《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§11. Volume exchange of grain (5)

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{barrel~(斛)} &= 10 \unit{pecks~(斗)} \\ 1 \unit{peck~(斗)} &= 10 \unit{quarts~(升)}. \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
今有粟三千九百九十九斛九斗六升。凡粟九斗、易豆一斛。問計豆幾何。 Suppose there be grain three thousand nine hundred [and] ninety-nine barrels, nine pecks, [and] six quarts. Every nine pecks of grain, [is] exchanged for one barrel of beans. [We] ask, how much [in] beans amounteth [this] to?
  • Version C has 蚪 for both 斗, and erroneously has 栗 for the second 粟.
答曰、四千四百四十四斛四斗。 Answer saith: four thousand four hundred [and] forty-four barrels [and] four pecks.
術曰、置粟三千九百九十九斛九斗六升為實、以九斗為法。實如法、即得。 Method saith: put [down the] grain three thousand nine hundred [and] ninety-nine barrels, nine pecks, [and] six quarts as [the] dividend, [and] use [the] nine pecks as [the] divisor. [Taking the] dividend as [per the] divisor, [we] are done.
  • In modern notation:
    \frac{ V(\text{grain}) }{ V(\text{beans}) } = \frac{ 9 \unit{pecks} }{ 1 \unit{barrel} },
    hence
    \begin{aligned} V(\text{beans}) &= \frac{ V(\text{grain}) }{ 9 \unit{pecks} / {\unit{barrel}} } \\[\tallspace] &= \frac{ 3999.96 \unit{barrels} }{ 9 \unit{pecks} / {\unit{barrel}} } \\[\tallspace] &= 4444.4 \unit{barrels}. \end{aligned}

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §11". <https://yawnoc.github.io/sun-tzu/iii/11> Accessed yyyy-mm-dd.