《孫子算經卷下》 "Sun Tzŭ's Computational Classic: Volume III"
§33. Division utilising a rate (5)

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{mile~(里)} &= 300 \unit{paces~(步)} \\ 1 \unit{pace~(步)} &= 6 \unit{rules~(尺)} \\ 1 \unit{rod~(丈)} &= 10 \unit{rules~(尺)}. \\ \end{aligned}

See Vol. I §1 (Units of length).

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
今有長安洛陽、相去九百里。 Suppose there be Chʻang‑an [and] Lok‑yang, separated from each other [by] nine hundred miles.
車輪一匝一丈八尺、欲自洛陽至長安。問輪匝幾何。 Once around [a] chariot wheel [be] one rod [and] eight rules, [and we] wish to [go] from Lok‑yang unto Chʻang‑an. [We] ask, how many [times goeth the] wheel around?
答曰、九萬匝。 Answer saith: nine myriad [times] around.
術曰、置九百里、以三百步乘之、得二十七萬步。 Method saith: put [down the] nine hundred miles; multiplying it by three hundred paces, resulteth in twenty-seven myriad paces.
  • In modern notation, this is a conversion from miles to paces:
    900 \unit{miles} \times \frac{300 \unit{paces}}{\unit{mile}} = 270000 \unit{paces}.
又以六尺乘之、得一百六十二萬尺。 And multiplying it by six rules, resulteth in one hundred [and] sixty-two myriad rules.
  • Next we a conversion from paces to rules:
    270000 \unit{paces} \times \frac{6 \unit{rules}}{\unit{pace}} = 1620000 \unit{rules}.
以車輪一丈八尺為法。除之、即得。 Use [the] chariot wheel's one rod [and] eight rules as [the] divisor. Dividing them, [we] are done.
  • Finally we have the division determining the number of turns:
    \frac{1620000 \unit{rules}}{18 \unit{rules} / {\unit{turn}}} = 90000 \unit{turns}.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume III §33". <https://yawnoc.github.io/sun-tzu/iii/33> Accessed yyyy-mm-dd.