〔一〕今有共买物，人出八，盈三；人出七，不足四。问人数、物价各几何？
荅曰：七人，物价五十三。
假设现在有一群人共同购买物品，如果每个人出8元，那么会多出3元；如果每个人出7元，那么会少了4元。问这群人有多少人，物品的价格是多少？
答案是：7人，物品的价格是53元。
Suppose there is a group of people buying an item together. If each person contributes 8 units of currency, they will have an excess of 3 units; if each person contributes 7 units, they will be short by 4 units. How many people are there in the group, and what is the price of the item? The answer is: 7 people, with the price of the item being 53 units of currency.
〔二〕今有共买鸡，人出九，盈十一；人出六，不足十六。问人数、鸡价各几何？
荅曰：九人，鸡价七十。
假设现在有人共同购买鸡，每个人出9文钱，总共多出了11文；每个人出6文钱，总共少了16文。问人数和鸡的价格各是多少？
答案是：9人，鸡的价格是70文。
Suppose there are people buying chickens together, if each person contributes 9 wen, they have a surplus of 11 wen; if each person contributes 6 wen, they are short by 16 wen. Question: How many people are there and what is the price of the chicken? The answer is: 9 people, the price of the chicken is 70 wen.
〔三〕今有共买璡，人出半，盈四；人出少半，不足三。问人数、璡价各几何？
荅曰：四十二人，
璡价十七。
假设现在有一群人共同购买璡，如果每个人出半价，那么会多出4元；如果每个人出少半价，那么会少了3元。问这群人有多少人，璡的价格是多少？
答案是：42人，璡的价格是17元。
Suppose there is a group of people buying something together. If each person pays half the price, they will have an excess of 4 units of currency; if each person pays a bit less than half the price, they will be short by 3 units of currency. How many people are there in the group, and what is the price of the item? The answer is: 42 people, with the price of the item being 17 units of currency.
〔四〕今有共买牛，七家共出一百九十，不足三百三十；九家共出二百七十，盈三十。问家数、牛价各几何？
荅曰：一百二十六家，
牛价三千七百五十。
盈不足术曰：置所出率，盈、不足各居其下。令维乘所出率，并以为实。并盈、不足为法。实如法而一。有分者，通之。盈不足相与同其买物者，置所出率，以少减多，余，以约法、实。实为物价，法为人数。其一术曰：并盈不足为实。以所出率以少减多，余为法。实如法得一人。以所出率乘之，减盈、增不足即物价。
假设现在有人共同购买牛，七家共出一百九十文钱，不足三百三十文；九家共出二百七十文钱，多出三十文。问共有多少家，以及牛的价格是多少？
答案是：共有一百二六家，牛的价格是三千七百五十文。
盈不足术的方法是：列出每家的出资率，将盈余和不足的数量分别放在出资率的下面。然后乘以出资率，将结果相加作为被除数（实）。将盈余和不足的量相加作为除数（法）。用实除以法得到一家的出资率。如果有分数，就进行通分。对于共同购买物品的盈与不足，将出资率相减，用少的减去多的，余下的用来约简实和法。实就是物品的价格，法就是参与的人数。另一种方法是：将盈与不足的量相加作为被除数（实）。用出资率相减，余下的部分作为除数（法）。用实用法得到一人的出资率。用出资率乘以这个数，减去盈余或增加不足的部分，就得到了物品的价格。
Suppose there are people jointly purchasing cattle; if each of the seven families contributes 190 wen, they are short by 330 wen; if each of the nine families contributes 270 wen, they have a surplus of 30 wen. Question: How many families are there in total, and what is the price of the cattle? The answer is: There are 126 families, and the price of the cattle is 3,750 wen. The method of solving excess and deficiency is as follows: List the rate of contribution for each family, with the amounts of surplus and deficiency placed below the rate of contribution. Then multiply by the rate of contribution, adding the results together as the dividend (shi). Add the surplus and deficiency together as the divisor (fa). Divide shi by fa to get the rate of contribution for one family. If there are fractions, they should be converted to a common denominator. For the surplus and deficiency associated with the joint purchase of goods, subtract the rates of contribution, using the smaller amount minus the larger, and the remainder is used to simplify shi and fa. Shi represents the price of the goods, and fa represents the number of participants. Another method is: Add the surplus and deficiency together as the dividend (shi). Subtract the rates of contribution, and the remainder becomes the divisor (fa). Dividing shi by fa gives the rate of contribution for one person. Multiply by this rate, subtracting the surplus or adding the deficiency to obtain the price of the goods.
〔五〕今有共买金，人出四百，盈三千四百；人出三百，盈一百。问人数、金价各几何？
荅曰：三十三人。金价九千八百。
假设现在有一群人共同购买黄金，如果每个人出400元，那么会多出3400元；如果每个人出300元，那么会多出100元。问这群人有多少人，黄金的价格是多少？
答案是：33人，黄金的价格是9800元。
Suppose there is a group of people buying gold together. If each person contributes 400 units of currency, they will have an excess of 3400 units; if each person contributes 300 units, they will have an excess of 100 units. How many people are there in the group, and what is the price of the gold? The answer is: 33 people, with the price of the gold being 9800 units of currency.
〔六〕今有共买羊，人出五，不足四十五；人出七，不足三。问人数、羊价各几何？
荅曰：二十一人，
羊价一百五十。
两盈、两不足术曰：置所出率，盈、不足各居其下。令维乘所出率，以少减多，余为实。两盈、两不足以少减多，余为法。实如法而一。有分者通之。两盈、两不足相与同其买物者，置所出率，以少减多，余，以约法实，实为物价，法为人数。
其一术曰：置所出率，以少减多，余为法。两盈、两不足，以少减多，余为实。实如法而一得人数。以所出率乘之，减盈、增不足，即物价。
假设现在有一群人共同购买羊，如果每个人出5元，那么会少了45元；如果每个人出7元，那么会少了3元。问这群人有多少人，羊的价格是多少？
答案是：21人，羊的价格是150元。
使用“两盈、两不足”的方法来解这个问题：
将每个人出的金额列出来，盈和不足的情况分别放在它们的下面。
让每个人的出资率（即每人出的钱数）互相乘以对应的盈和不足的差额。
用较小的盈或不足的金额减去较大的盈或不足的金额，剩下的作为实际值（实）。
如果有两个盈和两个不足，就用较小的那个减去较大的那个，剩下的作为标准（法）。
实际值除以标准值得到人数。如果涉及到分数，要进行通分。
当处理两个盈和两个不足一起参与购买物品的情况时，将每个人的出资率相减后剩下的数用来约简法和实，其中实代表物价，法代表人数。
另一种方法是：
将每个人的出资率相减，得到的差值作为标准（法）。
对于两个盈和两个不足，用较小的盈或不足的金额减去较大的盈或不足的金额，剩下的作为实际值（实）。
实际值除以标准值得到人数。
再用每个人出的金额乘以人数，减去盈的金额或增加不足的金额，得到的结果就是物价。
Suppose there is a group of people buying sheep together. If each person contributes 5 units of currency, they will be short by 45 units; if each person contributes 7 units, they will be short by 3 units. How many people are there in the group, and what is the price of the sheep? The answer is: 21 people, with the price of the sheep being 150 units of currency. Using the method of "two surpluses and two deficits" to solve this problem: List out the amount each person contributes, with the surpluses and deficits placed below them. Multiply the rates at which each person contributes (the amount of money each person gives) by the corresponding differences in surpluses and deficits. Subtract the larger surplus or deficit from the smaller one, and the remainder serves as the actual value (real). If there are two surpluses and two deficits, subtract the larger from the smaller, and the remainder serves as the standard (rule). Divide the actual value by the standard to get the number of people. If fractions are involved, adjust the denominators accordingly. When dealing with cases where two surpluses and two deficits are involved together in purchasing an item, use the difference after subtracting the rates at which each person contributes to simplify the rule and the real, where the real represents the price of the goods, and the rule represents the number of people. Another method is: Subtract the rates at which each person contributes, and the resulting difference serves as the standard (rule). For two surpluses and two deficits, subtract the larger surplus or deficit from the smaller one, and the remainder serves as the actual value (real). Divide the actual value by the standard to get the number of people. Then multiply the amount each person contributes by the number of people, subtracting the surplus amount or adding the deficit amount to get the price of the goods.
〔七〕今有共买豕，人出一百，盈一百；人出九十，適足。问人数、豕价各几何？
荅曰：一十人，
豕价九百。
现在有人共同购买猪，如果每人出一百元，则多出一百元；如果每人出九十元，则恰好足够。问人数和猪的价格各是多少？。
回答说：十个人，猪价九百元
Now, there are people who jointly purchase a pig. If each person contributes one hundred yuan, there is an excess of one hundred yuan; if each person contributes ninety yuan, it is just enough. How many people are there and what is the price of the pig? The answer is: ten people, the price of the pig is nine hundred yuan.
〔八〕今有共买犬，人出五，不足九十；人出五十，適足。问人数、犬价各几何？
荅曰：二人，
犬价一百。盈、適足，不足、適足术曰：以盈及不足之数为实。置所出率，以少减多，余为法。实如法得一人。其求物价者，以適足乘人数得物价。
假设现在有一群人共同购买狗，如果每个人出5元，那么会少90元；如果每个人出50元，刚好够。问这群人有多少人，狗的价格是多少？
答案是：2人，狗的价格是100元。
使用“盈、适足，不足、适足”的方法来解这个问题：
将盈和不足的金额作为实际值（实）。
列出每个人出的金额，用较少的金额减去较多的金额，剩下的作为标准（法）。
实际值除以标准值得到人数。
在求物价时，用适足的金额乘以人数得到物价。
Suppose there is a group of people buying a dog together. If each person contributes 5 units of currency, they will be short by 90 units; if each person contributes 50 units, it will be just enough. How many people are there in the group, and what is the price of the dog? The answer is: 2 people, with the price of the dog being 100 units of currency. Using the method of "surplus, just enough, deficit, just enough": Take the surplus and deficit amounts as the real value (real). List out the rates at which each person contributes, subtracting the larger from the smaller, leaving the remainder as the rule (rule). Divide the real value by the rule to get the number of people. To find the price of the goods, multiply the just enough amount by the number of people to get the price of the goods.
〔九〕今有米在十斗桶中，不知其数。满中添粟而舂之，得米七斗。问故米几何？
荅曰：二斗五升。
术曰：以盈不足术求之，假令故米二斗，不足二升。令之三斗，有余二升。
现在有一个能装十斗的桶里面装着一些米，具体数量未知。在桶里加满粟后一起舂，最后得到七斗米。问原来有多少斗米？
回答说：二斗五升。
There is rice in a barrel with a capacity of ten tou (a traditional Chinese unit of volume), and the exact amount is unknown. After filling the barrel with millet and then pounding them together, seven tou of rice are obtained. How much rice was there originally? The answer is: two tou and five sheng (another traditional Chinese unit of volume). Mathematical Solution: Use the method of excess and deficiency to solve it. Assume that originally there were two tou of rice, which would be deficient by two sheng. If we assume three tou, there would be an excess of two sheng.
〔十〕今有垣高九尺。瓜生其上，蔓日长七寸。瓠生其下，蔓日长一尺。问几何日相逢？瓜、瓠各长几何？
荅曰：五日、十七分日之五。瓜长三尺七寸、十七分寸之一，
瓠长五尺二寸、十七分寸之十六。
术曰：假令五日，不足五寸。令之六日，有余一尺二寸。
假设现在有一道墙高9尺。瓜藤从墙顶生长，每天长7寸。瓠藤从墙底生长，每天长1尺。问多少天后两者相遇？相遇时瓜和瓠各长多少？
答案是：5天又5/17天。瓜长3尺7寸又1/17寸，瓠长5尺2寸又16/17寸。
使用的方法：
假设5天后相遇，但这样会少5寸。
假设6天后相遇，但这样会多出1尺2寸（即12寸）。
Suppose there is a wall that is 9 feet high. A melon vine grows from the top of the wall, growing 7 inches each day. A gourd vine grows from the bottom of the wall, growing 1 foot each day. How many days will it take for them to meet? And how long will the melon and gourd be when they meet? The answer is: 5 days and5/17of a day. The melon will be 3 feet and 7 inches plus1/17of an inch, and the gourd will be 5 feet and 2 inches plus16/17 of an inch. The method used: Assuming they meet after 5 days, but this would be short by 5 inches. Assuming they meet after 6 days, but this would be an excess of 1 foot and 2 inches (that is, 12 inches).
〔一一〕今有蒲生一日，长三尺。莞生一日，长一尺。蒲生日自半。莞生日自倍。问几何日而长等？
荅曰：二日、十三分日之六。
各长四尺八寸、十三分寸之六。
术曰：假令二日，不足一尺五寸。令之三日，有余一尺七寸半。
现在有蒲草，它每天长三尺。莞草每天长一尺。蒲草每天的增长速度是前一天的一半，而莞草每天的增长速度是前一天的两倍。问多少天后它们的长度相等？
回答说：两天又十三分之六天，它们各长四尺八寸又十三分之六寸。
There are two types of plants, one is called "pu" which grows three feet in a day, and the other is called "guan" which grows one foot in a day. The growth rate of "pu" halves each day, while the growth rate of "guan" doubles each day. How many days will it take for them to be of equal height? The answer is: two days and six out of thirteen days, at which point they will each be four feet and eight inches, plus six out of thirteen inches. Mathematical Solution: Assuming two days, they would be short by one foot and five inches. If we assume three days, there would be an excess of one foot and seven and a half inches.
〔一二〕今有垣厚五尺，两鼠对穿。大鼠日一尺，小鼠亦日一尺。大鼠日自倍，小鼠日自半。问几何日相逢？各穿几何？
荅曰：二日、十七分日之二。
大鼠穿三尺四寸、十七分寸之十二，
小鼠穿一尺五寸、十七分寸之五。
术曰：假令二日，不足五寸。令之三日，有余三尺七寸半。
假设现在有20斗质量不好的谷子，舂磨后得到9斗粗米。现在想要得到10斗精米，问需要多少斗质量不好的谷子？
答案是：24斗加上6升、81分之74升。
计算方法如下：将9斗粗米乘以9作为标准。也将10斗精米乘以10，再将20斗质量不好的谷子乘以这个结果，得出实际需要的谷子数量。实际需要的谷子数量除以标准得出每斗的数量。
Suppose there are 20 pecks of poor-quality grain, which after milling yield 9 pecks of coarse rice. Now, to obtain 10 pecks of fine rice, how many pecks of poor-quality grain are needed? The answer is: 24 pecks plus 6 liters and 74/81 liters. The method is as follows: Multiply the 9 pecks of coarse rice by 9 to establish a standard. Also multiply the 10 pecks of fine rice by 10, then multiply again by the 20 pecks of poor-quality grain to find the actual amount of grain needed. The actual amount of grain needed divided by the standard gives the quantity per peck.
〔一三〕今有醇酒一斗，直钱五十；行酒一斗，直钱一十。今將钱三十，得酒二斗。问醇、行酒各得几何？
荅曰：醇酒二升半，
行酒一斗七升半。
术曰：假令醇酒五升，行酒一斗五升，有余一十。令之醇酒二升，行酒一斗八升，不足二。
现在有浓度高的酒一斗，价值五十钱；稀释后的酒一斗，价值十钱。现在用三十钱可以得到二斗酒。问浓度高的酒和稀释后的酒各得到了多少？
回答说：浓度高的酒得到了二升半，稀释后的酒得到了一斗七升半。
There is a quantity of high-quality wine that costs fifty coins per dou (a traditional Chinese unit of volume), and there is a quantity of diluted wine that costs ten coins per dou. Now, with thirty coins, one can obtain two dou of wine. How much high-quality wine and diluted wine are obtained, respectively? The answer is: two and a half sheng (another traditional Chinese unit of volume) of high-quality wine, and one dou and seven and a half sheng of diluted wine. Mathematical Solution: Assuming five sheng of high-quality wine and one dou and five sheng of diluted wine, there would be an excess of ten coins. If we assume two sheng of high-quality wine and one dou and eight sheng of diluted wine, there would be a shortage of two coins.
〔一四〕今有大器五、小器一容三斛；大器一、小器五容二斛。问大、小器各容几何？
荅曰：大器容二十四分斛之十三，小器容二十四分斛之七。
术曰：假令大器五斗，小器亦五斗，盈一十斗。令之大器五斗五升，小器二斗五升，不足二斗。
假设现在有5个大容器和1个小容器，总共能容纳3斛；1个大容器和5个小容器，总共能容纳2斛。问大容器和小容器各自的容量是多少？
答案是：大容器的容量是13/24斛，小容器的容量是7/24斛。使用的方法：假设每个大容器容量为5斗，每个小容器也为5斗，那么总容量会多出10斗。
假设每个大容器容量为5斗5升（即5.5斗），每个小容器为2斗5升（即2.5斗），那么总容量会少2斗。
Suppose there are 5 large vessels and 1 small vessel that together hold 3 hu; and 1 large vessel with 5 small vessels together hold 2 hu. What is the individual capacity of the large and small vessels? The answer is: The large vessel holds 13/24 of a hu, and the small vessel holds 7/24of a hu.The method used:Assuming each large vessel has a capacity of 5 dou, and each small vessel also has 5 dou, then the total capacity would be 10 dou too much. Assuming each large vessel has a capacity of 5 dou and 5 sheng (which is 5.5 dou), and each small vessel has 2 dou and 5 sheng (which is 2.5 dou), then the total capacity would be short by 2 dou.
〔一五〕今有漆三得油四，油四和漆五。今有漆三斗，欲令分以易油，还自和余漆。问出漆、得油、和漆各几何？
荅曰：出漆一斗一升、四分升之一，
得油一斗五升，
和漆一斗八升，四分升之三。
术曰：假令出漆九升，不足六升。令之出漆一斗二升，有余二升。
现在有比例，用三份漆可换得四份油，四份油可以和五份漆混合。现在有三斗漆，想要分出一部分去换油，然后拿换得的油和剩下的漆混合。问分别要分出多少漆、换得多少油以及混合后的漆总量是多少？
回答说：分出漆一斗一升又一分之四升，换得油一斗五升，混合后的漆总量是一斗八升又三分之四升。
Given a ratio where three parts of lacquer can be exchanged for four parts of oil, and four parts of oil can be mixed with five parts of lacquer. Now, there are three dou (a traditional Chinese unit of volume) of lacquer, and one wants to separate a portion to exchange for oil, then mix the obtained oil with the remaining lacquer. How much lacquer should be separated, how much oil is obtained, and what is the total amount after mixing? The answer is: separate one dou and one sheng (another traditional Chinese unit of volume) plus one quarter of a sheng of lacquer, obtain one dou and five sheng of oil, and have a total of one dou and eight sheng plus three quarters of a sheng of lacquer after mixing. Mathematical Solution: Assuming we separate nine sheng of lacquer, it would be six sheng short. If we separate twelve sheng of lacquer, there would be an excess of two sheng.
〔一六〕今有玉方一寸，重七两；石方一寸，重六两。今有石立方三寸，中有玉，并重十一斤。问玉、石重各几何？
荅曰：玉一十四寸，重六斤二两。石一十三寸，重四斤十四两。
术曰：假令皆玉，多十三两。令之皆石，不足十四两。不足为玉，多为石。各以一寸之重乘之，得玉石之积重。
假设现在有一块玉，每方寸重7两；一块石头，每方寸重6两。现在有一块3立方寸的石头，中间含有玉，总重量是11斤。问玉和石各自的重量是多少？
答案是：玉14方寸，重6斤2两。石13方寸，重4斤14两。
使用的方法：
假设都是玉，则多出13两。
假设都是石，则不足14两。
不足的部分是玉的重量，多出的部分是石的重量。
分别用每方寸的重量乘以各自的体积，得到玉石的体积重量。
Suppose there is a piece of jade that weighs 7 liang per cubic inch, and a stone that weighs 6 liang per cubic inch. Now there is a stone with a volume of 3 cubic inches containing jade, with a total weight of 11 jin. What are the respective weights of the jade and the stone? The answer is: The jade has a volume of 14 cubic inches and weighs 6 jin and 2 liang. The stone has a volume of 13 cubic inches and weighs 4 jin and 14 liang. The method used: Assuming it's all jade, there would be an excess of 13 liang. Assuming it's all stone, there would be a shortfall of 14 liang. The shortfall represents the weight of the jade, and the excess represents the weight of the stone. Multiply each by the weight per cubic inch to get the total weight of the jade and stone.
〔一七〕今有善田一亩，价三百；恶田七亩，价五百。今并买一顷，价钱一万。问善、恶田各几何？
荅曰：善田一十二亩半，
恶田八十七亩半。术曰：假令善田二十亩，恶田八十亩，多一千七百一十四钱、七分钱之二。令之善田一十亩，恶田九十亩，不足五百七十一钱、七分钱之三。
假设现在有优质田地，每亩价值300钱；劣质田地，每亩价值500钱。现在要一起购买1顷（即100亩）的田地，总价为1万钱。问优质田地和劣质田地各有多少亩？
答案是：优质田地12亩半，劣质田地87亩半。
使用的方法：
假设购买20亩优质田地和80亩劣质田地，那么会多出1714钱又2/7钱。
假设购买10亩优质田地和90亩劣质田地，那么会不足571钱又3/7钱。
Suppose there are 20 pecks of poor-quality grain, which after milling yield 9 pecks of coarse rice. Now, to obtain 10 pecks of fine rice, how many pecks of poor-quality grain are needed? The answer is: 24 pecks plus 6 liters and 74/81 liters. The method is as follows: Multiply the 9 pecks of coarse rice by 9 to establish a standard. Also multiply the 10 pecks of fine rice by 10, then multiply again by the 20 pecks of poor-quality grain to find the actual amount of grain needed. The actual amount of grain needed divided by the standard gives the quantity per peck.
〔一八〕今有黄金九枚，白银一十一枚，称之重適等。交易其一，金轻十三两。问金、银一枚各重几何？
荅曰：金重二斤三两一十八銖，银重一斤十三两六銖。
术曰：假令黄金三斤，白银二斤、一十一分斤之五，不足四十九，於右行。令之黄金二斤，白银一斤、一十一分斤之七，多一十五於左行。以分母各乘其行內之数，以盈不足维乘所出率，并以为实。并盈不足为法。实如法，得黄金重。分母乘法以除，得银重。约之得分也。
假设现在有9枚黄金和11枚白银，它们的重量恰好相等。如果交换其中的一枚，黄金就会轻13两。问一枚黄金和一枚白银各重多少？
答案是：黄金重2斤3两18铢，白银重1斤13两6铢。
使用的方法：
假设黄金每枚重3斤，白银每枚重2斤加上 5/11 斤，这样计算出来的总重量不足49两，记在右侧。假设黄金每枚重2斤，白银每枚重1斤加上7/11斤，这样计算出来的总重量多出15两，记在左侧。
将分母乘以各自行内的数，用盈缺的差乘以所出现的比例，合并起来作为被除数。将盈缺相加作为除数。
用实际的除法得到黄金的重量。再用分母乘以除数来除，得到白银的重量。通过约分得到分数。
Suppose there are 9 pieces of gold and 11 pieces of silver, which weigh exactly the same. If one is exchanged, the gold will be light by 13 taels. How much does one piece of gold and one piece of silver weigh, respectively? The answer is: Gold weighs 2 jin, 3 liang, and 18 zhu, while silver weighs 1 jin, 13 liang, and 6 zhu. The method used: Assuming each gold piece weighs 3 jin, and each silver piece weighs 2 jin plus 5/11jin, this results in a total weight that is less than 49 taels, recorded on the right side. Assuming each gold piece weighs 2 jin, and each silver piece weighs 1 jin plus 7/11jin, this results in a total weight that exceeds by 15 taels, recorded on the left side. Multiply the denominator by the numbers within each row, use the difference between the surplus and the deficit multiplied by the emerging ratios, combine them as the dividend. Add the surplus and the deficit together as the divisor. Use the actual division to obtain the weight of the gold. Then divide again using the denominator multiplied by the divisor to get the weight of the silver. Obtain the fraction through reduction.
〔一九〕今有良马与駑马发长安至齐。齐去长安三千里。良马初日行一百九十三里，日增十三里。駑马初日行九十七里，日减半里。良马先至齐，復还迎駑马。问几何日相逢及各行几何？
荅曰：一十五日、一百九十一分日之一百三十五而相逢。良马行四千五百三十四里、一百九十一分里之四十六。
駑马行一千四百六十五里、一百九十一分里之一百四十五。术曰：假令十五日，不足三百三十七里半。令之十六日，多一百四十里。以盈、不足维乘假令之数，并而为实。并盈不足为法。实如法而一，得日数。不尽者，以等数除之而命分。
现在有良马和駑马从长安出发去齐国。齐国离长安三千里。良马第一天行一百九十三里，每天增加十三里。駑马第一天行九十七里，每天减少半里。良马先到达齐国，然后返回迎接駑马。问它们会在多少天后相遇，以及各自走了多少路程？
回答说：在第15天、第191天的135/191时相遇。良马走了4534里、第191天的46/191。駑马走了1465里、第191天的145/191。
There are two horses, one fast and one slow, departing from Chang'an to Qi. The distance to Qi from Chang'an is three thousand li. The fast horse travels one hundred and ninety-three li on the first day, increasing its daily distance by thirteen li each day. The slow horse travels ninety-seven li on the first day, with its daily distance decreasing by half a li each day. The fast horse arrives in Qi first and then returns to meet the slow horse. How many days will it take for them to meet, and how far will each have traveled? The answer is: they will meet on the 15th day, at 135/191 of the 191st day. The fast horse will have traveled 4534 li, which is 46/191 of the total distance for that day. The slow horse will have traveled 1465 li, which is 145/191 of the total distance for that day. Mathematical Solution: Assuming they meet on the fifteenth day, there would be a shortfall of three hundred and thirty-seven and a half li. If we assume they meet on the sixteenth day, there would be an excess of one hundred and forty li. We use the method of "ying and buzu" (surplus and deficiency) to deal with this problem. We multiply the surplus and deficiency by the assumed number of days and add them together as the actual total distance. Then we add up all the surpluses and deficiencies to form a 'method' (or standard). Dividing the actual total by this 'method', we find the number of days. Whatever is not accounted for in this division is dealt with by using the 'remainder' to calculate the fractional part of the day.
〔二０〕今有人持钱之蜀，贾利十三。初返归一万四千，次返归一万三千，次返归一万二千，次返归一万一千，后返归一万。凡五返归钱，本利俱尽。问本持钱及利各几何？
荅曰：本三万四百六十八钱、三十七万一千二百九十三分钱之八万四千八百七十六。利二万九千五百三十一钱、三十七万一千二百九十三分钱之二十八万六千四百一十七。
术曰：假令本钱三万，不足一千七百三十八钱半。令之四万，多三万五千三百九十钱八分。
现在有人拿钱去蜀地经商，获得的利润是本金的13%。第一次返回带回一万四千钱，第二次返回带回一万三千钱，第三次返回带回一万二千钱，第四次返回带回一万一千钱，最后一次返回带回一万块钱。总共五次返回，带回来的钱包括本金和利润都没有剩余。问最开始拿的本金和利润各是多少？
回答说：本金是三万四百六十八钱，利息是二万九千五百三十一钱。
Now, someone takes money to Shu for business, earning a profit of 13%. The first return brings back 14,000 cash, the second return brings back 13,000 cash, the third return brings back 12,000 cash, the fourth return brings back 11,000 cash, and the final return brings back 10,000 cash. In total, after five returns, all the money including the principal and interest has been brought back. How much was the initial principal and the profit? The answer is: the principal is 34,668 cash, and the interest is 29,531 cash. Mathematical Solution: Assuming the principal is 30,000 cash, there would be a shortfall of 1,738 cash and a half. If we assume 40,000 cash, there would be an excess of 35,390 cash and eight-tenths.