《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§19. Solving a linear equation (3)

This section gives a worked example of solving a linear equation. (More accurately, it inverts a sequence of discounts.)

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}

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
今有器中米、不知其數。	Suppose there be grain amidst [a] vessel, [and we] know not its number.
前人取半、中人三分取一、後人四分取一、餘米一斗五升。問本米幾何。	[The] former person taking [an] half, [the] middle person taking one of three shares, [and the] latter person taking one of four shares, [there] remaineth grain one peck [and] five quarts. [We] ask, how much [be the] original grain?	In modern notation, we seek V such that V \cdot \roundbr{1 - \frac{1}{2}} \roundbr{1 - \frac{1}{3}} \roundbr{1 - \frac{1}{4}} = 1.5 \unit{pecks}.
答曰、六斗。	Answer saith: six pecks.
術曰、置餘米一斗五升、以六乘之、得九斗。以二除之、得四斗五升。	Method saith: put [down the] remaining grain one peck [and] five quarts; multiplying it by six, resulteth in nine pecks. Dividing it by two, resulteth in four pecks [and] five quarts.
以四乘之、得一斛八斗。以三除之、即得。	Multiplying it by four, resulteth in one barrel [and] eight pecks. Dividing it by three, [we] are done.	In modern notation, \begin{aligned} V &= 1.5 \unit{pecks} \times \frac{6}{2} \times \frac{4}{3} \\[\tallspace] &= 6 \unit{pecks}. \end{aligned}

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §19". <https://yawnoc.github.io/sun-tzu/iii/19> Accessed yyyy-mm-dd.