Systematic Treatise on Arithmetic, Scroll 13

Page 2

Answer: 18 large boats, carrying 3,000 yin (引): a unit of weight used specifically for government-monopolized goods like salt of salt. 18 small boats, carrying 1,350 yin of salt.

Method: Set out the capacities of 4 boats and 3 boats.

3 boats 4 boats
\ /
Ｘ
/
500 300

First, take the upper-left 3 and mutually multiply (互乘) by the lower-right 300 to get 900. Next, take the upper-right 4 and mutually multiply by the lower-left 500 to get 2,000. Combine these to get 2,900, which serves as the Divisor (法). ○ Set out 12 This represents the ratio of boats or a constant derived from the problem's conditions and multiply by the total salt 4,350 yin to obtain 52,200 as the Dividend (實). Divide this by the Divisor.

The result is 18, which is the number of large boats. First, multiply this by the large boat capacity of 500 to get 9,000, then reduce (歸): a division technique by the 3 boats to obtain 3,000 yin. Additionally, take the 18 small boats and multiply by the capacity of 300 to get 5,400, then reduce by the 4 boats to obtain 1,350 yin of salt. This matches the inquiry.

Song of Increased Money and Judging Depth

Dan Now written as 擔 or 儋; both are interchangeable with 担 [a carrying pole or a unit of weight]

The Lu family has grain, and there is a loud dispute;
They meet and ask how this came to be.