hu

Table 3. Weight Measure (衡 heng)

10 絫 = 1 铢 24 铢 = 1 两 16 两 = 1 斤 30 斤 = 1 钧 4 钧 = 1 石

10 lei = 1 zhu 24 zhu = 1 liang 16 liang = 1 jin 30 jin = 1 jun 4 jun = 1 dan*

* The character 石 is pronounced shi for stone but the unit of measurement is pronounced dan.

Problem Study 1: Exchanging Grains

The following problem from the Jiuzhang Suanshu (Chapter 2, Problem 1) is an exercise of unit conversions and the jinyou method.

[2.01] 今有粟一斗，欲为粝米。问：得几何？ 答曰：为粝米六升。 术曰：以粟求粝米，三之，五而一。

[2.01] Suppose the is 1 dou of millet, and one wishes to exchange it for hulled millet. Question: How much should there be? Answer: 6 sheng of husked millet are exchanged. Method: Using [the given amount of] millet to determine [the sought amount of] hulled millet, multiply by 3, and divide by 5.

Solution for [2.01]

Let $x$ denote the sought amount of hulled millet. From the table of capacity measures (Table 2), we find that 10 sheng (升) makes 1 dou (斗). Thus, the given amount is 10 sheng of millet. Referencing the market rates from Chapter 2 of the Jiuzhang Suanshu, we find that the exchange rate for millet is 50, and the exchange rate for hulled millet is 30. Substituting the given rates and the given amount of millet into the jinyou rule yields:

$x = \frac{30 \times 10 \; sheng}{50}$

$x = \frac{3 \times 10 \; sheng}{5}$

$x = 6 \; sheng$

Problem Study 2: Application with Unit Conversions

The following problem from the Sunzi Suanjing (Chapter 3, Problem 33) is a classic unit conversions problem.

[3.33] 今有长安、洛阳相去九百里。车轮一匝一丈八尺。欲自洛阳至长安，问：轮匝几何？ 答曰：九万匝。                        术曰：置九百里，以三百步乘之，得二十七万步。又以六尺乘之，得一百六十二万尺。以车轮一丈八尺为法。除之，即得。

[3.33] Suppose the distance between Chang’an and Luoyang is 900 li. One rotation of the cart’s wheel measures 1 zhang 8 chi. Question: If one wishes to travel from Luoyang to Chang’an, how many rotations will the wheel make? Answer: 90,000 rotations. Method: Put down 900 li, multiply by 300 bu to obtain 270,000 bu. Next multiply by 6 chi to obtain 1,620,000 chi. Let [the circumference of ] the wheel, 1 zhang 8 chi, be the divisor (法 fa). Divide to get the answer.

Solution for [3.33]

(1) Convert the distance between Luoyang (洛阳) and Chang’an (长安) from the unit li to chi.

1 li = 300 bu

$900 \; li \left(\frac{300 \; bu}{1 \; li} \right) = 270000 \; bu$

1 bu = 6 chi

$270000 \; bu \left(\frac{6 \; chi}{1 \; bu} \right) = 1620000 \; chi$

(2) Convert one rotation of the wheel into chi.

$\begin{align} 1 \; zhang \; 8 \; chi &= 10 \;chi + 8 \; chi \\ &= 18 \; chi \end{align}$

(3) Divide the distance between Luoyang and Chang’an by the circumference of one rotation.

$\frac{1620000 \; chi}{18 \;chi/rotations} = 90000 \; rotations$

References

[1] 钱宝琮主编.《中国数学史》，北京：科学出版社，1964年。

[2] 纪志刚《孙子算经、张邱建算经、夏侯阳算经导读》， 武汉：湖北教育出版社，1999年。

[3] 郭书春《九章算术译注》，上海：上海古籍出版社，2009年。

[4] 李兆华《中国数学史基础》，天津：天津教育出版社，2010年。

[5] Martzloff, Jean-Claude. A History of Chinese Mathematics (English Translation). Spinger-Verlag. 1997, 2006.

[6] Yong, Lam Lay, & Ang, Tian Se. Fleeting Footsteps: Tracing the Conception of Arithmetic and Algebra in Ancient China, Revised Edition. World Scientific Publishing Company. 2004

[7] Wilkinson, Endymion. Chinese History: A New Manual, 4^th Edition. Harvard University Press. 2015.