TCM Arithmetic 3: Rates and Unit Conversions

By: Tao Steven Zheng (郑涛)

Rates: Theory and Applications

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

The concept of “rate” (率 lü) was central in Chinese mathematics. It will crop up in arithmetic, equations, and even geometry.

One important method discussed in Chapter two, titled "Grains" (粟米 su mi), of the Jiuzhang Suanshu is the jin you method (今有术 jin you shu). The jin you 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.

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 Jin you 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 jin you method describes is the 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 jin you 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 jin you method. Aryabhata called it the Trairāśika (त्रैराशिक, literally "rule of three quantities"). Europeans would later call it the "Rule of Three".

Unit Conversions

Equipped with knowledge of rates, it is easy to extend the method of calculating exchange to the conversion of units.