《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§3. Subtracting fractions

This section gives a worked example of subtracting fractions.

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
今有九分之八、減其五分之一。問餘幾何。
答曰、四十五分之三十一。		 Suppose there be eight of nine shares, [and] subtract of it one of five shares. [We] ask, how much remaineth?
Answer saith: thirty-one of forty-five shares.
  • b 分之 a: lit. a of b shares

    I have rendered fractions literally, since there are references to the numerator "之 a" ("a of"), and the denominator "b 分" ("b shares"), in the lines to follow.

術曰、置九分五分在右方、之八之一在左方。 Method saith: put [the] nine shares [and the] five shares on [the] right; [the] eight of [and the] one of, on [the] left.
母互乘子、五分之一得九、九分之八得四十、以少減多、餘三十一為實。母相乘、得四十五為法。 [The] denominators [being] mutually multiplied with [the] numerators, [for the] one of five shares [there] resulteth nine, [and for the] eight of nine shares [there] resulteth forty; subtracting of [the] greater by [the] lesser, [there] remaineth thirty-one as [the] dividend. [The] denominators multiplied with each other, result in forty-five as [the] divisor.
  • Version C erroneously has 王十一 for 三十一.
  • In modern notation:
    \frac{8}{9} - \frac{1}{5} = \frac{8 \times 5 - 1 \times 9}{9 \times 5} = \frac{31}{45}.
不滿法、以法命之、即得。 [The dividend] reacheth not [the] divisor, [and] naming it [for a fraction] using [the] divisor, [we] are done.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §3". <https://yawnoc.github.io/sun-tzu/ii/3> Accessed yyyy-mm-dd.