《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§4. Redistributing fractions for equality

This section gives a worked example of redistributing three fractions for equality.

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、三分之二、四分之三。問減多益少、幾何而平。 Suppose there be one of three shares, two of three shares, [and] three of four shares. [We] ask, subtracting of [the] greater [to] benefit [the] lesser, how much [that they be] level?
  • 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.

答曰、減四分之三者二、三分之二者一、并以益三分之一、而各平於一十二分之七。 Answer saith: that subtracted of [the] three of four shares, [is] two; that of [the] two of three shares, one; combined to benefit [the] one of three shares, with each [being] level at seven of twelve shares.
  • In modern notation:
    \frac{3}{4} - \colr{\frac{2}{12}} = \frac{2}{3} - \colv{\frac{1}{12}} = \frac{1}{3} + \roundbr{\colg{\frac{2}{12} + \frac{1}{12}}} = \colb{\frac{7}{12}}.
  • To see why Sun Tzŭ's algorithm works, it is helpful to first work through the problem once ourselves. More generally, consider redistributing the fractions a/A, b/B and c/C for equality. Their mean is
    m = \frac{L}{3 ABC},
    where
    L = aBC + bCA + cAB,
    so the fractions must be changed by the quantities
    m - \left( \frac{a}{A}, \frac{b}{B}, \frac{c}{C} \right) = \frac{L - (3 aBC, 3 bCA, 3 cAB)}{3 ABC}
    respectively. Of course Sun Tzŭ did not have the luxury of modern mathematical notation:
術曰、置三分三分四分在右方、之一之二之三在左方。母互乘子、副并得六十三、置右為平實。 Method saith: put [the] three shares, [the] three shares, [and the] four shares on [the] right; [the] one of, [the] two of, [and the] three of, on [the] left. [The] denominators mutually multiplied with [the] numerators, combined subsidiarily, result in sixty-three, put [unto the] right as [the] level dividend.
  • Version C erroneously has 母互乘于 for 母互乘子.
  • 平實： level dividend

    L = aBC + bCA + cAB = 63.

  • Altogether:
    \roundbr{ \begin{alignedat}{3.5} a &= 1 \qquad & A &= 3 \qquad & aBC &= 12 \\ b &= 2 \qquad & B &= 3 \qquad & bCA &= 24 \\ c &= 3 \qquad & C &= 4 \qquad & cAB &= 27 \\ \hline & & & & L &= 63 \end{alignedat} }.
母相乘、得三十六為法。 [The] denominators multiplied with each other, result in thirty-six as [the] divisor.
  • In modern notation:
    ABC = 3 \times 3 \times 4 = 36.
以列數三乘未并者及法。 Multiply those not yet combined, and [the] divisor, by [the] number of rows three.
  • 未并者： those not yet combined

    These are the individual terms aBC, bCA and cAB, before they were combined subsidiarily to make the level dividend L. We then obtain

    \begin{aligned} 3 aBC &= 36 \\ 3 bCA &= 72 \\ 3 cAB &= 81 \\ 3 ABC &= 108. \end{aligned}
等數得九、約訖。 [For their] equal number [there] resulteth nine; finish reducing [them].
  • 等數： equal number

    Greatest common divisor (GCD) is called an "equal number" because it is determined by repeated subtraction until an equal quantity is obtained (see §1).

    In the current example, we have

    \gcd (L, 3 aBC, 3 bCA, 3 cAB, 3 ABC) = 9,
    and therefore
    \begin{aligned} \frac{L - (3 aBC, 3 bCA, 3 cAB)}{3 ABC} &= \frac{63 \div 9 - (36, 72, 81) \div 9}{108 \div 9} \\[\tallspace] &= \frac{7 - (4, 8, 9)}{12} \\[\tallspace] &= \frac{(+3, -1, -2)}{12}. \end{aligned}
減四分之三者二、減三分之二者一、并以益三分之一、各平於一十二分之七。 That subtracted of [the] three of four shares, [is] two; that subtracted of [the] two of three shares, one; combined to benefit [the] one of three shares, each [being] level at seven of twelve shares.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §4". <https://yawnoc.github.io/sun-tzu/ii/4> Accessed yyyy-mm-dd.