衰分术曰：各置列衰，副并为法，以所分乘未并者各自为实，实如法而一。不满法者，以法命之。
衰分术说：分别设置各自的分配比例，将它们相加作为法则，用要分配的总数乘以每个分配者尚未合并的比例作为各自的实际分配数，实际分配数按照法则进行统一。如果不满法则的数量，就按照法则来命名。
The method of proportional division states: Set up the respective proportions for each party, add them together to form a standard (method), multiply the total amount to be divided by each party's proportion before it is combined to get the actual amounts for each, and make the actual amounts uniform according to the standard. For those who do not meet the standard quantity, name them according to the standard.
〔一〕今有大夫、不更、簪裹、上造、公士，凡五人，共猎得五鹿。欲以爵次分之，问各得几何？
荅曰：
大夫得一鹿、三分鹿之二。
不更得一鹿、三分鹿之一。簪裹得一鹿。
上造得三分鹿之二。
公士得三分鹿之一。
术曰：列置爵数，各自为衰，副并为法。以五鹿乘未并者，各自为实。实如法得一鹿。
现在有大夫、不更、簪裹、上造、公士，总共五人，他们一起猎得了五只鹿。他们想要根据爵位的等级来分配这些鹿，问每个人能得到多少？
答案是：
大夫得到一只鹿和剩下的鹿的三分之二。
不更得到一只鹿和剩下的鹿的三分之一。
簪裹得到一只鹿。
上造得到剩下的鹿的三分之二。
公士得到剩下的鹿的三分之一。
解题方法：列出各个爵位的等级，分别作为比例的分母，然后将它们相加作为总的比例（法）。用五只鹿乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每个人得到的鹿的数量。
Now there are five people: Grandee, Bushi, Zanmao, Shangzao, and Gongshi, who together hunted down five deer. They wish to divide the deer according to their rank. How many does each person get? The answer is: The Grandee gets one deer and two-thirds of the remaining deer. Bushi gets one deer and one-third of the remaining deer. Zanmao gets one deer. Shangzao gets two-thirds of the remaining deer. Gongshi gets one-third of the remaining deer. Solution method: List the ranks of each title as the denominators of ratios, then add them up to form the total ratio (law). Multiply the five deer by the respective ratio denominators (before adding), to get each person's actual score (real). Dividing the actual score by the total ratio gives the number of deer each person receives.
〔二〕今有牛、马、羊食人苗。苗主责之粟五斗。羊主曰：「我羊食半马。」马主曰：「我马食半牛。」今欲衰偿之，问各出几何？
荅曰：
牛主出二斗八升、七分升之四。
马主出一斗四升、七分升之二。羊主出七升、七分升之一。
术曰：置牛四、马二、羊一，各自为列衰，副并为法。以五斗乘未并者各自为实。实如法得一斗。
现在有牛、马、羊吃了人的庄稼。庄稼的主人要求赔偿五斗粟。羊的主人说：“我的羊吃的一半是马吃的。”马的主人说：“我的马吃的一半是牛吃的。”现在想要按比例赔偿，问每个人应该赔多少？
答案是：
牛主人出二斗八升又七分之四升。
马主人出一斗四升又七分之二升。
羊主人出七升又七分之一升。
解题方法：列出牛、马、羊的比例为4:2:1，然后将它们相加作为总的比例（法）。用五斗乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每个人应该赔的斗数。
Now there are oxen, horses, and sheep that have eaten someone's crops. The owner of the crops demands five pecks of millet as compensation. The owner of the sheep says, "My sheep ate half what the horse ate." The owner of the horse says, "My horse ate half what the ox ate." Now they want to compensate proportionally, asking how much should each person pay? The answer is: The ox owner pays two pecks and eight sheng, plus four parts of seven sheng. The horse owner pays one peck and four sheng, plus two parts of seven sheng. The sheep owner pays seven sheng, plus one part of seven sheng. Solution method: Set up the proportions for the oxen, horses, and sheep as 4:2:1, then add them up to form the total ratio (law). Multiply the five pecks by the respective ratio denominators (before adding), to get each person's actual score (real). Dividing the actual score by the total ratio gives the number of pecks each person should pay.
〔三〕今有甲持钱五百六十，乙持钱三百五十，丙持钱一百八十，凡三人俱出关，关税百钱。欲以钱数多少衰出之，问各几何？
荅曰：
甲出五十一钱、一百九分钱之四十一。乙出三十二钱、一百九分钱之一十二。
丙出一十六钱、一百九分钱之五十六。
术曰：各置钱数为列衰，副并为法，以百钱乘未并者，各自为实，实如法得一钱。
现在有甲持有560钱，乙持有350钱，丙持有180钱，总共三人一起出关，关税是100钱。想要根据他们持有的钱数比例来分摊关税，问每个人应该出多少？
答案是：
甲出51钱又一百九分之四十一。
乙出32钱又一百九分之十二。
丙出16钱又一百九分之五十六。
解题方法：列出各自持有的钱数作为比例的分母，然后将它们相加作为总的比例（法）。用100钱乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每个人应该出的关税。
Now, A has 560 coins, B has 350 coins, and C has 180 coins. All three are passing through a customs checkpoint together, and the customs fee is 100 coins. They wish to pay the fee in proportion to their amount of money. How much should each person pay? The answer is: A pays 51 coins and 41/109 of a coin. B pays 32 coins and 12/109 of a coin. C pays 16 coins and 56/109 of a coin. Solution method: List the amounts of money each person holds as the denominators of ratios, then add them up to form the total ratio (law). Multiply the 100 coins by the respective ratio denominators (before adding), to get each person's actual score (real). Dividing the actual score by the total ratio gives the number of coins each person should pay.
〔四〕今有女子善织，日自倍，五日织五尺。问日织几何？
荅曰：初日织一寸、三十一分寸之十九。
次日织三寸、三十一分寸之七。
次日织六寸、三十一分寸之十四。
次日织一尺二寸、三十一分寸之二十八。
次日织二尺五寸、三十一分寸之二十五。术曰：置一、二、四、八、十六为列衰，副并为法，以五尺乘未并者，各自为实，实如法得一尺
现在有一位擅长织布的女子，每天的织布量是前一天的两倍，五天内总共织了五尺布。问每天分别织了多少？
答案是：
第一天织了一寸又三十一分之十九。
第二天织了三寸又三十一分之七。
第三天织了六寸又三十一分之十四。
第四天织了一尺二寸又三十一分之二十八。
第五天织了二尺五寸又三十一分之二十五。
解题方法：列出数字1、2、4、8、16作为比例的分母，然后将它们相加作为总的比例（法）。用五尺乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每天织的布的长度。
Now there is a woman who is skilled at weaving, and the amount she weaves doubles each day. Over five days, she has woven a total of five chi (units of length). How much does she weave each day? The answer is: On the first day, she weaves one inch and 19/31 of an inch. On the second day, she weaves three inches and 7/31 of an inch. On the third day, she weaves six inches and 14/31 of an inch. On the fourth day, she weaves one foot and two inches plus 28/31 of an inch. On the fifth day, she weaves two feet and five inches plus 25/31 of an inch. Solution method: Set up the numbers 1, 2, 4, 8, 16 as the denominators of ratios, then add them up to form the total ratio (law). Multiply the five chi by the respective ratio denominators (before adding), to get each day's actual score (real). Dividing the actual score by the total ratio gives the length of cloth woven each day.
〔五〕今有北乡算八千七百五十八，西乡算七千二百三十六，南乡算八千三百五十六，凡三乡，发傜三百七十八人。欲以算数多少衰出之，问各几何？
荅曰：
北乡遣一百三十五人、一万二千一百七十五分人之一万一千六百三十七。
西乡遣一百一十二人、一万二千一百七十五分人之四千四。
南乡遣一百二十九人、一万二千一百七十五分人之八千七百九。术曰：各置算数为列衰，副并为法，以所发傜人数乘未并者，各自为实，实如法得一人。
现在北乡的算数是8758，西乡的算数是7236，南乡的算数是8356，总共三个乡要派出徭役共378人。想要根据算数的多少按比例分配，问每个乡应该分别派出多少人？
答案是：
北乡派出135人又12175分之11637。
西乡派出112人又12175分之44。
南乡派出129人又12175分之8709。
解题方法：列出各自乡的算数作为比例的分母，然后将它们相加作为总的比例（法）。用所要派出的徭役人数乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每个乡应该派出的人数。
Now the North District has a computation of 8,758, the West District has 7,236, and the South District has 8,356. There are three districts in total, and they need to dispatch a total of 378 people for corvée labor. They wish to allocate the number of people according to the computation figures. How many should each district send? The answer is: The North District sends 135 people plus 11,637 out of 12,175. The West District sends 112 people plus 44 out of 12,175. The South District sends 129 people plus 8,709 out of 12,175. Solution method: List the computation figures for each district as the denominators of ratios, then add them up to form the total ratio (law). Multiply the number of people to be dispatched by the respective ratio denominators (before adding), to get each district's actual score (real). Dividing the actual score by the total ratio gives the number of people each district should send.
〔六〕今有稟粟，大夫、不更、簪裹、上造、公士，凡五人，一十五斗。今有大夫一人后来，亦当稟五斗。仓无粟，欲以衰出之，问各几何？
荅曰：
大夫出一斗、四分斗之一。
不更出一斗。
簪褭出四分斗之三。
上造出四分斗之二。
公士出四分斗之一。术曰：各置所稟粟斛斗数，爵次均之，以为列衰，副并而加后来大夫亦五斗，得二十以为法。以五斗乘未并者各自为实。实如法得一斗。
现在有分配的粟米，大夫、不更、簪裹、上造、公士，共五人，总共15斗。现在有一个大夫后来加入，也应当分得5斗。但是仓库里没有多余的粟米了，想要按比例减少他们的份额，问每个人应该减少多少？
答案是：
大夫减少1斗又四分之一斗。
不更减少1斗。
簪褭减少四分之三斗。
上造减少四分之二斗。
公士减少四分之一斗。
解题方法：列出各自应得的粟米数量，按照爵位顺序均分，作为比例的分母，然后将它们相加，并加上后来的大夫的5斗，得到20作为总的比例（法）。用5斗乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每个人应该减少的斗数。
Now there is a distribution of millet, for the noblemen titles Dafu, Bugeng, Zanhao, Shangzao, Gongshi, totaling five people, amounting to 15 dou (units of dry measure). Now an additional Dafu comes later and is also supposed to receive 5 dou. However, the granary has no more millet, and it is desired to reduce their shares proportionally. How much should each person's share be reduced? The answer is: Dafu reduces by 1 dou and one-fourth of a dou. Bugeng reduces by 1 dou. Zanhao reduces by three-fourths of a dou. Shangzao reduces by two-fourths of a dou. Gongshi reduces by one-fourth of a dou. Solution method: List the number of dou of millet each person should receive, evenly divided according to rank order, as the denominators of ratios, then add them up and add the later-coming Dafu's 5 dou, obtaining 20 as the total ratio (law). Multiply by 5 dou with the respective ratio denominators (before adding), to get each person's actual score (real). Dividing the actual score by the total ratio gives the number of dou each person's share should be reduced by.
〔七〕今有稟粟五斛，五人分之，欲令三人得三，二人得二。问各几何？
荅曰：
三人，人得一斛一斗五升、十三分升之五。
二人，人得七斗六升、十三分升之十二。术曰：置三人，人三；二人，人二，为列衰。副并为法。以五斛乘未并者，各自为实。实如法得一斛。
返衰术曰：列置衰而令相乘，动者为不动者衰。
现在有5斛粟米，分给五个人，想让其中三个人得到3份，两个人得到2份。问每个人分别应该得到多少？
答案是：
三个人中，每个人得到1斛1斗5升又13分之5升。
两个人中，每个人得到7斗6升又13分之12升。
解题方法：列出三个人每人得3份，两个人每人得2份的比例关系。然后将它们相加作为总的比例（法）。用5斛乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每个人应该得到的粟米数量。
返衰术的方法是：列出比例并让它们相互乘，动数成为不动数的衰数。
Now there are 5 hu (units of dry measure) of millet to be divided among five people, with the wish that three people receive three shares each and two people receive two shares each. How much should each person get? The answer is: For the three people, each person gets 1 hu, 1 dou, and 5 sheng plus 5/13 of a sheng. For the two people, each person gets 7 dou, 6 sheng, and 12/13 of a sheng. Solution method: Set up the ratio so that each of the three people gets 3 shares and each of the two people gets 2 shares. Add these together to form the total ratio (law). Multiply the 5 hu by the respective ratio denominators (before adding), to get each person's actual score (real). Dividing the actual score by the total ratio gives the amount of millet each person should receive. The method for returning to the original ratio is as follows: List the ratios and multiply them so that the moving number becomes the decay number for the stationary number.
〔八〕今有大夫、不更、簪褭、上造、公士，凡五人，共出百钱。欲令高爵出少，以次渐多，问各几何？
荅曰：
大夫出八钱、一百三十七分钱之一百四。不更出一十钱、一百三十七分钱之一百三十。
簪褭出一十四钱、一百三十七分钱之八十二。上造出二十一钱、一百三十七分钱之一百二十三。
公士出四十三钱、一百三十七分钱之一百九。
术曰：置爵数各自为衰，而返衰之，副并为法。以百钱乘未并者各自为实。实如法得一钱。
现在有大夫、不更、簪褭、上造、公士，共五人，共同出100钱。想让爵位高的人出的钱少，依次递增，问每个人应该出多少钱？
答案是：
大夫出8钱又137分之14。
不更出10钱又137分之13。
簪褭出14钱又137分之82。
上造出21钱又137分之123。
公士出43钱又137分之109。
Now there are five people with titles Dafu, Bugeng, Zanhao, Shangzao, and Gongshi, who together contribute 100 coins. It is desired that those with higher ranks contribute less, with the amount gradually increasing in order. How much should each person contribute? The answer is: Dafu contributes 8 coins plus 14/137 of a coin. Bugeng contributes 10 coins plus 13/137 of a coin. Zanhao contributes 14 coins plus 82/137 of a coin. Shangzao contributes 21 coins plus 123/137 of a coin. Gongshi contributes 43 coins plus 109/137 of a coin.
〔九〕今有甲持粟三升，乙持糲米三升，丙持糲饭三升。欲令合而分之，问各几何？
荅曰：
甲二升、一十分升之七。
乙四升、一十分升之五。
丙一升、一十分升之八。
术曰：以粟率五十、糲米率三十、糲饭率七十五为衰、而返衰之，副并为法。以九升乘未并者各自为实。实如法得一升。
现在甲持有3升粟米，乙持有3升糙米，丙持有3升糙米饭。想要将它们合并后平均分配，问每个人各应得多少？
答案是：
甲得到2升又7/10升。
乙得到4升又5/10升。
丙得到1升又8/10升。
解题方法：以粟米的换算率为50、糙米的换算率为30、糙米饭的换算率为75来设置比例，然后反转这些比例，将它们相加作为总的比例（法）。用9升乘以各自的比例分母（未相加的），得到各自的实际得分（实）。实际得分除以总的比例就得到每个人应该得到的升数。
Now, A holds 3 sheng of millet, B holds 3 sheng of unpolished rice, and C holds 3 sheng of unpolished rice meal. They wish to combine them and divide equally. How much should each person get? The answer is: A gets 2 sheng and 7/10 of a sheng. B gets 4 sheng and 5/10 of a sheng. C gets 1 sheng and 8/10 of a sheng. Solution method: Set the exchange rates for millet at 50, unpolished rice at 30, and unpolished rice meal at 75 as the ratios, then reverse these ratios, adding them up to form the total ratio (law). Multiply by 9 sheng with the respective ratio denominators (before adding), to get each person's actual score (real). Dividing the actual score by the total ratio gives the number of sheng each person should receive.
〔一０〕今有丝一斤，价直二百四十。今有钱一千三百二十八，问得丝几何？
荅曰：五斤八两一十二銖、三分銖之四。
术曰：以一斤价数为法，以一斤乘今有钱数为实，实如法得丝数。
现在有丝一斤，价值240钱。现在有1328钱，问可以买多少斤丝？
答案是：可以买5斤8两12铢又3/4铢的丝。
解题方法：以一斤丝的价格为比例（法），用这个价格乘以现有的钱数作为实际得分（实）。实际得分除以比例得到可以买的丝的数量。
Now, one jin (a unit of weight) of silk is valued at 240 coins. If one has 1328 coins now, how much silk can be bought? The answer is: 5 jin, 8 liang, 12 zhu, and 4/3 zhu of silk can be purchased. Solution method: Use the price per jin as the ratio (law), multiply this price by the number of coins currently available to get the actual amount (real). Dividing the actual amount by the ratio yields the quantity of silk that can be bought.
〔一一〕今有丝一斤价直三百四十三。今有丝七两一十二銖，问得钱几何？
荅曰：一百六十一钱、三十二分钱之二十三。
术曰：以一斤銖数为法，以一斤价数，乘七两一十二銖为实。实如法得钱数。
现在有丝一斤，价值343钱。现在有丝7两12铢，问可以卖多少钱？
答案是：可以卖161钱又23/32钱。
解题方法：以一斤丝的铢数为比例（法），用一斤丝的价格乘以现有的丝的重量（7两12铢）作为实际得分（实）。实际得分除以比例得到可以卖的钱的数量。
Now, one jin (a unit of weight) of silk is valued at 343 coins. If one has 7 liang and 12 zhu of silk now, how much money can be made? The answer is: 161 coins and 23/32 of a coin can be obtained. Solution method: Use the number of zhu per jin as the ratio (law), multiply this price by the weight of silk available (7 liang and 12 zhu) to get the actual amount (real). Dividing the actual amount by the ratio yields the amount of money that can be made.
〔一二〕今有縑一丈价直一百二十六。今有縑一匹九尺五寸，问得钱几何？
荅曰：六百三十三钱、五分钱之三。
术曰：以一丈寸数为法，以价钱数乘今有縑寸数为实，实如法得钱数。
现在有绢一丈，价值126钱。现在有绢一匹，长度为9尺5寸，问可以卖多少钱？
答案是：可以卖633钱又3/5钱。
解题方法：以一丈的寸数为比例（法），用这个价格乘以现有的绢的长度（9尺5寸）作为实际得分（实）。实际得分除以比例得到可以卖的钱的数量。
Now, one zhang (a unit of length) of silk fabric is valued at 126 coins. If one has a piece of silk fabric measuring 9 chi and 5 cun (ancient Chinese units of length), how much money can be made? The answer is: 633 coins and 3/5 of a coin can be obtained. Solution method: Use the number of cun per zhang as the ratio (law), multiply this price by the length of silk fabric available (9 chi and 5 cun) to get the actual amount (real). Dividing the actual amount by the ratio yields the amount of money that can be made.
〔一三〕今有布一匹，价直一百二十三。今有布二丈七尺，问得钱几何？
荅曰：八十四钱、分钱之三。
术曰：以一匹尺数为法，今有布尺数乘价钱为实，实如法得钱数。
现在有布一匹，价值123钱。现在有布2丈7尺，问可以卖多少钱？
答案是：可以卖84钱又3/10钱。
解题方法：以一匹布的尺数为比例（法），用这个价格乘以现有的布的长度（2丈7尺）作为实际得分（实）。实际得分除以比例得到可以卖的钱的数量。
Now, one pi (a unit of length) of cloth is valued at 123 coins. If one has 2 zhang and 7 chi of cloth, how much money can be made? The answer is: 84 coins and 3/10 of a coin can be obtained. Solution method: Use the number of chi per pi as the ratio (law), multiply this price by the length of cloth available (2 zhang and 7 chi) to get the actual amount (real). Dividing the actual amount by the ratio yields the amount of money that can be made.
〔一四〕今有素一匹一丈，价直六百二十五。今有钱五百，问得素几何？
荅曰：
术曰：以价直为法，以一匹一丈尺数乘今有钱数为实。实如法得素数。
现在有素（一种布）一匹一丈，价值625钱。现在有500钱，问可以买多少素？
答案是：可以买2/5匹。
解题方法：以素的价格为比例（法），用这个价格乘以现有的钱数作为实际得分（实）。实际得分除以比例得到可以买的素的数量。
Now, one pi and one zhang of plain silk fabric is valued at 625 coins. If one has 500 coins now, how much plain silk fabric can be bought? The answer is: 2/5 pi can be purchased. Solution method: Use the price as the ratio (law), multiply this price by the number of coins available to get the actual amount (real). Dividing the actual amount by the ratio yields the quantity of plain silk fabric that can be bought.
〔一五〕今有与人丝一十四斤，约得縑一十斤。今与人丝四十五斤八两，问得縑几何？
荅曰：三十二斤八两。
术曰：以一十四斤两数为法，以一十斤乘今有丝两数为实，实如法得縑数。
现在与人交换丝14斤，约定可以得到绢10斤。现在与人交换丝45斤8两，问可以得到多少斤绢？
答案是：可以得到32斤8两。
解题方法：以14斤的两数为比例（法），用这个比例乘以现有的丝的两数（45斤8两）作为实际得分（实）。实际得分除以比例得到可以得到的绢的数量。
Now, in exchange for 14 jin of silk, it is agreed that one can obtain 10 jin of silk fabric. If one exchanges 45 jin and 8 liang of silk now, how much silk fabric can be obtained? The answer is: 32 jin and 8 liang can be obtained. Solution method: Use the number of jin and liang as the ratio (law), multiply this ratio by the number of jin and liang of silk available to get the actual amount (real). Dividing the actual amount by the ratio yields the quantity of silk fabric that can be obtained.
〔一六〕今有丝一斤，耗七两。今有丝二十三斤五两，问耗几何？
荅曰：一百六十三两四銖半。术曰：以一斤展十六两为法，以七两乘今有丝两数为实，实如法得耗数。
现在有丝一斤，损耗7两。现在有丝23斤5两，问损耗多少？
答案是：损耗163两4铢半。
解题方法：以一斤展开为16两为比例（法），用这个比例乘以现有的丝的两数（23斤5两）作为实际得分（实）。实际得分除以比例得到损耗的数量。
Now, one jin of silk incurs a loss of 7 liang. If one has 23 jin and 5 liang of silk now, how much is the loss? The answer is: the loss is 163 liang and 4 zhu and a half. Solution method: Use the conversion of one jin to 16 liang as the ratio (law), multiply this ratio by the number of liang of silk available to get the actual amount (real). Dividing the actual amount by the ratio yields the amount of loss.
〔一七〕今有生丝三十斤，干之，耗三斤十二两。今有干丝一十二斤，问生丝几何？
荅曰：一十三斤一十一两十銖、七分銖之二。术曰：置生丝两数，除耗数，余，以为法。三十斤乘干丝两数为实。实如法得生丝数。
现在有生丝30斤，晾干后损耗了3斤12两。现在有干丝12斤，问原来有多少生丝？答案是：13斤11两10铢，七分之二。
解题方法：将生丝的两数减去损耗的两数，余下的作为比例（法）。用30斤乘以干丝的两数作为实际得分（实）。实际得分除以比例得到生丝的数量。
Now, there are 30 jin of raw silk, which loses 3 jin and 12 liang after drying. If one has 12 jin of dried silk now, how much raw silk was there originally? The answer is: 13 jin, 11 liang, and 10 zhu with two-sevenths of a zhu. Solution method: Subtract the amount lost from the number of jin of raw silk, and the remainder is used as the ratio (law). Multiply 30 jin by the number of jin of dried silk to get the actual amount (real). Dividing the actual amount by the ratio yields the number of jin of raw silk.
〔一八〕今有田一亩，收粟六升、太半升。今有田一顷二十六亩一百五十九步，问收粟几何？
荅曰：八斛四斗四升、一十二分升之五。
术曰：以亩二百四十步为法，以六升、太半升乘今有田积步为实，实如法得粟数。
现在有田一亩，收获粟米6升和3/4升。现在有田1顷26亩159步，问可以收获多少粟米？
答案是：8斛4斗4升加上12分之5升。
解题方法：以一亩240步为比例（法），用6升和3/4升乘以现有的田地的步数作为实际得分（实）。实际得分除以比例得到可以收获的粟米数量。
Now, one mu of land yields 6 sheng and three-quarters of a sheng of millet. If one has 1 qing, 26 mu, and 159 bu of land now, how much millet can be harvested? The answer is: 8 hu, 4 dou, 4 sheng, and five-twelfths of a sheng. Solution method: Use the number of bu per mu as the ratio (law), multiply this by the number of bu of land available to get the actual amount (real). Dividing the actual amount by the ratio yields the amount of millet that can be harvested.
〔一九〕今有取保一岁，价钱二千五百。今先取一千二百，问当作日几何？
荅曰：一百六十九日、二十五分日之二十三。
术曰：以价钱为法，以一岁三百五十四日乘先取钱数为实，实如法得日数。
现在有人担保一年，总价2500钱。现在先支付1200钱，问应该工作多少天？
答案是：169天又25分之23天。
解题方法：以一年的总价为比例（法），用354天乘以先支付的钱数作为实际得分（实）。实际得分除以比例得到应该工作的天数。
Now, a guarantee for one year costs 2500 coins. If one pays 1200 coins upfront now, how many days should one work? The answer is: 169 days and twenty-three twenty-fifths of a day. Solution method: Use the total price as the ratio (law), multiply this by the number of days in one year (354 days) to get the actual amount (real). Dividing the actual amount by the ratio yields the number of days that should be worked.
〔二０〕今有贷人千钱，月息三十。今有贷人七百五十钱，九日归之，问息几何？
荅曰：六钱、四分钱之三。
术曰：以月三十日，乘千钱为法。以息三十乘今所贷钱数，又以九日乘之，为实。实如法得一钱。
现在有人借了1000钱，月息30钱。现在有人借了750钱，9天后归还，问利息是多少？
答案是：6钱又4分之3钱。
解题方法：以一个月30天，乘以1000钱作为比例（法）。用30钱的利息乘以现在借的钱数，再乘以9天，作为实际得分（实）。实际得分除以比例得到每一钱的利息。
Now, someone borrows 1000 coins with a monthly interest of 30 coins. If someone else borrows 750 coins and returns them in 9 days, how much is the interest? The answer is: 6 coins and three-fourths of a coin. Solution method: Use the number of days in a month (30 days) multiplied by 1000 coins as the ratio (law). Multiply the interest of 30 coins by the amount of coins borrowed now, and then multiply by 9 days to get the actual amount (real). Dividing the actual amount by the ratio yields the interest per coin.