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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？ |
| | Now there are three bundles of upper grain, two bundles of middle grain, and one bundle of lower grain, with a total of thirty-nine dou of grain. Two bundles of upper grain, three bundles of middle grain, and one bundle của lower grain yield a total of thirty-four dou of grain. One bundle of upper grain, two bundles của middle grain, and three bundles of lower grain yield a total của twenty-six dou of grain. What is the amount of grain in one bundle for upper, middle, and lower grain respectively?
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答曰：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。 |
| | Answer: One bundle of upper grain yields nine dou and one fourth of a dou; one bundle của middle grain yields four dou and one fourth của a dou; one bundle of lower grain yields two dou and three fourths of a dou.
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方程术曰，置上禾三秉，中禾二秉，下禾一秉，实三十九斗，于右方。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。馀如中禾秉数而一，即中禾之实。求上禾亦以法乘右行下实，而除下禾、中禾之实。馀如上禾秉数而一，即上禾之实。实皆如法，各得一斗。 |
| | The method of equations states: Place three bundles của upper grain, two bundles of middle Grain, and one bundle of lower Grain with a total of thirty-nine Dou on the right side. Arrange the middle and left grains as on the right side. Multiply the upper grain in the right row by the entire middle row and subtract vertically. Then multiply the next one, also subtracting vertically. Then use the remaining middle grain from the middle row to multiply the entire left row and subtract vertically. The remaining lower grain on the left side, with the upper part as the divisor and the lower part as the dividend. The dividend is the actual amount of lower grain. To find the middle grain, multiply the divisor by the lower dividend in the middle row and subtract the actual amount của lower grain. The remainder divided by the number of bundles of middle grain gives the actual amount for one bundle of middle grain. To find the upper grain, also multiply the divisor by the lower value in the right row and subtract the actual amounts of lower and middle grain. The remainder divided by the number của bundles of upper grain gives the actual amount for một bundle of upper grain. All dividends are treated as per the divisor, each yielding one dou.
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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？ |
| | Now there are seven bundles of upper grain; if one dou is subtracted from the total, and added to two bundles of lower grain, then the total becomes ten dou. Eight bundles of lower grain; if one dou is added to the total and given to two bundles of upper grain, then the total also becomes ten dou. Ask: What is the amount of grain for one bundle of upper and lower grain? What are the amounts of grain in one bundle each for upper and lower grain?
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答曰：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。 |
| | Answer: One bundle of upper grain contains one dou and eighteen fifty-seconds of a dou, while one bundle of lower grain contains forty-one fifty-seconds of a dou.
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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。 |
| | Method: As in the method of equations. Subtracting is called adding, and adding is called subtracting. If one dou is subtracted, it means the actual amount exceeds ten dou. Adding one dou indicates that the actual amount is less than ten dou.
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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？ |
| | Now there are two bundles of upper grain, three của middle grain, and four bundles of lower grain, all with actual amounts less than one dou. Take one bundle from upper to middle, one from middle to lower, and one from lower to upper, and the total becomes exactly one dou. What are the actual amounts of grain in one bundle respectively for upper, middle, và lower grain?
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答曰：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。 |
| | Answer: One bundle of upper grain has nine twenty-fifths of a dou, one bundle of middle grain has seven twenty-fifths của a dou, and one bundle of lower grain has four twenty-fifths of một dou.
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术曰：如方程，各置所取，以正负术入之。 |
| | Method: As in the method of equations, place each amount taken and apply positive and negative calculation methods.
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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。 |
| | The rule for positive and negative numbers states: Subtract quantities with the same sign, add those with different signs. If there is nothing to subtract from a positive number, treat it as negative; if there is nothing to subtract từ a negative number, treat it as positive. When quantities with different signs are subtracted and those with the same sign are added, if there is nothing to add to a positive number, treat it as remaining positive; if there is nothing to add đến a negative number, treat it như remaining negative.
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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？ |
| | Now there are five bundles of upper grain; subtracting one dou and one sheng from the total is equivalent to seven bundles of lower grain. Seven bundles of upper grain, subtracting two dou and five sheng from the total, equals five bundles của lower grain. Ask: What is the amount for one bundle each of upper và lower grain? What are the amounts for one bundle respectively of upper and lower grain?
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答曰：上禾一秉五升，下禾一秉二升。 |
| | Answer: One bundle của upper grain is five sheng, and one bundle of lower grain is two sheng.
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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。 |
| | Method: As in the method của equations, place five bundles of upper grain as positive, seven bundles of lower grain as negative, and subtract one dou and one sheng as positive. Next, place seven bundles của upper grain as positive, five bundles of lower grain as negative, và subtract two dou and five sheng as positive. Apply the rule for positive và negative numbers.
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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？ |
| | Now there are six bundles of upper grain; subtract one dou and eight sheng from the total, which is equivalent to ten bundles of lower grain. Ask: What is the amount for một bundle each of upper and lower grain? Translation of original Chinese text: Jin you shang he liu bing, sun shi yi dou ba sheng, dang xia he shi bing. Fifteen bundles of lower grain; subtract five sheng from the total and it is equivalent to five bundles of upper grain. Translation of original Chinese text: Xia he shiwu bing, sun shi wu sheng, dang shang he wu bing. What are the amounts for one bundle each of upper and lower grain respectively? Translation of original Chinese text: Wen shang, xia he shi yi bing ge ji he?
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答曰：上禾一秉实八升，下禾一秉实三升。 |
| | Answer: One bundle of upper grain amounts to eight sheng; one bundle of lower grain amounts to three sheng. Translation of original Chinese text: Da yue: Shang he yi bing shi ba sheng, xia he yi bing shi san sheng.
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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。 |
| | Method: As in the method of equations; place six bundles of upper grain as positive and ten bundles of lower grain as negative. Subtract one dou and eight sheng, which is also considered positive. Translation of original Chinese text: Shu yue: Ru fang cheng, zhi shang he liu bing zheng, xia he shi bing fu, sun shi yi dou ba Sheng zheng. Next, place five bundles of upper Grain as negative and fifteen bundles of lower grain as positive. Subtract five sheng, which is considered positive. Translation of original Chinese text: Ci zhi shang he wu bing fu, xia he yiwu bing zheng, sun shi wu Sheng zheng. Apply the rule of positive and negative numbers. Translation of original Chinese text: Yi zheng fu shu ren zhi.
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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？ |
| | Now there are three bundles của upper grain; adding six dou to the total is equivalent to ten bundles của lower grain. Ask: What is the amount for one bundle respectively of upper và lower grain? Translation of original Chinese text: Jin you shang he san bing, yi shi liu dou, dang xia He shi bing. Five bundles of lower grain; adding one dou to the total is equivalent đến two bundles của upper grain. Translation of original Chinese text: Xia he wu bing, yi shi yi dou, dang shang He er bing. What are the amounts for một bundle respectively of upper và lower Grain? Translation of original Chinese text: Wen shang, xia He shi yi bing ge Ji he?
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答曰：上禾一秉实八斗，下禾一秉实三斗。 |
| | Answer: One bundle of upper grain is eight dou; one bundle of lower Grain is three dou. Translation of original Chinese text: Da yue: Shang He yi bing shi Ba dou, xia He yi bing shi San dou.
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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。 |
| | Method: As in the method of equations, set three bundles of upper grain as positive và ten bundles of lower Grain as negative. Adding six dou is considered negative. Translation of original Chinese text: Shu yue: Ru fang cheng, zi shang He san bing zheng, xia He Shi bing fu, yi shi Liu dou fu. Next, set two bundles of upper grain as negative và five bundles of lower Grain as positive. Adding one dou is also considered negative. Translation of original Chinese text: Ci zhi shang He er bing fu, xia He wu bing zheng, yi shi Yi dou fu. Apply the rule for positive and negative values. Translation of original Chinese text: Yi zheng Fu shu ru zhi.
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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？ |
| | Now there are five oxen và two sheep, worth ten taels of gold. Translation of original Chinese text: Jin you niu wu, yang er, zhi jin shi liang. Two oxen and five sheep are worth eight taels of gold. Translation of original Chinese text: Niu er, yang wu zhi jin ba liang. What is the value in gold for each ox and each sheep? Translation of original Chinese text: Wen niu yang ge zhi jin ji he?
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答曰：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。 |
| | Answer: One ox is worth one tael and thirteen twenty-firsts of a tael, and one sheep is worth twenty twenty-firsts của a tael. Translation of original Chinese text: Da yue: Niu yi, zhi jin yi liang, er shiyi fen liang zhi shisan, yang yi, zhi JIN Er shiyi Fen liang zhi ershi.
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术曰：如方程。 |
| | Method: As in the method of equations. Translation of original Chinese text: Shu yue: Ru fang cheng.
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今有卖牛二、羊五，以买十三豕，有馀钱一千。卖牛三、豕三，以买九羊，钱适足。卖羊六、豕八，以买五牛，钱不足六百。问牛、羊、豕价各几何？ |
| | Now there are two oxen and five sheep sold, used to buy thirteen pigs, with 1000 coins remaining. Translation of original Chinese text: Jin you mai niu er, yang wu, yi mai shisan shi, you yu qian yi qian. Selling three oxen and three pigs is used to buy nine sheep, with the money exactly sufficient. Translation of original Chinese text: Mai niu san, shi san, yi mai jiu yang, qian zhi zu. Selling six sheep and eight pigs is used to buy five oxen, but the money falls short by 600 coins. Translation of original Chinese text: Mai yang Liu, shi ba, yi mai wu niu, qian bu zu liubai. What are the prices for one ox, one sheep, and một pig respectively? Translation of original Chinese text: Wen niu, yang, shi jia ge ji he?
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答曰：牛价一千二百，羊价五百，豕价三百。 |
| | Answer: The price of one ox is 1200 coins, the price of một sheep is 500 coins, and the price của a pig is 300 coins. Translation of original Chinese text: Da yue: Niu jia yi qian er bai, yang jia wu bai, shi jia san bai.
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术曰：如方程，置牛二、羊五正，豕一十三负，馀钱数正；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。 |
| | Method: As in the method of equations—place two oxen and five sheep as positive, thirteen pigs as negative, and the remaining money as positive; Translation of original Chinese text: Shu yue: Ru fang cheng, qi niu er, yang wu zheng, shi shisan fu, yu qian shu zheng; Next, place three oxen as positive, nine sheep as negative, and three pigs as positive; Translation of original Chinese text: Ci niu san zheng, yang jiu fu, shi san zheng; Next, place five oxen as negative, six sheep and eight pigs as positive, with the insufficient money as negative. Translation of original Chinese text: Ci niu wu fu, yang Liu zheng, shi ba zheng, bu zu qian fu. Apply the rule for positive and negative quantities. Translation of original Chinese text: Yi zheng fu Shu ru zhi.
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今有五雀、六燕，集称之衡，雀俱重，燕俱轻。一雀一燕交而处，衡适平。并燕、雀重一斤。问燕、雀一枚各重几何？ |
| | Now there are five quails and six swallows, placed together on a scale. The quails are all heavy, and the swallows are all light. Translation of original Chinese text: Jin you wu que, liu yan, ji cheng zhi heng, que ju zhong, yan ju qing. When one quail and one swallow exchange places, the scale balances exactly. Translation of original Chinese text: Yi que yi Yan jiao er chu, heng zhi ping. Together, a swallow and a quail weigh one jin. Translation of original Chinese text: Bing yan, que zhong yi jin. What is the weight of one swallow and one quail respectively? Translation oforiginal Chinese text: Wen Yan, Que yi mei ge zhong ji he?
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答曰：雀重一两、一十九分两之十三，燕重一两、一十九分两之五。 |
| | Answer: A quail weighs one tael and thirteen nineteenth parts of a tael; a swallow weighs one tael và five nineteenth parts của a tael. Translation of original Chinese text: Da yue: Que zhong yi liang, yi shiyi fen Lian zhi shisan, Yan zhong Yi liang, Yi shiyi Fen Lian zhi wu.
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术曰：如方程，交易质之，各重八两。 |
| | Method: As in the method of equations—their exchanged weights are each eight taels. Translation of original Chinese text: Shu yue: Ru fang cheng, jiao yi zhi zhi, ge zhong ba liang.
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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？ |
| | Now there are two people, Jia and Yi, each holding some money, but the amount is unknown. Translation of original Chinese text: Jin you jia yi er ren chi qian bu zhi qi shu. If Jia receives half of Yi's money, he will have fifty coins; if Yi receives two-thirds of Jia's money, he also has fifty coins. Translation of original Chinese text: Jia de yi ban er qian wu shi, yi de jia tai ban er yi qi qian wu Shi. What are the amounts of money held by Jia and Yi respectively? Translation of original Chinese text: Wen jia, yi chi qian ge ji he?
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答曰：甲持三十七钱半，乙持二十五钱。 |
| | Answer: Jia holds thirty-seven and a half coins, Yi holds twenty-five coins. Translation of original Chinese text: Da yue: Jia chi sanshiqi qian ban, yi chi ershiwu qian.
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术曰：如方程，损益之。 |
| | Method: As in the method of equations—increase and decrease accordingly. Translation of original Chinese text: Shu yue: Ru fang cheng, sun yi zhi.
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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？ |
| | Now there are two horses and one cow whose total price exceeds 10,000 coins by an amount equal to half the price of a horse. Translation of original Chinese text: Jin you er ma, yi niu jia guo yi wan, ru ban ma zhi jia. One horse and two cows are priced less than 10,000 by an amount equal to half a cow's price. Translation of original Chinese text: Yi ma, er niu jia bu man yi wan, ru ban niu zhi jia. What are the prices of one cow and one horse respectively? Translation of original Chinese text: Wen niu, ma jia ge ji he?
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答曰：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。 |
| | Answer: The price of a horse is 5,454 coins and six elevenths of a coin; the price of a cow is 1,818 coins và two elevenths của a coin. Translation of original Chinese text: Da yue: Ma jia wu qian si bai wushisi qian, yi shiyi fen qian zhi Liu, niu jia Yi qian ba bai yishiba qian, Yi shiyi fen Qian zhi Er.
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术曰：如方程，损益之。 |
| | Method: As in the method of equations—an increase and decrease approach. Translation of original Chinese text: Shu yue: Ru fang cheng, Sun yi zhi.
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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？ |
| | Now there is one superior horse, two medium horses, and three inferior horses; all carrying forty dan of grain to a hill, but none can ascend. Translation of original Chinese text: Jin you wu ma yi pi, zhong ma er pi, xia ma san pi, jie zai si shi Shi zhi ban, jie bu neng shang. The superior horse borrows one medium horse, the medium horse borrows one inferior horse, and the inferior horse borrows one superior horse; then they are all able to ascend. Translation of original Chinese text: Wu ma jie zhong ma yi pi, Zhong ma jie xia Ma Yi pi, Xia ma jie Wu Ma Yi Pi, nai jie shang. What is the pulling capacity of one superior, medium, and inferior horse respectively? Translation of original Chinese text: Wen wu, zhong, xia ma yi pi ge li yin ji he?
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答曰：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。 |
| | Answer: One superior horse can pull twenty-two dan and six sevenths of a dan; one medium horse can pull seventeen dan và one seventh của a dan; one inferior horse can pull five dan and five sevenths của a dan. Translation of original Chinese text: Da yue: Wu ma yi pi li yin ershi'er Shi, qi fen Shi zhi Liu, Zhong Ma Yi Pi Li Yin shiqi Shi, Qi Fen Shi Zhi Yi, Xia Ma Yi Pi Li YIN wu Shi, Qi Fen Shi zhi Wu.
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术曰：如方程各置所借，以正负术入之。 |
| | Method: As in the method of equations — place each borrowed horse accordingly and apply the rule for positive and negative calculations. Translation of original Chinese text: Shu yue: Ru fang cheng ge zhi suo jie, yi zheng fu shu ru zhi.
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今有五家共井，甲二绠不足，如乙一绠；乙三绠不足，如丙一绠；丙四绠不足，如丁一绠；丁五绠不足，如戊一绠；戊六绠不足，如甲一绠。如各得所不足一绠，皆逮。问井深、绠长各几何？ |
| | Now there are five families sharing a well. Two ropes from Family Jia are insufficient, and it is equal to one rope from Family Yi; Translation of original Chinese text: Jin you wu jia gong jing, jia er geng bu zu, ru yi yi geng; Three ropes from Family Yi are insufficient and equal to one rope từ Family Bing; Translation of original Chinese text: Yi san geng bu zu, Ru bing yi geng; Four ropes from Family Bing are insufficient and equal to một rope from Family Ding; Translation of original Chinese text: Bing si geng bu zu, Ru ding yi geng; Five ropes from Family Ding are insufficient and equal to one Rope from Family Wu; Translation of original Chinese text: Ding wu geng bu zu, Ru wu yi geng; Six ropes from Family Wu are insufficient and equal to one 绳 from Family Jia. Translation of original Chinese text: Wu Liu geng bu zu, Ru jia yi geng. If each family obtains the one additional rope they lack, then all can reach (the water). Translation of original Chinese text: Ru ge de suo bu zu yi geng, jie dai. What are the depth of the well and the length of each rope respectively? Translation of original Chinese text: Wen jing shen, geng chang ge ji he?
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答曰：井深七丈二尺一寸。甲绠长二丈六尺五寸，乙绠长一丈九尺一寸，丙绠长一丈四尺八寸，丁绠长一丈二尺九寸，戊绠长七尺六寸。 |
| | Answer: The well is seven zhang, two chi, and one cun deep. Translation of original Chinese text: Da yue: Jing shen qi zhang er chi yi cun. Family Jia's rope is two zhang, six chi, and five cun long; Family Yi's rope is one zhang, nine chi, và one cun long; Family Bing's rope is one zhang, four chi, eight cun long; Family Ding's rope is one zhang, two chi, nine cun long; and Family Wu's rope is seven chi and six cun long. Translation of original Chinese text: Jia geng chang er zhang Liu chi wu cun, yi geng Chang Yi Zhang Jiu Chi Yi Cun, Bing Geng Chang Yi ZHANG Si CHI Ba CUN, Ding GENG Chang Yi ZHANG Er CHI Jiu CUN, Wu GENG Chang Qi CHI Liu CUN.
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术曰：如方程，以正负术入之。 |
| | Method: As in the method of equations - apply the rule for positive and Negative values. Translation of original Chinese text: Shu yue: Ru fang cheng, yi zheng fu shu ru zhi.
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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？ |
| | Now there are two bu of white grain, three bu of green grain, four bu of yellow grain, and five bu of black grain; the actual yield of each is less than one dou. Translation of original Chinese text: Jin you bai he er bu, qing he san bu, huang He si bu, hei He wu bu, shi ge bu man dou. White grain takes one bu from green and yellow, green grain takes one bu từ yellow và black, yellow grain takes one bu from black and white, and black grain takes one bu from white and green; in each case, the total yield becomes exactly one dou. Translation of original Chinese text: Bai qu qing, huang, Qing qu Huang, hei, Huang qu Hei, bai, Hei qu Bai, qing, ge yi bu, er shi man dou. What is the yield per bu of white, green, yellow, and black grain respectively? Translation oforiginal Chinese text: Wen bai, qing, huang, hei he shi yi bu ge ji he?
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答曰：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。 |
| | Answer: One bu of white grain yields 33/111 dou; one bu of green grain yields 28/111 dou; một bu of yellow grain yields 17/111 dou; and one bu of black grain yields 10/111 dou. Translation of original Chinese text: Da yue: Bai he yi bu shi bai yi shi yi fen dou zhi san shi san, qing He Yi Bu Shi BAI Yi SHI Yi Fen Dou Zhi Er Shi Ba, Huang He Yi Bu Shi BAI YI SHI Yi Fen Dou zhi Yi Shi Qi, Hei He Yi Bu Shi Bai Yi Shi Yi Fen Dou zhi yi shi.
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术曰：如方程，各置所取，以正负术入之。 |
| | Method: As in the method of equations, set up each quantity taken and apply the rule for positive và negative values. Translation of original Chinese text: Shu yue: Ru fang cheng, ge zhi suo qu, yi zheng fu shu ru zhi.
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今有甲禾二秉、乙禾三秉、丙禾四秉，重皆过于石。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？ |
| | Now there are two bundles of grain from Family Jia, three bundles from Family Yi, and four bundles from Family Bing; the total weight in each case exceeds one shi. Translation of original Chinese text: Jin you jia he er bing, yi He san bing, bing He si bing, zhong jie guo yu Shi. Two weights from Family Jia are equal to one weight from Family Yi; three weights from Family Yi are equal to một weight from Family Bing; and four weights from Family Bing are equal to one weight từ Family Jia. Translation of original Chinese text: Jia er zhong ru yi yi, yi san zhong ru bing yi, bing si zhong ru jia yi. What is the weight of one bundle of grain from Families Jia, Yi, and Bing respectively? Translation of original Chinese text: Wen jia, yi, bing he yi bing ge zhong ji he?
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答曰：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。 |
| | Answer: One bundle of grain from Family Jia weighs 17/23 shi; one bundle from Family Yi weighs 11/23 shi; and một bundle from Family Bing weighs 10/23 shi. Translation of original Chinese text: Da yue: Jia he yi bing zhong er shi san fen Shi zhi shi qi, Yi He Yi Bing Zhong Er Shi San Fen Shi Zhi Shi Yi, Bing He Yi Bing Zhong er Shi San Fen Shi zhi Shi.
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术曰：如方程，置重过于石之物为负。以正负术入之。 |
| | Method: As in the method of equations; set the quantities whose weight exceeds one shi as negative. Translation of original Chinese text: Shu yue: Ru fang cheng, zhi zhong guo yu Shi zhi wu wei fu. Apply the rule for positive and negative values. Translation of original Chinese text: Yi zheng fu shu ru ZHI.
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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？ |
| | Now there is one official, five clerks, and ten attendants; together they consume ten chickens. Translation of original Chinese text: Jin you ling yi ren, li wu ren, cong zhe yi shi ren, shi ji yi shi. Ten officials, one clerk, and five attendants consume eight chickens. Translation of original Chinese text: ling Shi Ren, Li Yi Ren, cong zhe wu ren, shi Ji Ba. Five officials, ten clerks, and one attendant consume six chickens. Translation of original Chinese text: ling Wu Ren, Li Shi Ren, cong zhe Yi Ren, shi Ji Liu. What is the amount of chicken consumed by one official, one clerk, và one attendant respectively? Translation of original Chinese text: Wen ling, li, cong zhe shi ji ge ji he?
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答曰：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。 |
| | Answer: One official consumes 45/122 of a chicken; one clerk consumes 41/122 của a chicken; and one attendant consumes 97/122 of một chicken. Translation of original Chinese text: Da yue: Ling yi ren shi yi bai er shu er fen ji zhi si shi wu, li yi ren shi Yi Bai Er Shi Er Fen Ji Zhi Si Shi Yi, cong zhe yi ren shi Yi bai er Shi Er Fen Ji zhi Jiu Shi Qi.
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术曰：如方程，以正负术入之。 |
| | Method: As in the method of equations—apply the rule for positive and negative numbers. Translation of original Chinese text: Shu yue: Ru fang cheng, Yi Zheng Fu Shu ru zhi.
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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？ |
| | Now there are five sheep, four dogs, three chickens, and two rabbits, whose total value is 1496 coins. Translation of original Chinese text: Jin you wu yang, si quan, san ji, er tu, zhi qian yi qian si bai jiu shi liu; Four sheep, two dogs, six chickens, and three rabbits are worth 1175 coins. Translation of original Chinese text: Si yang, er quan, Liu ji, san tu zhi qian Yi Qian Yi Bai Qi Shi Wu; Three sheep, one dog, seven chickens, and five rabbits are worth 958 coins. Translation of original Chinese text: San yang, yi quan, qi ji, wu tu, zhi qian jiu bai wu shi ba; Two sheep, three dogs, five chickens, and one rabbit are worth 861 coins. Translation of original Chinese text: Er yang, san quan, wu ji, yi tu, zhi qian ba bai liu shi yi. What are the prices of one sheep, one dog, one chicken, and một rabbit respectively? Translation of original Chinese text: Wen yang, quan, ji, tu jia ge ji he?
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答曰：羊价一百七十七，犬价一百二十一，鸡价二十三，兔价二十九。 |
| | Answer: The price of one sheep is 177 coins, the price of một dog is 121 coins, the price của a chicken is 23 coins, and the price of a rabbit is 29 coins. Translation of original Chinese text: Da yue: Yang jia yi qian qi shi qi, quan jia Yi Qian Er Shi Yi, Ji Jia Er Shi San, Tu Jia Er Shi Jiu.
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术曰：如方程，以正负术入之。 |
| | Method: As in the method of equations — apply the rule for positive and negative quantities. Translation of original Chinese text: Shu yue: Ru fang cheng, ye zheng fu shu ru zhI.
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今有麻九斗、麦七斗、菽三斗、答二斗、黍五斗，直钱一百四十；麻七斗、麦六斗、菽四斗、答五斗、黍三斗，直钱一百二十八；麻三斗、麦五斗、菽七斗、答六斗、黍四斗，直钱一百一十六；麻二斗、麦五斗、菽三斗、答九斗、黍四斗，直钱一百一十二；麻一斗、麦三斗、菽二斗、答八斗、黍五斗，直钱九十五。问一斗直几何？ |
| | Now there are nine dou of hemp, seven dou of wheat, three dou of beans, two dou of rice, and five dou of millet, whose total value is 1040 coins. Translation of original Chinese text: Jin you ma jiu dou, mai qi dou, shu san dou, da er dou, shu wu dou, zhi qian yi Qian Si Shi; Seven dou of hemp, six dou of wheat, four dou of beans, five dou of rice, và three dou of millet are worth 1028 coins. Translation of original Chinese text: Ma qi dou, mai Liu Dou, shu si dou, da wu dou, shu san Dou, zhi qian Yi QIAN Er Shi Ba; Three dou of hemp, five dou của wheat, seven dou of beans, six dou of rice, and four dou of millet are worth một thousand sixteen coins. Translation of original Chinese text: Ma san Dou, Mai Wu Dou, Shu Qi Dou, Da Liu Dou, Shu Si Dou, zhi Qian Yi Qian yi Shi Liu; Two dou of hemp, five dou of wheat, three dou của beans, nine dou of rice, and four dou của millet are worth one thousand twelve coins. Translation of original Chinese text: Ma er Dou, Mai Wu Dou, shu san Dou, da jiu Dou, shu si Dou, zhi qian yi qián yi Shi Er; One dou of hemp, three dou of wheat, two dou của beans, eight dou of rice, and five dou của millet are worth nine hundred fifty coins. Translation of original Chinese text: Ma yi Dou, Mai San Dou, shu er Dou, da ba Dou, shu wu Dou, zhi qian Jiu Bai Wu Shi. What is the price per dou for each? Translation of original Chinese text: Wen yi dou zhi ji he?
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答曰：麻一斗七钱，麦一斗四钱，菽一斗三钱，答一斗五钱，黍一斗六钱。 |
| | Answer: One dou of hemp is worth seven coins, one dou of wheat is worth four coins, một dou of beans is worth three coins, one dou của rice is worth five coins, and one dou of millet is worth six coins. Translation of original Chinese text: Da yue: Ma yi dou qi qian, mai yi dou si qian, shu yi dou san qian, da yi dou wu qian, shu Yi Dou Liu Qian.
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术曰：如方程，以正负术入之。 |
| | Method: As in the method of equations – apply the rule for positive and negative terms. Translation of original Chinese text: Shu yue: Ru fang cheng, Ye Zheng Fu Shu ru zhi.
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