A History of Mathematics From Mesopotamia to Modernity

ChineseMathematics 99

Solutions to exercises

1. (a) We would probably call the original cerealx. Then after the three taxes have been exacted,
   there is left^23 ·^45 ·^67 x=^1635 x(which we are told is 5). Hencex =^17516 = 101516 dou; and
   15
   16 dou=

150

16 sheng=9

3

8 sheng. TheNine Chaptersof course does not use ‘x’, but proceeds

inversely, telling you to multiply the remaining 5 by 3, 5, and 7 and divide by 2, 4, and 6 (‘the

remainders’). The result is the same.

(b) TheNine Chapterssolution is given above in Section 4. We would call the number of

contributorsx(say) and the pricey, and arrive at simultaneous equations:

9 x− 11 =y (1)

6 x+ 16 =y (2)

It is then usual to subtract, and get 3x=27, sox=9; from whichyfollows by substitution.

2. (a) We have:

ay−b=x (3)

a 1 y+b 1 =x (4)

From which, subtracting,(a−a 1 )y=b+b 1. Note thata>a 1 , because it gives the excess!

Hence,

y=

b+b 1

a−a 1

Here, b+b 1 is the ‘divisor’, and a−a 1 (greater contribution minus smaller) is the

‘lesser assumption’.

Substituting back (see 1 (b)), and simplifying, we get

x=

ab 1 +a 1 b

a−a 1

and nowa 1 b+ab 1 , the result of cross-multiplying, is the ‘dividend’. (b) In this case, the

dividend is^12 × 3 +^13 × 4 =^176 ; the divisor is 4+ 3 =7, and the lesser assumption is

1

2 −

1

3 =

1

6. This gives 42 for the number of people, and 17 for the price. (TheNine Chapters

has rules for dealing with fractions, but I have assumed we do not need them.)

3. No solution required.


4. We have the matrix as simplified, and now we simplify it further. Following thefangcheng
   method, multiply the first column by 3 and subtract the third column. After which, you
   multiply the new first column by 5 (the first non-zero entry in the second column) and subtract
   4 ×the second column. Result:
   
   
   
   

003

452

811

39 24 39







,









003

052

36 1 1

99 24 39







