"Sun Tzŭ's Computational Classic: Volume III" 《孫子算經卷下》 §16

《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§16. Division utilising a rate (3)

This section gives a word problem where division by a rate is used to determine an amount.

The relevant unit conversions for length are

\begin{aligned} 1 \unit{length~(匹)} &= 40 \unit{rules~(尺)} \\ 1 \unit{rod~(丈)} &= 10 \unit{rules~(尺)}, \end{aligned}

and, for capacity,

\begin{aligned} 1 \unit{barrel~(斛)} &= 10 \unit{pecks~(斗)} \\ 1 \unit{peck~(斗)} &= 10 \unit{quarts~(升)} \\ 1 \unit{quart~(升)} &= 10 \unit{gills~(合)}. \end{aligned}

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
今有粟一十二萬八千九百四十斛九斗三合、出與人買絹、一匹直粟三斛五斗七升。問絹幾何。	Suppose there be grain twelve myriad eight thousand nine hundred [and] forty barrels, nine pecks, [and] three gills, supplied to [a] person [to] buy strong-silk, [each] one length [thereof] worth grain three barrels, five pecks, [and] seven quarts. [We] ask, how much strong-silk [be bought]?
答曰、三萬六千一百一十七匹三丈六尺。	Answer saith: three myriad six thousand one hundred [and] seventeen lengths, three rods, [and] six rules.
術曰、置粟一十二萬八千九百四十斛九斗三合為實、以三斛五斗七升為法。	Method saith: put [down the] grain twelve myriad eight thousand nine hundred [and] forty barrels, nine pecks, [and] three gills as [the] dividend, [and] use [the] three barrels, five pecks, [and] seven quarts as [the] divisor.
除之、得匹、餘、四十之、所得又以法除之、即得。	Dividing them, resulteth in [the] lengths; [the] remainder, quadragintuple it, [and of] that which resulteth, again dividing it by [the] divisor, [we] are done.	In modern notation, \begin{aligned} \frac{ 128940.903 \unit{barrels} }{ 3.57 \unit{barrels} / {\unit{length}} } &= 36117 \unit{lengths} + \frac{3.213 \unit{lengths}}{3.57} \times \frac{40 \unit{rules}}{\unit{length}} \\[\tallspace] &= 36117 \unit{lengths} + 36 \unit{rules}. \end{aligned}

Source text

Target text

Notes

今有粟一十二萬八千九百四十斛九斗三合、出與人買絹、一匹直粟三斛五斗七升。問絹幾何。

Suppose there be grain twelve myriad eight thousand nine hundred [and] forty barrels, nine pecks, [and] three gills, supplied to [a] person [to] buy strong-silk, [each] one length [thereof] worth grain three barrels, five pecks, [and] seven quarts. [We] ask, how much strong-silk [be bought]?

答曰、三萬六千一百一十七匹三丈六尺。

Answer saith: three myriad six thousand one hundred [and] seventeen lengths, three rods, [and] six rules.

術曰、置粟一十二萬八千九百四十斛九斗三合為實、以三斛五斗七升為法。

Method saith: put [down the] grain twelve myriad eight thousand nine hundred [and] forty barrels, nine pecks, [and] three gills as [the] dividend, [and] use [the] three barrels, five pecks, [and] seven quarts as [the] divisor.

除之、得匹、餘、四十之、所得又以法除之、即得。

Dividing them, resulteth in [the] lengths; [the] remainder, quadragintuple it, [and of] that which resulteth, again dividing it by [the] divisor, [we] are done.

In modern notation,
\begin{aligned} \frac{ 128940.903 \unit{barrels} }{ 3.57 \unit{barrels} / {\unit{length}} } &= 36117 \unit{lengths} + \frac{3.213 \unit{lengths}}{3.57} \times \frac{40 \unit{rules}}{\unit{length}} \\[\tallspace] &= 36117 \unit{lengths} + 36 \unit{rules}. \end{aligned}

《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III" §16. Division utilising a rate (3)

Translation

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《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§16. Division utilising a rate (3)