〔一〕今有上禾三秉，中禾二秉，下禾一秉，实三十九斗；上禾二秉，中禾三秉，下禾一秉，实三十四斗；上禾一秉，中禾二秉，下禾三秉，实二十六斗。问上、中、下禾实一秉各几何？
荅曰：
上禾一秉，九斗、四分斗之一，
中禾一秉，四斗、四分斗之一，
下禾一秉，二斗、四分斗之三。方程术曰，置上禾三秉，中禾二秉，下禾一秉，实三十九斗，於右方。中、左禾列如右方。以右行上禾遍乘中行而以直除。又乘其次，亦以直除。然以中行中禾不尽者遍乘左行而以直除。左方下禾不尽者，上为法，下为实。实即下禾之实。求中禾，以法乘中行下实，而除下禾之实。余如中禾秉数而一，即中禾之实。求上禾亦以法乘右行下实，而除下禾、中禾之实。余如上禾秉数而一，即上禾之实。实皆如法，各得一斗。
假设有上等禾三捆，中等禾两捆，下等禾一捆，总共打出39斗粮食；上等禾两捆，中等禾三捆，下等禾一捆，总共打出34斗粮食；上等禾一捆，中等禾两捆，下等禾三捆，总共打出26斗粮食。问上、中、下等禾每捆各能打多少斗粮食？
答案是：
上等禾每捆能打9又1/4斗，
中等禾每捆能打4又1/4斗，
下等禾每捆能打2又3/4斗。
解法是：
将“上等禾三捆，中等禾两捆，下等禾一捆，总共打出39斗粮食”列在右侧。中等和下等禾的排列方式与上等禾相同。用右侧上等禾的数量去乘中等禾所在的行，然后用竖式计算。接着乘以下一行，同样用竖式计算。然后用中等禾所在行未能整除的部分去乘下等禾所在的行，再用竖式计算。左侧下等禾未能整除的部分，上面作为除数，下面作为被除数。被除数就是下等禾的实际产量。求中等禾时，用除数乘中等禾所在行的下面的实际产量，然后除以下等禾的实际产量。余数按照中等禾的捆数来算，就是中等禾的实际产量。求上等禾也是用除数乘上等禾所在行的下面的实际产量，然后除以下等禾和中等禾的实际产量。余数按照上等禾的捆数来算，就是上等禾的实际产量。实际产量都按照这个算法来计算，每种类型的禾每捆都能打出整数斗的粮食。
Suppose there are three bundles of superior grain, two bundles of medium grain, and one bundle of inferior grain, yielding a total of 39 bushels; two bundles of superior grain, three bundles of medium grain, and one bundle of inferior grain, yielding a total of 34 bushels; and one bundle of superior grain, two bundles of medium grain, and three bundles of inferior grain, yielding a total of 26 bushels. What is the yield per bundle for each grade of grain? The answer is: Superior grain yields 9 and 1/4 bushels per bundle, Medium grain yields 4 and 1/4 bushels per bundle, Inferior grain yields 2 and 3/4 bushels per bundle. The solution method is: List "three bundles of superior grain, two bundles of medium grain, and one bundle of inferior grain, yielding a total of 39 bushels" on the right side. Arrange the medium and inferior grain in the same way as the superior grain. Multiply the row of medium grain by the number of superior grain on the right and perform vertical division. Then multiply the next row and perform vertical division as well. Next, multiply the left-hand side with the remainder from the medium grain row that did not divide evenly, and perform vertical division. The part of the inferior grain on the left that does not divide evenly is used as the divisor above and the dividend below. The dividend represents the actual yield of the inferior grain. To find the medium grain, multiply the divisor by the actual yield below the medium grain row and then divide by the actual yield of the inferior grain. The remainder, when divided by the number of medium grain bundles, gives the actual yield of the medium grain. The same process is applied to find the superior grain, using the divisor to multiply the actual yield below the superior grain row and then dividing by the actual yields of the inferior and medium grain. The remainder, when divided by the number of superior grain bundles, gives the actual yield of the superior grain. The actual yields are calculated according to this method, ensuring that each type of grain yields an integer number of bushels per bundle.
〔二〕今有上禾七秉，损实一斗，益之下禾二秉，而实一十斗。下禾八秉，益实一斗与上禾二秉，而实一十斗。问上、下禾实一秉各几何？
荅曰：
上禾一秉实一斗、五十二分斗之一十八，下禾一秉实五十二分斗之四十一。术曰：如方程。损之曰益，益之曰损。损实一斗者，其实过一十斗也。益实一斗者，其实不满一十斗也。
今有上等的禾七把，减少其实际容量一斗，加上下等的禾两把，而实际容量为十斗。下等的禾八把，增加其实际容量一斗与上等的禾两把，而实际容量为十斗。问上、下等的禾每把的实际容量各是多少？
答案是：
上等的禾每把的实际容量是1斗又52分之18，下等的禾每把的实际容量是52分之41。方法如下：依照方程式。减少的称为益，增加的称为损。减少实际容量一斗的，其实际容量超过10斗。增加实际容量一斗的，其实际容量不满10斗。
Now, there are seven measures of superior grain that, when reduced by one pint in actual content, combined with two measures of inferior grain, the total actual content becomes ten pints. For eight measures of inferior grain, if we add one pint to its actual content and combine it with two measures of superior grain, the total actual content will also be ten pints. What is the actual content for each measure of the superior and inferior grain? Answer: The actual content for one measure of superior grain is 1 pint and 18/52nds of a pint; the actual content for one measure of inferior grain is 41/52nds of a pint. The method is as follows: according to the equation. A reduction is termed an increase, and an increase is termed a reduction. Those that have one pint reduced in actual content exceed ten pints in reality. Those that have one pint added in actual content do not fill up ten pints in reality.
〔三〕今有上禾二秉，中禾三秉，下禾四秉，实皆不满斗。上取中，中取下，下取上各一秉而实满斗。问上、中、下禾实一秉各几何？
荅曰：上禾一秉实二十五分斗之九，
中禾一秉实二十五分斗之七，
下禾一秉实二十五分斗之四。术曰：如方程，各置所取，以正负术入之。
正负术曰：同名相除，异名相益，正无入负之，负无入正之。其异名相除，同名相益，正无入正之，负无入负之。
现在有上等的禾二捆，中等的禾三捆，下等的禾四捆，它们的籽粒都不满一斗。从上等的取中等的，中等的取下等的，下等的取上等的每一捆，然后籽粒就满了一斗。问上、中、下等的禾每一捆绑满斗时的籽粒分别是多少？
答案是：上等的禾每一捆满斗时是二十五分之九斗，中等的禾每一捆满斗时是二十五分之七斗，下等的禾每一捆满斗时是二十四分之一斗。方法是：像方程一样，把各自所取的量放好，用正负术计算。
正负术的规则是：相同的数相减，不同的数相加，正数不入负数，负数不入正数；如果它们符号不同则相减，相同则相加，正数不入正数，负数不入负数。
Now there are two bundles of superior grain, three bundles of medium grain, and four bundles of inferior grain; none of them fill a bushel. Taking one bundle from the superior, the medium, and the inferior respectively makes exactly one full bushel. What is the amount of grain per bundle for the superior, medium, and inferior grain that fills a bushel? Answer: One bundle of the superior grain equals nine-twentieths of a bushel, medium grain equals seven-twentieths of a bushel, and inferior grain equals one-quarter of a bushel. The method is like solving an equation, placing the taken amounts accordingly and using the rule of positive and negative numbers. The rule for positive and negative numbers states: subtract numbers with the same sign, add numbers with different signs; positive does not enter negative, negative does not enter positive. If their signs differ, subtract; if the same, add; positive does not enter positive, negative does not enter negative.
〔四〕今有上禾五秉，损实一斗一升，当下禾七秉。上禾七秉，损实二斗五升，当下禾五秉。问上、下禾实一秉各几何？
荅曰：
上禾一秉五升，
下禾一秉二升。术曰：如方程，置上禾五秉正，下禾七秉负，损实一斗一升正。次置上禾七秉正，下禾五秉负，损实二斗五升正。以正负术入之。
现在有上等的禾五捆，减少籽粒一斗一升，相当于下等的禾七捆。上等的禾七捆，减少籽粒二斗五升，相当于下等的禾五捆。问上、下等的禾每一捆绑满斗时的籽粒分别是多少？
答案是：上等的禾每一捆是五升，下等的禾每一捆是二升。方法是：像方程一样，把上等的五捆绑满斗时设为正数，下等的七捆设为负数，减少的籽粒一斗一升设为正数。再将上等的七捆绑满斗时设为正数，下等的五捆设为负数，减少的籽粒二斗五升设为正数。然后用正负术计算。
Now there are five bundles of superior grain, which, when reduced by one peal and one liter of grain, equals seven bundles of inferior grain. Seven bundles of superior grain, when reduced by two peals and five liters of grain, equals five bundles of inferior grain. What is the amount of grain per bundle for the superior and inferior grain? Answer: Each bundle of the superior grain contains five liters, and each bundle of the inferior grain contains two liters. The method is to set up the equation as follows: assign a positive value to five bundles of the superior grain, a negative value to seven bundles of the inferior grain, and a positive value to the reduction of one peal and one liter of grain. Then assign a positive value to seven bundles of the superior grain, a negative value to five bundles of the inferior grain, and a positive value to the reduction of two peals and five liters of grain. Then calculate using the rule of positive and negative numbers.
〔五〕今有上禾六秉，损实一斗八升，当下禾一十秉。下禾十五秉，损实五升，当上禾五秉。问上、下禾实一秉各几何？
荅曰：
上禾一秉实八升，
下禾一秉实三升。
术曰：如方程，置上禾六秉正，下禾一十秉负，损实一斗八升正。次置上禾五秉负，下禾一十五秉正，损实五升正。以正负术人之。
现有上等禾六把，减少其实际容量一斗八升，相当于下等禾十把。下等禾十五把，减少其实际容量五升，相当于上等禾五把。问上、下等禾每把的实际容量各是多少？
答案是：
上等禾每把的实际容量是八升，
下等禾每把的实际容量是三升。
方法是：按照方程，将上等禾六把设为正数，下等禾十把设为负数，减少的实际容量一斗八升也设为正数。然后将上等禾五把设为负数，下等禾十五把设为正数，减少的实际容量五升也设为正数。用正负数的方法计算得出结果。
Now, there are six measures of superior grain that, when reduced by 1 pint and 8 liters in actual content, is equivalent to ten measures of inferior grain. For fifteen measures of inferior grain, if we reduce its actual content by 5 liters, it is equivalent to five measures of superior grain. What is the actual content for each measure of the superior and inferior grain? Answer: The actual content for one measure of superior grain is 8 liters, and the actual content for one measure of inferior grain is 3 liters. The method is as follows: according to the equation, set six measures of superior grain as positive, ten measures of inferior grain as negative, with a reduction of 1 pint and 8 liters as positive. Then set five measures of superior grain as negative, fifteen measures of inferior grain as positive, with a reduction of 5 liters also as positive. Use the method of positive and negative numbers to calculate the result.
〔六〕今有上禾三秉，益实六斗，当下禾十秉。下禾五秉，益实一斗，当上禾二秉。问上、下禾实一秉各几何？
荅曰：
上禾一秉实八斗，
下禾一秉实三斗。
术曰：如方程，置上禾三秉正，下禾一十秉负，益实六斗负。次置上禾二秉负，下禾五秉正，益实一斗负。以正负术入之。
现在有上等的禾三捆，增加籽粒六斗，相当于下等的禾十捆。下等的禾五捆，增加籽粒一斗，相当于上等的禾两捆。问上、下等的禾每一捆绑满斗时的籽粒分别是多少？
答案是：上等的禾每一捆是八斗，下等的禾每一捆是三斗。方法是：像方程一样，把上等的三捆绑满斗时设为正数，下等的十捆设为负数，增加的籽粒六斗设为负数。再将上等的二捆绑满斗时设为负数，下等的五捆设为正数，增加的籽粒一斗设为负数。然后用正负术计算。
Now there are three bundles of superior grain, which, when increased by six peals of grain, equals ten bundles of inferior grain. Five bundles of inferior grain, when increased by one peal of grain, equals two bundles of superior grain. What is the amount of grain per bundle for the superior and inferior grain? Answer: Each bundle of the superior grain contains eight peals, and each bundle of the inferior grain contains three peals. The method is to set up the equation as follows: assign a positive value to three bundles of the superior grain, a negative value to ten bundles of the inferior grain, and a negative value to the increase of six peals of grain. Then assign a negative value to two bundles of the superior grain, a positive value to five bundles of the inferior grain, and a negative value to the increase of one peal of grain. Then calculate using the rule of positive and negative numbers.
〔七〕今有牛五、羊二，直金十两。牛二、羊五直金八两。问牛羊各直金几何？
荅曰：
牛一，直金一两、二十一分两之一十三，
羊一，直金二十一分两之二十。
术曰：如方程。
现在有五头牛和两只羊，价值十两金。两头牛和五只羊价值八两金。问一头牛和一只羊各值多少金？
答案是：
一头牛的价值是1两金和21分之13，
一只羊的价值是21分之20。
方法是：按照方程来解决。
Now, there are five oxen and two sheep valued at ten taels of gold. Two oxen and five sheep are valued at eight taels of gold. What is the value of one ox and one sheep in gold? Answer: One ox is valued at 1 tael and 21/13 of a tael of gold, One sheep is valued at 20/21 of a tael of gold. The method is as follows: solve it according to the equation.
〔八〕今有卖牛二、羊五，以买十三豕，有余钱一千。卖牛三、豕三，以买九羊，钱適足。卖羊六、豕八，以买五牛，钱不足六百。问牛、羊、豕价各几何？
荅曰：
牛价一千二百，
羊价五百，豕价三百。
术曰：如方程，置牛二、羊五正，豕一十三负，余钱数正；次牛三正，羊九负，豕三正；次牛五负，羊六正，豕八正，不足钱负。以正负术入之。
现在卖二头牛、五只羊，用来买十三头猪，还剩下一千钱。卖三头牛、三头猪，用来买九只羊，钱刚好够。卖六只羊、八头猪，用来买五头牛，钱不够，差六百钱。问牛、羊、猪的价格分别是多少？
答案是：牛的价格是一千二百钱，羊的价格是五百钱，猪的价格是三百钱。方法是：像方程一样，把卖二头牛、五只羊设为正数，买十三头猪设为负数，剩下的钱数设为正数；然后卖三头牛设为正数，买九只羊设为负数，买三头猪设为正数；再卖五头牛设为负数，买六只羊设为正数，买八头猪设为正数，不够的钱设为负数。然后用正负术计算。
Now selling two cows and five sheep to buy thirteen pigs leaves a surplus of one thousand coins. Selling three cows and three pigs to buy nine sheep just covers the cost. Selling six sheep and eight pigs to buy five cows is not enough, with a shortfall of six hundred coins. What are the prices for the cow, sheep, and pig, respectively? Answer: The price of the cow is one thousand two hundred coins, the price of the sheep is five hundred coins, and the price of the pig is three hundred coins. The method is to set up the equation as follows: assign positive values to selling two cows and five sheep, negative values to buying thirteen pigs, and a positive value to the surplus money; then assign a positive value to selling three cows, a negative value to buying nine sheep, and a positive value to buying three pigs; then assign a negative value to selling five cows, a positive value to buying six sheep, and a positive value to buying eight pigs, with a negative value for the shortfall of money. Then calculate using the rule of positive and negative numbers.
〔九〕今有五雀、六燕，集称之衡，雀俱重，燕俱轻。一雀一燕交而处，衡適平。并燕、雀重一斤。问燕、雀一枚各重几何？
荅曰：
雀重一两、一十九分两之十三，
燕重一两、一十九分两之五。
术曰：如方程，交易质之，各重八两。
现在有五只雀和六只燕，它们聚集在称重的天平上。所有的雀都比燕重，所有的燕都比雀轻。一只雀和一只燕交替站在天平的两边时，天平恰好平衡。所有燕和雀的总重量是一斤（16两）。问一只燕和一只雀各重多少？
答案是：
一只雀的重量是1两又19分之13，
一只燕的重量是1两又19分之5。
方法是：按照方程来解决，将它们交换位置来确定它们的准确重量，每边总重8两。
Now, there are five sparrows and six swallows that gather on a balance scale. All the sparrows are heavier than the swallows. When one sparrow and one swallow switch places, the balance scale is exactly balanced. The combined weight of all the swallows and sparrows is one jin (which is 16 taels). How much does each swallow and sparrow weigh? Answer: One sparrow weighs 1 tael and 13/19 of a tael, One swallow weighs 1 tael and 5/19 of a tael. The method is as follows: solve it according to the equation by trading their positions to determine their exact weights, with each side having a total weight of 8 taels.
〔十〕今有甲乙二人持钱不知其数。甲得乙半而钱五十，乙得甲太半而亦钱五十。问甲、乙持钱各几何？
荅曰：
甲持三十七钱半，乙持二十五钱。术曰：如方程，损益之。
现在有甲乙两个人，不知道他们各自持有多少钱。甲得到乙的一半的钱后剩下五十钱，乙得到甲的太半（即三分之一）的钱后也剩下五十钱。问甲、乙各自持有多少钱？
答案是：甲持有三十七钱半，乙持有二十五钱。方法是：像方程一样，对甲乙的钱进行损益处理。
Now there are two people, A and B, who have an unknown amount of money. Person A, after receiving half of B's money, has fifty coins left. Person B, after receiving one-third (tooban) of A's money, also has fifty coins left. How much money do A and B each have? Answer: Person A has thirty-seven and a half coins, and person B has twenty-five coins. The method is to adjust the amounts of money for A and B as if solving an equation.
〔一一〕今有二马、一牛价过一万，如半马之价。一马、二牛价不满一万，如半牛之价。问牛、马价各几何？
荅曰：马价五千四百五十四钱、一十一分钱之六，牛价一千八百一十八钱、一十一分钱之二。
术曰：如方程，损益之
现在有两匹马和一头牛的价格超过一万，且等于半匹马的价格。一匹马和两头牛的价格不足一万，且等于半头牛的价格。问一头牛和一匹马的价格各是多少？
答案是：
一匹马的价格是5454钱又1/11钱的6分之一，
一头牛的价格是1818钱又1/11钱的2分之一。
方法是：按照方程来解决，通过增加或减少来确定价格。
Now, the price of two horses and one ox exceeds ten thousand, which is equivalent to half the price of a horse. The price of one horse and two oxen is less than ten thousand, which is equivalent to half the price of an ox. What are the prices of an ox and a horse? Answer: The price of a horse is 5454 qian and 1/11 qian divided by six, The price of an ox is 1818 qian and 1/11 qian divided by two. The method is as follows: solve it according to the equation by adjusting the values accordingly.
〔一二〕今有武马一匹，中马二匹，下马三匹，皆载四十石至阪，皆不能上。武马借中马一匹，中马借下马一匹，下马借武马一匹，乃皆上。问武、中、下马一匹各力引几何？
荅曰：
武马一匹力引二十二石、七分石之六，
中马一匹力引十七石、七分石之一，下马一匹力引五石、七分石之五。
术曰：如方程各置所借，以正负术入之。
现在有一匹武马，两匹中马，三匹下马，都载着四十石的货物到山坡上，都不能上去。武马借了一匹中马，中马借了一匹下马，下马借了一匹武马，这样它们就都能上山坡了。问武马、中马、下马每一匹各自能拉多少石的货物？
答案是：武马一匹能拉二十二石又七分之一石的重量，中马一匹能拉十七石又七分之一石的重量，下马一匹能拉五石又七分之五石的重量。
方法是：像方程一样，把借来的马的数量设为未知数，用正负术计算。
Now there is one war horse, two middle horses, and three lower horses, all carrying forty stone loads to the hillside, none of them able to ascend. The war horse borrows one middle horse, the middle horse borrows one lower horse, and the lower horse borrows one war horse, then they all manage to climb up. How much weight can each type of horse pull? Answer: One war horse can pull twenty-two and one sixth of a stone, one middle horse can pull seventeen and one seventh of a stone, and one lower horse can pull five and five sevenths of a stone. The method is to set up the equation like setting unknowns for the number of horses borrowed and calculate using the rule of positive and negative numbers.
〔一三〕今有五家共井，甲二綆不足，如乙一綆；乙三綆不足，如丙一綆；丙四綆不足，如丁一綆；丁五綆不足，如戊一綆；戊六綆不足，如甲一綆。如各得所不足一綆，皆逮。问井深、綆长各几何？
荅曰：井深七丈二尺一寸。
甲綆长二丈六尺五寸，
乙綆长一丈九尺一寸，
丙綆长一丈四尺八寸，
丁綆长一丈二尺九寸，
戊綆长七尺六寸。
术曰：如方程，以正负术入之。
现在有五家人共用一口井，甲的两根绳索不够深，差一根绳索；乙的三根绳索不够深，差一根绳索；丙的四根绳索不够深，差一根绳索；丁的五根绳索不够深，差一根绳索；戊的六根绳索不够深，差一根绳索。如果每家都得到所差的那根绳索，就都能达到井底。问井有多深，各家的绳索各有多长？
答案是：
井深7丈2尺1寸。
甲的绳索长2丈6尺5寸，
乙的绳索长1丈9尺1寸，
丙的绳索长1丈4尺8寸，
丁的绳索长1丈2尺9寸，
戊的绳索长7尺6寸。
方法是：按照方程来解决，用正负数的方法计算。
Now, there are five households sharing a well. Household A's two ropes are not long enough, lacking the length of one rope to reach the bottom; Household B's three ropes are not long enough, lacking one rope; Household C's four ropes are not long enough, lacking one rope; Household D's five ropes are not long enough, lacking one rope; Household E's six ropes are not long enough, lacking the length of one rope to reach the bottom. If each household obtains the length of the missing rope, they will all reach the bottom of the well. What is the depth of the well and the length of each household's rope? Answer: The depth of the well is 7 zhang, 2 chi, and 1 cun. Household A's rope is 2 zhang, 6 chi, and 5 cun long, Household B's rope is 1 zhang, 9 chi, and 1 cun long, Household C's rope is 1 zhang, 4 chi, and 8 cun long, Household D's rope is 1 zhang, 2 chi, and 9 cun long, Household E's rope is 7 chi and 6 cun long. The method is as follows: solve it according to the equation by using the method of positive and negative numbers.
〔一四〕今有白禾二步、青禾三步、黄禾四步、黑禾五步，实各不满斗。白取青、黄，青取黄、黑，黄取黑、白，黑取白、青，各一步，而实满斗。问白、青、黄、黑禾实一步各几何？
荅曰：
白禾一步实一百一十一分斗之三十三，
青禾一步实一百一十一分斗之二十八，
黄禾一步实一百一十一分斗之一十七，
黑禾一步实一百一十一分斗之一十。
术曰：如方程，各置所取，以正负术入之。
现在有白禾两步、青禾三步、黄禾四步、黑禾五步，各自的籽粒都不满一斗。白禾从青禾和黄禾那里各取一步，青禾从黄禾和黑禾那里各取一步，黄禾从黑禾和白禾那里各取一步，黑禾从白禾和青禾那里各取一步，这样它们各自的籽粒就满一斗了。问白禾、青禾、黄禾、黑禾每走一步各自的籽粒有多少？
答案是：白禾每走一步的籽粒是一百十一分之三十三斗，青禾每走一步的籽粒是一百十一分之二十八斗，黄禾每走一步的籽粒是一百十一分之十七斗，黑禾每走一步的籽粒是一百十一分之十斗。
方法是：像方程一样，把各自取出的籽粒数量设为未知数，用正负术计算。
Now there are two steps of white grain, three steps of green grain, four steps of yellow grain, and five steps of black grain, each with less than a full bushel of grain. The white grain takes one step each from the green and yellow grains, the green grain takes one step each from the yellow and black grains, the yellow grain takes one step each from the black and white grains, and the black grain takes one step each from the white and green grains, after which they each have a full bushel of grain. How much grain does each type of grain produce per step? Answer: White grain produces thirty-three parts out of one hundred and eleven per step, green grain produces twenty-eight parts out of one hundred and eleven per step, yellow grain produces seventeen parts out of one hundred and eleven per step, and black grain produces ten parts out of one hundred and eleven per step. The method is to set up the equation like setting unknowns for the amount taken from each type of grain and calculate using the rule of positive and negative numbers.
〔一五〕今有甲禾二秉、乙禾三秉、丙禾四秉，重皆过於石。甲二重如乙一，乙三重如丙一，丙四重如甲一。
问甲、乙、丙禾一秉各重几何？荅曰：
甲禾一秉重二十三分石之十七，乙禾一秉重二十三分石之十一，
丙禾一秉重二十三分石之十。
术曰：如方程，置重过於石之物为负。以正负术入之。
现在有甲禾两束、乙禾三束、丙禾四束，它们的重量都超过一石。甲禾的两倍重量等于乙禾的一倍重量，乙禾的三倍重量等于丙禾的一倍重量，丙禾的四倍重量等于甲禾的一倍重量。问甲禾、乙禾、丙禾每束各重多少？
答案是：甲禾一束重二十三分之十七石，乙禾一束重二十三分之十一石，丙禾一束重二十三分之十石。
方法是：像方程一样，把超过一石的重量设为负数。用正负术计算。
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.
〔一六〕今有令一人、吏五人、从者一十人，食鸡一十；令一十人、吏一人、从者五人，食鸡八；令五人、吏一十人、从者一人，食鸡六。问令、吏、从者食鸡各几何？
荅曰：
令一人食一百二十二分鸡之四十五，
吏一人食一百二十二分鸡之四十一，
从者一人食一百二十二分鸡之九十七。术曰：如方程，以正负术入之。
现在有一位令、五位吏和十位随从，他们一共吃了10只鸡；有十位令、一位吏和五位随从，他们一共吃了8只鸡；有五位令、十位吏和一位随从，他们一共吃了6只鸡。问一位令、一位吏和一位随从各吃了多少只鸡？
答案是：一位令吃了122/45只鸡，一位吏吃了122/41只鸡，
一位随从吃了 122/97只鸡。
方法是：按照方程来解决，用正负数的方法计算。
Now, there is one official, five clerks, and ten attendants who together eat ten chickens; ten officials, one clerk, and five attendants eat eight chickens together; five officials, ten clerks, and one attendant eat six chickens together. How many chickens does an official, a clerk, and an attendant each eat? Answer:One official eats 122/45 chickens,One clerk eats 122/41 chickens,One attendant eats 122/97 chickens. The method is as follows: solve it according to the equation by using the method of positive and negative numbers.
〔一七〕今有五羊、四犬、三鸡、二兔，直钱一千四百九十六；四羊、二犬、六鸡、三兔直钱一千一百七十五；三羊、一犬、七鸡、五兔，直钱九百五十八；二羊、三犬、五鸡、一兔，直钱八百六十一。问羊、犬、鸡、兔价各几何？
荅曰：
羊价一百七十七，
犬价一百二十一，
鸡价二十三，
兔价二十九。
术曰：如方程，以正负术入之。
现在有五只羊、四只狗、三只鸡、两只兔子，总共值一千四百九十六文钱；四只羊、两只狗、六只鸡、三只兔子总共值一千一百七十五文钱；三只羊、一只狗、七只鸡、五只兔子总共值九百五十八文钱；两只羊、三只狗、五只鸡、一只兔子总共值八百六十一文钱。问羊、狗、鸡、兔的价格各是多少？
答案是：羊每只一百七十七文钱，狗每只一百二十一文钱，鸡每只二十三文钱，兔子每只二十九文钱。
方法是：像方程一样，使用正负术进行计算。
Now there are five sheep, four dogs, three chickens, and two rabbits, valued at 1496 wen (a unit of currency); four sheep, two dogs, six chickens, and three rabbits valued at 1175 wen; three sheep, one dog, seven chickens, and five rabbits valued at 958 wen; two sheep, three dogs, five chickens, and one rabbit valued at 861 wen. What is the price of each sheep, dog, chicken, and rabbit? Answer: The price of a sheep is 177 wen, a dog is 121 wen, a chicken is 23 wen, and a rabbit is 29 wen. The method is to set up the equation like setting unknowns for the prices and calculate using the rule of positive and negative numbers.
〔一八〕今有麻九斗、麦七斗、菽三斗、荅二斗、黍五斗，直钱一百四十；麻七斗、麦六斗、菽四斗、荅五斗、黍三斗，直钱一百二十八；麻三斗、麦五斗、菽七斗、荅六斗、黍四斗，直钱一百一十六；麻二斗、麦五斗、菽三斗、荅九斗、黍四斗，直钱一百一十二；麻一斗、麦三斗、菽二斗、荅八斗、黍五斗，直钱九十五。问一斗直几何？
荅曰：
麻一斗七钱，
麦一斗四钱，
菽一斗三钱，
荅一斗五钱，
黍一斗六钱。
术曰：如方程，以正负术入之。
现有麻9斗、麦7斗、菽3斗、荅2斗、黍5斗，价值140钱；麻7斗、麦6斗、菽4斗、荅5斗、黍3斗，价值128钱；麻3斗、麦5斗、菽7斗、荅6斗、黍4斗，价值116钱；麻2斗、麦5斗、菽3斗、荅9斗、黍4斗，价值112钱；麻1斗、麦3斗、菽2斗、荅8斗、黍5斗，价值95钱。问每斗各值多少钱？
答案是：
麻一斗值7钱，
麦一斗值4钱，
菽一斗值3钱，
荅一斗值5钱，
黍一斗值6钱。
方法是：按照方程来解决，用正负数的方法计算。
Now, there are 9 dou of hemp, 7 dou of wheat, 3 dou of beans, 2 dou of vetch, and 5 dou of millet worth 140 qian; 7 dou of hemp, 6 dou of wheat, 4 dou of beans, 5 dou of vetch, and 3 dou of millet worth 128 qian; 3 dou of hemp, 5 dou of wheat, 7 dou of beans, 6 dou of vetch, and 4 dou of millet worth 116 qian; 2 dou of hemp, 5 dou of wheat, 3 dou of beans, 9 dou of vetch, and 4 dou of millet worth 112 qian; 1 dou of hemp, 3 dou of wheat, 2 dou of beans, 8 dou of vetch, and 5 dou of millet worth 95 qian. How much is each dou worth? Answer: One dou of hemp is worth 7 qian, One dou of wheat is worth 4 qian, One dou of beans is worth 3 qian, One dou of vetch is worth 5 qian, One dou of millet is worth 6 qian. The method is as follows: solve it according to the equation by using the method of positive and negative numbers.