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

This section gives a worked example of adding 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
今有三分之一、五分之二。問合之二得幾何。
答曰、一十五分之一十一。		 Suppose there be one of three shares, [and] two of five shares. [We] ask, joining them two resulteth in how much?
Answer saith: eleven of fifteen shares.
  • Version C has only 合之 for 合之二.
  • Version B has 十一 for 一十一.
  • 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、置三分五分在右方、之一之二在左方。 Method saith: put [the] three shares [and the] five shares on [the] right; [the] one of [and the] two of, on [the] left.
母互乘子、五分之二得六、三分之一得五、并之、得一十一為實。右方二母相乘、得一十五為法。 [The] denominators [being] mutually multiplied with [the] numerators, [for the] two of five shares [there] resulteth six, [and for the] one of three shares [there] resulteth five; combining them, resulteth in eleven as [the] dividend. [The] two denominators aright multiplied with each other, result in fifteen as [the] divisor.
  • In modern notation:
    \frac{1}{3} + \frac{2}{5} = \frac{1 \times 5 + 2 \times 3}{3 \times 5} = \frac{11}{15}.
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.
  • 不滿法： [The dividend] reacheth not [the] divisor

    This is saying that 11 < 15, i.e. the fraction 11/15 is proper (so no further division is necessary).

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §2". <https://yawnoc.github.io/sun-tzu/ii/2> Accessed yyyy-mm-dd.