粟米之法：粟率五十糲米三十粺米二十七凿米二十四 御米二十一小、十三半大。五十四糲饭七十五粺饭五十四凿饭四十八 御饭四十二菽、荅、麻、麦各四十五稻六十豉六十三 飧九十熟菽一百三半 櫱一百七十五 今有术曰：以所有数乘所求率为实，以所有率为法，实如法而一。
粟米之法： 粟的兑换比例是，每五十份粟可以换得三十份粝米，二十七份糲米，二十四份凿米。 御米的兑换比例是，每二十一分御米可以换得十三份半大麦，五十四份粝饭，七十五份粺饭，五十四份凿饭，四十八份御饭。 菽、荅（可能是某种古代豆类或谷物）、麻、麦的兑换比例各是四十五份。稻的兑换比例是六十份，豉（一种豆制品）的兑换比例是六十三份。 飧（可能指某种熟食）的兑换比例是九十份，熟菽的兑换比例是一百三份半。 櫱（可能是一种植物或物品）的兑换比例是一百七十五份。 今有术曰：以所有数乘所求率为实，以所有率为法，实如法而一。 现有一个方法说明：将所拥有的数量乘以所需的比率得到结果（实物），再将所拥有的比率作为规则（标准），结果如果符合规则就取其一。
The method for millet exchange is as follows: for every fifty parts of millet, one can exchange for thirty parts of coarse rice, twenty-seven parts of medium rice, and twenty-four parts of fine rice. For imperial rice, the exchange rate is such that for every twenty-one parts of imperial rice, one can exchange for thirteen and a half parts of barley, fifty-four parts of coarse cooked rice, seventy-five parts of medium cooked rice, fifty-four parts of fine cooked rice, and forty-eight parts of imperial cooked rice. The exchange rates for legumes, da (possibly a type of ancient legume or grain), hemp, and wheat are each forty-five parts. The exchange rate for rice is sixty parts, for fermented soybeans, it is sixty-three parts. For meals, the exchange rate is ninety parts, and for cooked legumes, it is one hundred and three and a half parts. For 櫱， the exchange rate is one hundred and seventy-five parts. There is a current method which states: Multiply the number of owned items by the desired rate to get the actual result, and take the owned rate as the standard; if the actual result complies with the standard, then take one. This translation into modern mathematical concepts is: There is a method that explains: Multiply the quantity you have by the rate you wish to obtain to get the actual outcome, and use the rate you possess as the criterion; if the actual outcome meets the criterion, then it is valid.
〔一〕今有粟一斗，欲为糲米。问得几何？
荅曰：为糲米六升。
术曰：以粟求糲米，三之，五而一。翻译成中文和英文
现有一斗粟米，想要将其加工成糲（粗）米。问可以得到多少？
答：可得到六升糲米。
方法说明：用一斗粟米制作糲米，数量上要乘以三，再除以五。
We have one dou of millet and want to process it into coarse rice. How much will we get? Answer: We will get six sheng of coarse rice. Method: To make coarse rice from millet, multiply by three and then divide by five.
〔二〕今有粟二斗一升，欲为粺米。问得几何？
荅曰：为粺米一斗一升、五十分升之十七。
术曰：以粟求粺米，二十七之，五十而一。
现有二斗一升粟米，想要将其加工成粺（精）米。问可以得到多少？
答：可得到一斗一升又17/50升粺米。
方法说明：用二斗一升粟米制作粺米，数量上要乘以二十七，再除以五十。
We have two dou and one sheng of millet and want to process it into refined rice. How much will we get? Answer: We will get one dou, one sheng, and17/50 sheng of refined rice. Method: To make refined rice from millet, multiply by twenty-seven and then divide by fifty.
〔三〕今有粟四斗五升，欲为凿米。问得几何？
荅曰：为凿米二斗一升、五分升之三。
术曰：以粟求凿米，十二之，二十五而一。
现有四斗五升粟米，想要将其加工成凿（整）米。问可以得到多少？
答：可得到二斗一升又3/5升凿米。
方法说明：用四斗五升粟米制作凿米，数量上要乘以十二，再除以二十五。
We have four dou and five sheng of millet and want to process it into whole grain rice. How much will we get? Answer: We will get two dou, one sheng, and3/5sheng of whole grain rice. Method: To make whole grain rice from millet, multiply by twelve and then divide by twenty-five.
〔四〕今有粟七斗九升，欲为御米。问得几何？
荅曰：为御米三斗三升、五十分升之九。
术曰：以粟求御米，二十一之，五十而一。
现有七斗九升粟米，想要将其加工成御（精选）米。问可以得到多少？
答：可得到三斗三升又9/50升御米。
方法说明：用七斗九升粟米制作御米，数量上要乘以二十一，再除以五十。
We have seven dou and nine sheng of millet and want to process it into select rice. How much will we get? Answer: We will get three dou, three sheng, and9/50sheng of select rice. Method: To make select rice from millet, multiply by twenty-one and then divide by fifty.
五〕今有粟一斗，欲为小。问得几何？
荅曰：为小二升、一十分升之七。术曰：以粟求小，二十七之，百而一。
现有一斗粟米，想要将其加工成小米。问可以得到多少？
答：可得到二升又七分之一升小。
方法说明：用一斗粟米制作小米，数量上要乘以二十七，再除以一百。
We have one dou of millet and want to process it into small rice (a specific type of processed millet). How much will we get? Answer: We will get two sheng and1/10sheng of small rice. Method: To make small from millet, multiply by twenty-seven and then divide by a hundred.
〔六〕今有粟九斗八升，欲为大。问得几何？
荅曰：为大一十斗五升、二十五分升之二十一。
术曰：以粟求大，二十七之，二十五而一。
现有九斗八升粟米，想要将其加工成大（特定加工的大米或粗粮）。问可以得到多少？
答：可得到十斗五升又二十一分之二十五升大●。
方法说明：用九斗八升粟米制作大，数量上要乘以二十七，再除以二十五。
We have nine dou and eight sheng of millet and want to process it into large (a specific type of processed rice or grain). How much will we get? Answer: We will get ten dou, five sheng, and 21/25sheng of large. Method： To make large from millet, multiply by twenty-seven and then divide by twenty-five.
〔七〕今有粟二斗三升，欲为糲饭。问得几何？
荅曰：为糲饭三斗四升半。
术曰：以粟求糲饭，三之，二而一。
现有二斗三升粟米，想要将其加工成粝（粗）饭。问可以得到多少？
答：可得到三斗四升半粝饭。
方法说明：用二斗三升粟米制作粝饭，数量上要乘以三，再除以二。
We have two dou and three sheng of millet and want to process it into coarse rice. How much will we get? Answer: We will get three dou, four sheng, and half a sheng of coarse rice. Method: To make coarse rice from millet, multiply by three and then divide by two.
〔八〕今有粟三斗六升，欲为粺饭。问得几何？
荅曰：为粺饭三斗八升、二十五分升之二十二。术曰：以粟求粺饭，二十七之，二十五而一。
现有三斗六升粟米，想要将其加工成粺（精）饭。问可以得到多少？
答：可得到三斗八升又二十二分之二十五升粺饭。
方法说明：用三斗六升粟米制作粺饭，数量上要乘以二十七，再除以二十五。。
We have three dou and six sheng of millet and want to process it into refined rice. How much will we get? Answer: We will get three dou, eight sheng, and22/25sheng of refined rice. Method: To make refined rice from millet, multiply by twenty-seven and then divide by twenty-five.
〔九〕今有粟八斗六升，欲为凿饭。问得几何？
荅曰：为凿饭八斗二升、二十五分升之一十四。
术曰：以粟求凿饭，二十四之，二十五而一。
现有八斗六升粟米，想要将其加工成凿（整）饭。问可以得到多少？
答：可得到八斗二升又十四分之二十五升凿饭。
方法说明：用八斗六升粟米制作凿饭，数量上要乘以二十四，再除以二十五。
We have eight dou and six sheng of millet and want to process it into whole grain rice. How much will we get? Answer: We will get eight dou, two sheng, and14/25sheng of whole grain rice. Method: To make whole grain rice from millet, multiply by twenty-four and then divide by twenty-five.
〔十〕今有粟九斗八升，欲为御饭。问得几何？
荅曰：为御饭八斗二升、二十五分升之八。
术曰：以粟求御饭，二十一之，二十五而一。
现有九斗八升粟米，想要将其加工成御（精选）饭。问可以得到多少？
答：可得到八斗二升又八分之二十五升御饭。
方法说明：用九斗八升粟米制作御饭，数量上要乘以二十一，再除以二十五。
We have nine dou and eight sheng of millet and want to process it into select rice. How much will we get? Answer: We will get eight dou, two sheng, and8/25sheng of select rice. Method: To make select rice from millet, multiply by twenty-one and then divide by twenty-five.
〔一一〕今有粟三斗少半升，欲为菽。问得几何？
荅曰：为菽二斗七升、一十分升之三。
现有三斗少半升粟米，想要将其加工成菽（豆类）。问可以得到多少？
答：可得到二斗七升又三分之一升菽。
We have three dou and a bit less than half a sheng of millet and want to process it into beans. How much will we get? Answer: We will get two dou, seven sheng, and1/10sheng of beans.
〔一二〕今有粟四斗一升、太半升，欲为荅。问得几何？
荅曰：为荅三斗七升半。
现有四斗一升又三分之二升粟米，想要将其加工成荅（一种食品，可能指豆腐或豆制品）。问可以得到多少？
答：可得到三斗七升半荅。
We have four dou, one sheng, and a bit more than half a sheng of millet and want to process it into 'da' (possibly tofu or another soy product). How much will we get? Answer: We will get three dou, seven sheng, and half a sheng of 'da'.
〔一三〕今有粟五斗、太半升，欲为麻。问得几何？
荅曰：为麻四斗五升、五分升之三。
〔一三〕现有五斗又三分之二升粟米，想要将其加工成麻（可能指某种特定处理的谷物）。问可以得到多少？
答：可得到四斗五升又三分之一升麻。
We have five dou and two-thirds of a sheng of millet and want to process it into hemp. How much will we get? Answer: We will get four dou, five sheng, and one-third of a sheng of hemp.
〔一四〕今有粟一十斗八升、五分升之二，欲为麦。问得几何？
荅曰：为麦九斗七升、二十五分升之一十四。
术曰：以粟求菽、荅、麻、麦，皆九之，十而一。
现有十斗八升又二分之五升粟米，想要将其加工成麦。问可以得到多少？
答：可得到九斗七升又十四分之二十五升麦。
术曰：以粟米加工成菽、荅、麻、麦等，都是乘以九，再除以十。
We have ten dou, eight sheng, and two-fifths of a sheng of millet and want to process it into wheat. How much will we get? Answer: We will get nine dou, seven sheng, and fourteen-twenty-fifths of a sheng of wheat. The method says: To make soybeans, 'da', hemp, or wheat from millet, multiply by nine and then divide by ten.
〔一五〕今有粟七斗五升、七分升之四，欲为稻。问得几何？
荅曰：为稻九斗、三十五分升之二十四。
术曰：以粟求稻，六之，五而一。
现有七斗五升又四分之七升粟米，想要将其加工成稻。问可以得到多少？
答：可得到九斗又二十四分之三十五升稻。
术曰：以粟米加工成稻，乘以六，再除以五。
We have seven dou, five sheng, and four-sevenths of a sheng of millet and want to process it into rice. How much will we get? Answer: We will get nine dou and twenty-four thirty-fifths of a sheng of rice. The method says: To make rice from millet, multiply by six and then divide by five.
〔一六〕今有粟七斗八升，欲为豉。问得几何？
荅曰：为豉九斗八升、二十五分升之七。
术曰：以粟求豉，六十三之，五十而一。
现有七斗八升粟米，想要将其加工成豉。问可以得到多少？
答：可得到九斗八升又七分之二十五升豉。
术曰：以粟米加工成豉，乘以六十三，再除以五十。
We have seven dou and eight sheng of millet and want to process it into fermented soybeans. How much will we get? Answer: We will get nine dou, eight sheng, and seven-twenty-fifths of a sheng of fermented soybeans. The method says: To make fermented soybeans from millet, multiply by sixty-three and then divide by fifty.
〔一七〕今有粟五斗五升，欲为飧。问得几何？
荅曰：为飧九斗九升。术曰：以粟求飧，九之，五而一。
现有五斗五升粟米，想要将其加工成飧（熟食）。问可以得到多少？
答：可得到九斗九升飧。
术曰：以粟米加工成飧，乘以九，再除以五。
We have five dou and five sheng of millet and want to process it into cooked rice. How much will we get? Answer: We will get nine dou and nine sheng of cooked rice. The method says: To make cooked rice from millet, multiply by nine and then divide by five.
〔〔一八〕今有粟四斗，欲为熟菽。问得几何？
荅曰：为熟菽八斗二升、五分升之四。术曰：以粟求熟菽，二百七之，百而一。
现有四斗粟米，想要将其加工成熟菽（熟的豆类）。问可以得到多少？
答：可得到八斗二升又四分之五升熟菽。
术曰：以粟米加工成熟菽，乘以二百零七，再除以一百。
We have four dou of millet and want to process it into cooked beans. How much will we get? Answer: We will get eight dou, two sheng, and four-fifths of a sheng of cooked beans. The method says: To make cooked beans from millet, multiply by two hundred and seven, then divide by one hundred.
〔一九〕今有粟二斗，欲为櫱。问得几何？
荅曰：为櫱七斗。
术曰：以粟求櫱，七之，二而一。
现有二斗粟米，想要将其加工成櫱。问可以得到多少？ 答：可得到七斗櫱。 术曰：以粟米加工成櫱，乘以七，再除以二。
We have two dou of millet and want to process it into 'nao' (possibly a type of fermented drink or specific food). How much will we get? Answer: We will get seven dou of 'nao'. The method says: To make 'nao' from millet, multiply by seven and then divide by two.
〔二０〕今有糲米十五斗五升、五分升之二，欲为粟。问得几何？
荅曰：为粟二十五斗九升。术曰：以糲米求粟，五之，三而一。
现有粝米（去壳未舂的米）十五斗五升又二分之五升，想要将其加工成粟米。问可以得到多少？
答：可得到二十五斗九升粟米。
术曰：以粝米加工成粟米，乘以五，再除以三。
We have fifteen dou, five sheng, and two-fifths of a sheng of coarse rice (unpolished rice that has been hulled but not pounded), and we want to process it into millet. How much will we get? Answer: We will get twenty-five dou and nine sheng of millet. The method says: To make millet from coarse rice, multiply by five and then divide by three.
〔二一〕今有粺米二斗，欲为粟。问得几何？
荅曰：为粟三斗七升、二十七分升之一。术曰：以粺米求粟，五十之，二十七而一。
现有粺米（精米）二斗，想要将其加工成粟米。问可以得到多少？
答：可得到三斗七升又二十七分之二升粟米。
术曰：以粺米加工成粟米，乘以五十，再除以二十七。
We have two dou of refined rice (polished rice) and want to process it into millet. How much will we get? Answer: We will get three dou, seven sheng, and one twenty-seventh of a sheng of millet. The method says: To make millet from refined rice, multiply by fifty and then divide by twenty-seven.
〔二二〕今有凿米三斗、少半升，欲为粟。问得几何？
荅曰：为粟六斗三升、三十六分升之七。术曰：以凿米求粟，二十五之，十三而一。
现有凿米（去壳并舂过的米）三斗少半升，想要将其加工成粟米。问可以得到多少？
答：可得到六斗三升又三十六分之七升粟米。
术曰：以凿米加工成粟米，乘以二十五，再除以十三。
We have three dou and a bit less than half a sheng of polished rice (rice that has been hulled, pounded, and polished), and we want to process it into millet. How much will we get? Answer: We will get six dou, three sheng, and seven thirty-sixths of a sheng of millet. The method says: To make millet from polished rice, multiply by twenty-five and then divide by thirteen.
〔二三〕今有御米十四斗，欲为粟。问得几何？
荅曰：为粟三十三斗三升、少半升。
术曰：以御米求粟，五十之，二十一而一。
现有御米（精选的优质米）十四斗，想要将其加工成粟米。问可以得到多少？
答：可得到三十三斗三升又少半升粟米。
术曰：以御米加工成粟米，乘以五十，再除以二十一。
We have fourteen dou of select rice (choice quality rice) and want to process it into millet. How much will we get? Answer: We will get thirty-three dou, three sheng, and a bit less than half a sheng of millet. The method says: To make millet from select rice, multiply by fifty and then divide by twenty-one.
〔二四〕今有稻一十二斗六升、一十五分升之一十四，欲为粟。问得几何？
荅曰：为粟一十斗五升、九分升之七。
术曰：以稻求粟，五之，六而一。
现有稻米十二斗六升又十五分之十四升，想要将其加工成粟米。问可以得到多少？
答：可得到十斗五升又九分之七升粟米。
术曰：以稻米加工成粟米，乘以五，再除以六。
We have twelve dou, six sheng, and fourteen fifteen-hundredths of a sheng of rice (unhulled rice) and want to process it into millet. How much will we get? Answer: We will get ten dou, five sheng, and seven nine-hundredths of a sheng of millet. The method says: To make millet from rice, multiply by five and then divide by six.
〔二五〕今有糲米一十九斗二升、七分升之一，欲为粺米。问得几何？
荅曰：为粺米一十七斗二升、一十四分升之一十三。
术曰：以糲米求粺米，九之，十而一。
现有粝米（去壳未舂的米）十九斗二升又七分之二升，想要将其加工成粺米（精米）。问可以得到多少？
答：可得到十七斗二升又十四分之十三升粺米。
术曰：以粝米加工成粺米，乘以九，再除以十。
We have nineteen dou, two sheng, and one seven-hundredth of a sheng of coarse rice (unpolished rice that has been hulled but not pounded), and we want to process it into refined rice. How much will we get? Answer: We will get seventeen dou, two sheng, and thirteen fourteen-hundredths of a sheng of refined rice. The method says: To make refined rice from coarse rice, multiply by nine and then divide by ten.
〔二六〕今有糲米六斗四升、五分升之三，欲为糲饭。问得几何？
荅曰：为糲饭一十六斗一升半。
术曰：以糲米求糲饭，五之，二而一。
现有粝米（去壳未舂的米）六斗四升又五分之三升，想要将其加工成粝饭。问可以得到多少？
答：可得到十六斗一升半的粝饭。
术曰：以粝米加工成粝饭，乘以五，再除以二。
We have six dou, four sheng, and three five-hundredths of a sheng of coarse rice (unpolished rice that has been hulled but not pounded), and we want to cook it into coarse rice meal. How much will we get? Answer: We will get sixteen dou and one and a half sheng of coarse rice meal. The method says: To make coarse rice meal from coarse rice, multiply by five and then divide by two.
〔二七〕今有糲饭七斗六升、七分升之四，欲为飧。问得几何？
荅曰：为飧九斗一升、三十五分升之三十一。术曰：以糲饭求飧，六之，五而一。
现有粝饭（熟的粗糙米饭）七斗六升又七分之四升，想要将其加工成飧（熟食）。问可以得到多少？
答：可得到九斗一升又三十五分之三十一升的飧。
术曰：以粝饭加工成飧，乘以六，再除以五。
We have seven dou, six sheng, and four seven-hundredths of a sheng of coarse rice meal, and we want to process it into cooked rice. How much will we get? Answer: We will get nine dou, one sheng, and thirty-one thirty-fifths of a sheng of cooked rice. The method says: To make cooked rice from coarse rice meal, multiply by six and then divide by five.
〔二八〕今有菽一斗，欲为熟菽。问得几何？
荅曰：为熟菽二斗三升。
术曰：以菽求熟菽，二十三之，十而一。
现有菽（豆类）一斗，想要将其加工成熟菽（熟的豆类）。问可以得到多少？
答：可得到二斗三升的熟菽。
术曰：用菽加工成熟菽，乘以二十三，再除以十。
We have one dou of beans and want to cook them. How much will we get? Answer: We will get two dou and three sheng of cooked beans. The method says: To make cooked beans from raw beans, multiply by twenty-three and then divide by ten.
〔二九〕今有菽二斗，欲为豉。问得几何？
荅曰：为豉二斗八升。
术曰：以菽求豉，七之，五而一。
现有菽（豆类）二斗，想要将其加工成豉（发酵的豆制品）。问可以得到多少？
答：可得到二斗八升的豉。
术曰：以菽加工成豉，乘以七，再除以五。
We have two dou of beans and want to make fermented bean paste. How much will we get? Answer: We will get two dou and eight sheng of fermented bean paste. The method says: To make fermented bean paste from beans, multiply by seven and then divide by five.
〔三０〕今有麦八斗六升、七分升之三，欲为小●，问得几何？
荅曰：为小●二斗五升、一十四分升之一十三。术曰：以麦求小●，三之，十而一。
现有麦八斗六升又七分之三升，想要将其加工成小●（一种食品）。问可以得到多少？
答：可得到二斗五升又十四分之十三升的小●。
术曰：以麦加工成小●，乘以三，再除以十。
We have eight dou, six sheng, and three seven-hundredths of a sheng of wheat and want to process it into 'xiao ●' (a type of food). How much will we get? Answer: We will get two dou and five sheng, plus thirteen fourteen-hundredths of a sheng of 'xiao ●'. The method says: To make 'xiao ●' from wheat, multiply by three and then divide by ten.
〔三一〕今有麦一斗，欲为大●。问得几何？
荅曰：为大●一斗二升。
术曰：以麦求大●，六之，五而一。
现有麦一斗，想要将其加工成大●（一种食品）。问可以得到多少？
答：可得到一斗二升的大●。
术曰：以麦加工成大●，乘以六，再除以五。
We have one dou of wheat and want to process it into 'da ●' (a type of food). How much will we get? Answer: We will get one dou and two sheng of 'da ●'. The method says: To make 'da ●' from wheat, multiply by six and then divide by five.
〔三二〕今有出钱一百六十，买瓴甓十八枚。问枚几何？
荅曰：一枚，八钱、九分钱之八。
现有出钱一百六十，买瓴甓（一种建筑材料）十八枚。问每枚多少钱？
答：一枚八钱又九分之八钱。
We have 160 coins to buy 18 bricks. How much is each brick? Answer: One brick costs eight coins and eight-ninths of a coin.
〔三三〕今有出钱一万三千五百，买竹二千三百五十箇。问箇几何？
荅曰：一箇，五钱、四十七分钱之三十五。
经率术曰：以所买率为法，所出钱数为实，实如法得一钱。
现有出钱一万三千五百，买竹二千三百五十个。问每个多少钱？
答：一个五钱又四十七分之三十五钱。
经率术曰：以所买的率为法（除数），所出的钱数为实（被除数），实除以法得到每一个单位的价格。
We have 13,500 coins to buy 2,350 bamboos. How much is each bamboo? Answer: One bamboo costs five coins and thirty-five forty-sevenths of a coin. The method says: Use the rate of purchase as the divisor and the amount of money spent as the dividend; divide the dividend by the divisor to get the price per unit.
〔三四〕今有出钱五千七百八十五，买漆一斛六斗七升、太半升。欲斗率之，问斗几何。
荅曰：一斗，三百四十五钱、五百三分钱之一十五。
现有出钱五千七百八十五，买漆一斛六斗七升又太半升。想要计算每斗的价格，问每斗多少钱？
答：一斗漆三百五十四钱又五百三分之十五钱。
We have 5,785 coins to buy 1.675 pecks of lacquer. We want to calculate the price per peck; how much is each peck? Answer: One peck costs 345 coins and fifteen one-hundred-and-five-hundredths of a coin.
〔三五〕今有出钱七百二十，买縑一匹二丈一尺。欲丈率之，问丈几何？
荅曰：一丈，一百一十八钱、六十一分钱之二。
现有出钱七百二十，买縑（一种丝织品）一匹二丈一尺。想要计算每丈的价格，问每丈多少钱？
答：一丈縑一百一十八钱又六十一分之二钱。
We have 720 coins to buy one piece of silk fabric that is two zhang and one chi long. We want to calculate the price per zhang; how much is each zhang? Answer: One zhang costs 118 coins and two sixty-firsts of a coin.
〔三六〕今有出钱二千三百七十，买布九匹二丈七尺。欲匹率之，问匹几何？
荅曰：一匹，二百四十四钱、一百二十九分钱之一百二十四。
现有出钱二千三百七十，买布九匹二丈七尺。想要计算每匹的价格，问每匹多少钱？
答：一匹布二百四十四钱又一百二十九分之一百二十四钱。
We have 2,370 coins to buy nine pieces of cloth that are two zhang and seven chi long. We want to calculate the price per piece; how much is each piece? Answer: One piece costs 244 coins and 124 out of 129 parts of a coin.
〔三七〕今有出钱一万三千六百七十，买丝一石二钧一十七斤。欲石率之，问石几何？
荅曰：一石，八千三百二十六钱、一百九十七分钱之一百七十八。
经术术曰：以所求率乘钱数为实，以所买率为法，实如法得一。
现有出钱一万三千六百七十，买丝一石二钧一十七斤。想要计算每石的价格，问每石多少钱？
答：一石丝八千三百二十六钱又一百九十七分之一百四十八钱。
经术术曰：以所求的率（单价）乘以钱数得到总价，以此作为被除数，以所买的量作为除数，用被除数除以除数得到单价。
We have 13,670 coins to buy one dan, two jun, and seventeen jin of silk. We want to calculate the price per dan; how much is each dan? Answer: One dan costs 8,326 coins and 148 out of 197 parts of a coin. The method says: Multiply the desired rate by the amount of money to get the total price as the dividend, and use the quantity bought as the divisor. Divide the dividend by the divisor to get the unit price.
〔三八〕今有出钱五百七十六，买竹七十八箇。欲其大小率之，问各几何？
荅曰：其四十八箇，箇七钱。
其三十箇，箇八钱。
现有出钱五百七十六，买竹七十八个。想要根据大小计算单价，问每个多少钱？ 答：其中四十八个个，每个七钱。另外三十个个，每个八钱。
We have 576 coins to buy 78 bamboos. We want to calculate the price based on size; how much is each? Answer: Forty-eight of them, each costs seven coins. The remaining thirty, each costs eight coins.
〔三九〕今有出钱一千一百二十，买丝一石二钧十八斤。欲其贵贱斤率之，问各几何？
荅曰：其二钧八斤，斤五钱。其一石一十斤，斤六钱。
现有出钱一千一百二十，买丝一石二钧十八斤。想要根据贵贱计算每斤的单价，问每斤多少钱？ 答：其中二钧八斤，每斤五钱。剩下的一石十斤，每斤六钱。
We have 1,120 coins to buy one dan, two jun, and eighteen jin of silk. We want to calculate the price per jin based on quality; how much is each jin? Answer: Two jun and eight jin, each jin costs five coins. The remaining one dan and ten jin, each jin costs six coins.
〔四０〕今有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。欲其贵贱石率之，问各几何？
荅曰：
其一钧九两一十二銖，石八千五十一钱。
其一石一钧二十七斤九两一十七銖，石八千五十二钱。
现有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。想要根据贵贱计算每石的单价，问每石多少钱？ 答：其中一钧九两十二銖，每石八千五十一钱。另外一石一钧二十七斤九两十七銖，每石八千五十二钱。
We have 13,970 coins to buy one dan, two jun, twenty-eight jin, three liang, and five zhu of silk. We want to calculate the price per dan based on quality; how much is each dan? Answer: For one jun, nine liang, and twelve zhu, each dan costs eight thousand fifty-one coins. For one dan, one jun, twenty-seven jin, nine liang, and seventeen zhu, each dan costs eight thousand fifty-two coins.
〔四一〕今有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。欲其贵贱钧率之，问各几何？
荅曰：其七斤一十两九銖，钧二千一十二钱。
其一石二钧二十斤八两二十銖，钧二千一十三钱。
现有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。想要根据贵贱计算每钧的单价，问每钧多少钱？ 答：其中七斤十两九銖，每钧二千一百二十二钱。另外一石二钧二十斤八两二十銖，每钧二千一百三十三钱。
We have 13,970 coins to buy one dan, two jun, twenty-eight jin, three liang, and five zhu of silk. We want to calculate the price per jun based on quality; how much is each jun? Answer: For seven jin, ten liang, and nine zhu, each jun costs two thousand one hundred and twelve coins. For one dan, two jun, twenty jin, eight liang, and twenty zhu, each jun costs two thousand one hundred and thirteen coins.
〔四二〕今有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。欲其贵贱斤率之，问各几何？
荅曰：
其一石二钧七斤十两四銖，斤六十七钱。
其二十斤九两一銖，斤六十八钱。
现有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。想要根据贵贱计算每斤的单价，问每斤多少钱？ 答：其中一石二钧七斤十两四銖，每斤六十七钱。另外二十斤九两一銖，每斤六十八钱。
We have 13,970 coins to buy one dan, two jun, twenty-eight jin, three liang, and five zhu of silk. We want to calculate the price per jin based on quality; how much is each jin? Answer: For one dan, two jun, seven jin, ten liang, and four zhu, each jin costs sixty-seven coins. For twenty jin, nine liang, and one zhu, each jin costs sixty-eight coins.
〔四三〕今有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。欲其贵贱两率之，问各几何？
荅曰：
其一石一钧一十七斤一十四两一銖，两四钱。
其一钧一十斤五两四銖，两五钱。
其率术曰：各置所买石、钧、斤、两以为法，以所率乘钱数为实，实如法而一。不满法者反以实减法，法贱实贵。
现有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。想要根据贵贱计算每两的单价，问每两多少钱？ 答：其中一石一钧十七斤十四两一銖，每两四钱。另外一钧十斤五两四銖，每两五钱。其率术曰：分别取所买的石、钧、斤、两作为除数，以所求的率乘以钱数作为被除数，用被除数除以除数得到每一单位的价格。如果被除数小于除数，则反过来用被除数减去除数，因为除数便宜而被除数贵。
We have 13,970 coins to buy one dan, two jun, twenty-eight jin, three liang, and five zhu of silk. We want to calculate the price per liang based on quality; how much is each liang? Answer: For one dan, one jun, seventeen jin, fourteen liang, and one zhu, each liang costs four coins. For one jun, ten jin, five liang, and four zhu, each liang costs five coins. The method says: Take the purchased amounts in dan, jun, jin, and liang as the divisor, multiply the desired rate by the total amount of money to get the dividend, and divide the dividend by the divisor to get the unit price. If the dividend is less than the divisor, subtract the divisor from the dividend instead, because the divisor is cheaper and the dividend is more expensive.
〔四四〕今有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。欲其贵贱銖率之，问各几何？
荅曰：
其一钧二十斤六两十一銖，五銖一钱。其一石一钧七斤一十二两一十八銖，六銖一钱。
现有出钱一万三千九百七十，买丝一石二钧二十八斤三两五銖。想要根据贵贱计算每銖的单价，问每銖多少钱？ 答：其中一钧二十斤六两十一銖，每五銖值一钱。另外一石一钧七斤十二两一十八銖，每六銖值一钱。
We have 13,970 coins to buy one dan, two jun, twenty-eight jin, three liang, and five zhu of silk. We want to calculate the price per zhu based on quality; how much is each zhu? Answer: For one jun, twenty jin, six liang, and eleven zhu, each zhu costs one coin out of every five. For one dan, one jun, seven jin, twelve liang, and eighteen zhu, each zhu costs one coin out of every six.
〔四五〕今有出钱六百一十，买羽二千一百翭。欲其贵贱率之，问各几何？
荅曰：
其一千一百四十翭，三翭一钱。
其九百六十翭，四翭一钱。
现有出钱六百一十，买羽二千一百翭。想要根据贵贱计算单价，问每翭多少钱？ 答：其中一千一百四十翭，三翭一钱。另外九百六十翭，四翭一钱。
We have 610 coins to buy 2,100 feathers. We want to calculate the price per feather based on quality; how much is each feather? Answer: For 1,140 feathers, the price is one coin for every three feathers. For the remaining 960 feathers, the price is one coin for every four feathers.
〔四六〕今有出钱九百八十，买矢簳五千八百二十枚。欲其贵贱率之，问各几何？
荅曰：
其三百枚，五枚一钱。
其五千五百二十枚，六枚一钱。
反其率术曰：以钱数为法，所率为实，实如法而一。不满法者反以实减法，法少，实多。二物各以所得多少之数乘法实，即物数。
现有出钱九百八十，买矢簳五千八百二十枚。想要根据贵贱计算单价，问每枚多少钱？ 答：其中三百枚，五枚一钱。另外五千五百二十枚，六枚一钱。 反其率术曰：以钱数为除数，所求率为被除数，用被除数除以除数得到一。如果被除数小于除数，则反过来用被除数减去除数，因为除数少而被除数多。两种物品各自用所得的多少之数乘以被除数和除数，就得到了物品数量。
We have 980 coins to buy 5,820 arrow shafts. We want to calculate the price per arrow shaft based on quality; how much is each arrow shaft? Answer: For 300 arrow shafts, the price is one coin for every five arrow shafts. For the remaining 5,520 arrow shafts, the price is one coin for every six arrow shafts. The method in reverse says: Take the amount of money as the divisor and the desired rate as the dividend, and divide the dividend by the divisor to get one. If the dividend is less than the divisor, subtract the divisor from the dividend instead, because the divisor is smaller and the dividend is larger. For both items, multiply the number obtained by the divisor and the dividend to get the quantity of items.