Chapter 12 Section 3 Proportion and Proportional Distribution

Proportion problems have been seen as early as the pre-Qin period. "Jiu Zhang" Millet Chapter Jin Youshu is a complete proportional algorithm: all numbers, all rates and the required rate are known, then the required number is The desired number = all the numbers × the desired rate ÷ all the rates. After this method was introduced to India and the West, it was called the rule of three.Liu Hui believes that Jinyoushu is a universal method.All the problems in the nine numbers, as long as the rate relationship can be found out, through the homogeneous transformation, all of them can be attributed to this technique.For example, the title of all chapters in "Nine Chapters": A guest forgot to bring his clothes when he left the hotel. After 1/3 of the day, the host found out, caught up with the guest on horseback and returned the clothes to him, and it was 3/4 of the day when he returned home.The guest's horse travels 300 miles a day, how much does the host's horse travel in a day?Liu Hui believes that 3/4-1/3=5/12 is the daily rate of the host chasing customers back and forth, 5/24 is the daily rate of the host chasing customers, 5/24+1/3=13/24 is the rate of guests being chased Step forward and use the daily rate.The daily rate used by the host is the guest horse rate, and the daily rate used by the guests is the owner’s horse rate. Therefore, the guest horse rate is 5, which is the own rate, the host horse rate is 13, which is the desired rate, and 300 miles is the total number.The owner's horse travels a day = 300 miles x 13÷5 = 780 miles.

The proportional distribution method was called the decline method in ancient times, and the ratio of each part was called the column decline. The method proposed in "Nine Chapters" is: assuming that the number to be divided is A, the number of columns is a, a...a, the sum of the numbers is the law, and a certain column number a (i=1, 2...) is multiplied by the divided number The number A is the reality, and the reality is one like the law, that is, a certain part of A=Aa÷(a+a...+a).Liu Hui believes that it can be attributed to Jinyoushu: the divided number A is all the numbers, the sum of the column decays is the total rate, each column decay is the desired rate, and a certain part is the desired number.For example, the title of Chapter 1 of "Nine Chapters": The cattle, horses, and sheep ate other people's young crops, and the seedling owners demanded compensation of 5 buckets of millet.The owner of the sheep said: My sheep only ate half of the horse; the owner of the horse said: My horse only ate half of the ox.How much is the compensation?According to the method of decay, the column decay is 4, 2, 1, then

Sheep: 50 liters×1÷(4+2+1)=7(1/7) liters, Horse: 50 liters×2÷(4+2+1)=14(2/7) liters, Cattle: 50 liters×4÷(4+2+1)=28(4/7) liters. If each part is distributed according to the ratio of 1/a, 1/a, ..., 1/a, "Nine Chapters" is called the anti-decay technique, and its formula is: A=A...aa...a÷(aa...a+aa... a+...+aa...a).Liu Hui said that this is "the one who moves for the sake of the immobile dies". ("Nine Chapters of Arithmetic Decline Chapter Notes") The government has to collect taxes, some of which are food, and some are corvee.Counties have different household registrations, distances, and food prices. How can the distribution be made to make the burden of each household fair and reasonable?This is the problem of equal loss, and it is also a problem of proportional distribution.It's just that the distribution ratio of each county is not predetermined, but has to be calculated according to the conditions of each county.Assuming that there is a total of grain A dendrobium payable in n counties, the number of households in each county is P, P, ... P, the distance is q, q, ... q, the price of each bucket of grain is r, r, ... r, and the labor price is m dendrobium for one vehicle. K money per mile, then the cost of transporting a dendrobium in county i is kq/m+r, and P/(kq/m+r) is the distribution ratio of county i.Liu Hui pointed out that this can make kq/m+r households pay a total of one dendrobium, and each household will be paid one coin, which is a fair burden.