《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§6. Volume exchange of grain (2)

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

The relevant unit conversion for capacity is

1 \unit{peck~(斗)} = 10 \unit{quarts~(升)}.

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
今有粟二斗一升。問為粺米幾何。
答曰、一斗一升五十分升之一十七。		 Suppose there be grain two pecks [and] one quart. [We] ask, how much be [this in] fine grain?
Answer saith: one peck, one quart, [and] seventeen fiftieths of [a] quart.
  • Version C erroneously has 稗 for 粺, both here and below.
  • 粺： fine(-grain)

    粺、 Cantonese: 《分韻撮要》 pʻei4, 《正韻》薄邁切 paai6, Mandarin: bài

術曰、置粟二十一升。以粺米率二十七乘之、得五百六十七升為實。以粟率五十為法。除之、不盡、以法而命分。 Method saith: put [down the] grain twenty-one quarts. Multiplying it by [the] fine grain rate twenty-seven, resulteth in five hundred [and] sixty-seven quarts as [the] dividend. Use [the] grain rate fifty as [the] divisor. Dividing them, [there be a] remainder; use [the] divisor and name [it for a] fraction.
  • 不盡： [there be a] remainder; lit. exhausteth not
  • In modern notation:
    \begin{aligned} V(\text{fine grain}) &= \frac{ V(\text{grain}) \cdot r(\text{fine grain}) }{ r(\text{grain}) } \\[\tallspace] &= \frac{21 \unit{quarts} \times 27}{50} \\[\tallspace] &= 11 \tfrac{17}{50} \unit{quarts}. \end{aligned}
    The rates 27 for fine grain 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 §6". <https://yawnoc.github.io/sun-tzu/ii/6> Accessed yyyy-mm-dd.