"Sun Tzŭ's Computational Classic: Volume II" 《孫子算經卷中》 §7

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
今有粟四斗五升。問為糳米幾何。答曰、二斗一升五分升之三。	Suppose there be grain four pecks [and] five quarts. [We] ask, how much be [this in] intricate grain? Answer saith: two pecks, one quart, [and] three fifths of [a] quart.	Version C erroneously has 槃 for 糳, both here and below.
術曰、置粟四十五升。以二約糳米率二十四、得一十二。乘之、得五百四十升為實。	Method saith: put [down the] grain forty-five quarts. Reducing [the] intricate grain rate twenty-four by two, resulteth in twelve. Multiplying them, resulteth in five hundred [and] forty quarts as [the] dividend.	以二約： reducing by two The intricate grain rate 24 here and the (regular) grain rate 50 below are reduced by their greatest common divisor, \gcd (24, 50) = 2.
以二約粟率五十、得二十五為法。	Reducing [the] grain rate fifty by two, resulteth in twenty-five as [the] divisor.
除之、不盡、以等數約之、而命分。	Dividing them, [there be a] remainder; reduce it by [the] equal number, and name [it for a] fraction.	不盡： [there be a] remainder; lit. exhausteth not 以等數約之： reduce it by [the] equal number The division 540 \div 25 leaves remainder 15/25. Their "equal number" is \gcd (15, 25) = 5 (see §1), so that the remainder is reduced to 3/5. In modern notation: \begin{aligned} V(\text{intricate grain}) &= \frac{ V(\text{grain}) \cdot r(\text{intricate grain}) }{ r(\text{grain}) } \\[\tallspace] &= \frac{45 \unit{quarts} \times 24 \div 2}{50 \div 2} \\[\tallspace] &= 21 \tfrac{3}{5} \unit{quarts}. \end{aligned} The rates 24 for intricate 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.

Source text

Target text

Notes

今有粟四斗五升。問為糳米幾何。
答曰、二斗一升五分升之三。

Suppose there be grain four pecks [and] five quarts. [We] ask, how much be [this in] intricate grain?
Answer saith: two pecks, one quart, [and] three fifths of [a] quart.

Version C erroneously has 槃 for 糳, both here and below.

術曰、置粟四十五升。以二約糳米率二十四、得一十二。乘之、得五百四十升為實。

Method saith: put [down the] grain forty-five quarts. Reducing [the] intricate grain rate twenty-four by two, resulteth in twelve. Multiplying them, resulteth in five hundred [and] forty quarts as [the] dividend.

以二約： reducing by two
The intricate grain rate 24 here and the (regular) grain rate 50 below are reduced by their greatest common divisor, \gcd (24, 50) = 2.

以二約粟率五十、得二十五為法。

Reducing [the] grain rate fifty by two, resulteth in twenty-five as [the] divisor.

除之、不盡、以等數約之、而命分。

Dividing them, [there be a] remainder; reduce it by [the] equal number, and name [it for a] fraction.

不盡： [there be a] remainder; lit. exhausteth not
以等數約之： reduce it by [the] equal number
The division 540 \div 25 leaves remainder 15/25. Their "equal number" is \gcd (15, 25) = 5 (see §1), so that the remainder is reduced to 3/5.
In modern notation:
\begin{aligned} V(\text{intricate grain}) &= \frac{ V(\text{grain}) \cdot r(\text{intricate grain}) }{ r(\text{grain}) } \\[\tallspace] &= \frac{45 \unit{quarts} \times 24 \div 2}{50 \div 2} \\[\tallspace] &= 21 \tfrac{3}{5} \unit{quarts}. \end{aligned}
The rates 24 for intricate 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.

《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§7. Volume exchange of grain (3)

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II" §7. Volume exchange of grain (3)

Translation

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《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§7. Volume exchange of grain (3)