##### Ratio and Proportion

Many of the problems in the early chapters of *Jiuzhang Suanshu* are simple ones involving ratios and proportions. They were designed to give the reader practice with the system of units, the use of basic arithmetic (including fractions), and the Rule of Three. The latter rule, versions of which were used in many world regions in ancient and medieval times, allows an unknown quantity to be calculated if it stands in a proportion with three known quantities. The following sample problems, drawn from Chapters 2 and 3, respectively, deal with quantities of lacquer and woven silk, classic products of China.

**Problem 1.** Now pay 5785 coins to purchase 1 *hu* 6 *dou* 7 2/3 *sheng* of lacquer. Tell: how much is 1 *dou*? (Shen et al., p. 151)

**Problem 2.** Now given 1 *jin* of silk costs 240 coins. Tell: given 1328 coins, how many *jin* of silk are obtained? (Shen et al., p. 169)

Your answer to Problem 2 should be given exactly, in units of *jin*, *liang*, *zhu*, and fractions of a *zhu*.

##### Small Plots and Huge Granaries

In feudal times, agricultural production was the main source of wealth. The vast majority of the people were poor peasants. They paid for things not with coins but with portions of their harvest, especially quantities of grain.

In China, millet was the most common grain, especially in the north. Rice was a comparative luxury enjoyed mostly by the upper classes, for it “takes a whole village” to tend the young shoots of rice, keep the paddies flooded, and manage the harvest.

Most peasants worked small plots of land owned by a landlord. In exchange, they were required every year to give the landlord a certain portion of the grain harvest, known as *rent*. This practice is reflected in another problem from Chapter 3 involving proportions:

**Problem 3.** Now given a field of 1 *mu*, 6 2/3 *sheng* of millet is collected [as rent]. Tell: given 1 *qing* 26 *mu* 159 [square] *bu* of field how much millet is collected? (Shen et al., p. 171)

Millet or other grains were also used to pay taxes to the central authorities every year. These taxes went to enrich the emperor and his court in the capital; to the network of local officials and military troops stationed across the empire and on its frontiers; and to government granaries that were maintained for times of famine. The sheer task of transporting tax grains to central repositories was a gargantuan one, but it was meted out proportionally to the various localities according to a system called *junshun* (“fair levies”). Localities might be levied in direct proportion to their number of households, in inverse proportion to their distance to the central bureau, and/or in inverse proportion to the local cost of grain. In solving problems such as the following, which is the first one in Chapter 6, rounding of the answers is necessary, since the carts used to transport grain taxes were always filled to capacity.

**Problem 4.** Now given the task of transporting tax millet is distributed among four counties. County A, 8 days from the tax bureau, has 10,000 households; County B, 10 days from the bureau, has 9,500 households; County C, 13 days from the bureau, has 12,350 households; County D, 20 days from the bureau, has 12,200 households. The total tax millet is 250,000 *hu* needing 10,000 carts. Assume the task is to be distributed in accordance with the distance from the bureau and the number of households. Tell: how much millet should each county transport? How many carts does each county employ? (Shen et al., p. 310)

The following problem from the same chapter deals with shipments of millet to Taicang, the national granary. This granary was located in Ch’ang-an (now called Xi’an), the first capital of China, completed during the Han Dynasty (206 BCE–220 CE). The source of the millet is Shanglin, the old Imperial Garden of the Qin Dynasty, lying to the west of Ch’ang-an. The problem amounts to finding the travel distance between these two sites.

**Problem 5.** Now someone transports provisions between two posts. An unloaded cart travels 70 *li* a day and a loaded one 50 *li* a day. Transporting millet from the National Granary to Shanglin. One makes 3 round trips in 5 days, how far is the distance between the two posts? (Shen et al., p. 325)

Download solutions to Problems here: Students Explore the *Nine Chapters* from China.