《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§8. Volume exchange of grain (4)

This section gives a worked example of volume exchange for trading grain using cross-multiplication.

The relevant unit conversions for capacity are

\begin{aligned} 1 \unit{peck~(斗)} &= 10 \unit{quarts~(升)} \\ 1 \unit{quart~(升)} &= 10 \unit{gills~(合)} \\ 1 \unit{gill~(合)} &= 10 \unit{ladles~(勺)}. \end{aligned}

See Vol. I §3 (Units of capacity).

Translation

Chinese source text: Version A, Version B, Version C, Version D.
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text Target text Notes
今有粟七斗九升。問為御米幾何。
答曰、三斗三升一合八勺。		 Suppose there be grain seven pecks [and] nine quarts. [We] ask, how much be [this in] poppy seed?
Answer saith: three pecks, three quarts, one gill, [and] eight ladles.
  • Version C erroneously has 栗 for 粟.
  • 御米： poppy seed; lit. imperial grain
術曰、置七斗九升。以御米率二十一乘之、得一千六百五十九升為實。以粟率五十除之、即得。 Method saith: put [down the] seven pecks [and] nine quarts. Multiplying it by [the] poppy seed rate twenty-one, resulteth in one thousand six hundred [and] fifty-nine quarts as [the] dividend. Dividing it by [the] grain rate fifty, [we] are done.
  • In modern notation:
    \begin{aligned} V(\text{poppy seed}) &= \frac{ V(\text{grain}) \cdot r(\text{poppy seed}) }{ r(\text{grain}) } \\[\tallspace] &= \frac{79 \unit{quarts} \times 21}{50} \\[\tallspace] &= 33.18 \unit{quarts}. \end{aligned}
    The rates 21 for poppy seed and 50 for (regular) grain appear to come from 《九章算術粟米》, "Nine Chapters [on] Computational Methods: Grain". See §5 commentary.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §8". <https://yawnoc.github.io/sun-tzu/ii/8> Accessed yyyy-mm-dd.