Demonstration in Chinese and Vietnamese mathematics 525
problem that the fl at-rate distribution method cannot be used to distribute
this amount, and the method of weighted distribution is proposed instead.
Th e ratio of the amounts to be given to the functionaries of the three
ranks is 7 : 5 : 2, and the numbers of functionaries of each rank are N (^) A = 8,
N (^) B = 20 and N (^) C = 300, respectively. Th ere are two questions: (1) to fi nd the
amount of silver to award each functionary of the categories A, B and C,
and (2) to fi nd the total amount of money allotted to each group of the
functionaries.
In modern terms, this is a problem on weighted distribution: one has to
fi nd the values x 1 , x 2 ,.. ., x (^) n given that x 1 + x 2 +.. .+ x (^) n = S and x 1 : x 2 :... : x (^) n ::
k 1 : k 2 :... : k (^) n for given weighting coeffi cients k 1 , k 2 ,.. ., k (^) n. Problems of this
type as well as the standard procedure for their solution equivalent to the
formula
(^) x
Sk
k
j
j
i
i
n
1
are found in a number of Chinese and Vietnamese mathematical treatises
beginning with the Chinese mathematical treatises Suan shu shu
(Writing on computations with counting rods) 48 a n d Jiu zhang suan shu.^49
However, the problem found in the Vietnamese treatise contains a par-
ticularity: it is known that there are three diff erent ranks of functionaries,
and for all functionaries of the same rank the weighting coeffi cients are the
same; in our notation, k 1 = k 2 =... = k 8 = k (^) A = 7, k 9 = k 10 =... = k 28 = k (^) B = 5,
k 29 = k 30 =... = k 328 = k (^) C = 2, and one is asked to fi nd the values x (^) A , x (^) B , x (^) C
( x (^) A = x 1 =... = x 8 , x (^) B = x 9 =... = x 28 , and x (^) C = x 29 =... = x 328 ) such that x (^) A :
x (^) B : x (^) C :: k (^) A : k (^) B : k (^) C , and N (^) A · x (^) A + N (^) B · x (^) B + N (^) C · x (^) C = S. Th e examinee is also asked
to fi nd the total amount of money allotted to each group of functionaries,
that is, to calculate the values X (^) A = x 1 +... + x 8 , X (^) B = x 9 +... + x 28 and X (^) C = x 29
+... + x 328.
In this chapter I use the term ‘aggregated weighted distribution’ to
identify the category of problems on weighted distribution in which the
‘sharers’ can be subdivided into groups A, B, C,... containing N (^) A , N (^) B , N (^) C ,..
. sharers, respectively, such that in each group the weighting coeffi cients are
the same and equal to k (^) A , k (^) B , k (^) C ,.... Any problem on aggregated weighted
distribution apparently can be solved with the classical algorithm cited
above, yet in several sources a modifi ed version of the method was used: the
48 Th e earliest extant Chinese mathematical treatise Suan shu shu was completed no later than the
early second century bce ; for English translations, see Cullen 2004 and Dauben 2008.
49 Cullen 2004 : 43–51, 54–6; Dauben 2008 : 114–21, 126–7; CG2004: 282–99.