《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§5. Volume exchange of grain (1)

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
今有粟一斗。問為糲米幾何。
答曰、六升。		 Suppose there be grain one peck. [We] ask, how much be [this in] coarse grain?
Answer saith: six quarts.
術曰、置粟一斗、十升。以糲米率三十乘之、得三百升為實。以粟率五十為法。除之、即得。 Method saith: put [down the] grain one peck, [even] ten quarts. Multiplying it by [the] coarse grain rate thirty, resulteth in three hundred quarts as [the] dividend. Use [the] grain rate fifty as [the] divisor. Dividing them, [we] are done.
  • In modern notation, with V for volume and r for rate:
    \begin{aligned} V(\text{coarse grain}) &= \frac{ V(\text{grain}) \cdot r(\text{coarse grain}) }{ r(\text{grain}) } \\[\tallspace] &= \frac{10 \unit{quarts} \times 30}{50} \\[\tallspace] &= 6 \unit{quarts}. \end{aligned}

Extended commentary

Here we have cross-multiplication. Vol. I §10 already gave a grain-to-coarse-grain exchange rate of 3/5, so it appears strange that here Sun Tzŭ uses 30/50.

I believe the rates 30 and 50 come from a separate text, the chapter 〈粟米〉 'Grain', of 《九章算術》 "Nine Chapters [on] Computational Methods". In this text, cross-multiplication is called 今有術, the "Suppose There Be method", after the incipit "Suppose there be" in all grain exchange rate problems. I think it is informative to include a translation of an excerpt here.

The source text for this excerpt is from 《四部叢刊初編》 (ctext.org library). Note that this excerpt is not a part of Sun Tzŭ's Computational Classic:

Source text Target text Notes
粟米之法、粟率五十、糲米三十、粺米二十七、糳米二十四、御米二十一、…… [In the] Method of Grain: [the] grain rate [be] fifty; coarse grain, thirty; fine grain, twenty-seven; intricate grain, twenty-four; poppy seed, twenty-one; …
  • 御米： poppy seed; lit. imperial grain
今有術曰、以所有數乘所求率為實、以所有率為法、實如法而一。 [The] Suppose There Be method saith: use [the] number of That There Be times [the] rate of That Sought as [the] dividend, [and] use [the] rate of That There Be as [the] divisor. [Take the] dividend as [per the] divisor [being] one.
  • In modern notation, with V for volume and r for rate:
    V(\text{that sought}) = \frac{ V(\text{that there be}) \cdot r(\text{that sought}) }{ r(\text{that there be}) }.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §5". <https://yawnoc.github.io/sun-tzu/ii/5> Accessed yyyy-mm-dd.