第六章   均输

现在有均输粮：甲县一万户，行道八天；乙县九千五百户，行道十天；丙县一途经一万二千三百五十户，行道十三日；丁县一万二千二百户，行道二十日，各到运输所。共四县赋，当输送二十五万斛，一个万乘用车。想把道路远近，户口数多少，穿出来的。向粮食、车各是多少？
回答说： 甲县粮食一万斛八干，四辆战车三千三百二十。 乙县粮食六万斛三千一百七十五，七辆车二千五百二十。 丙县粮食六万斛三千一百七十五，七辆车二千五百二十。 丁县粮食四万五百五十斛，两辆车一千六百二十。 均输术说：让县户数，各如其本行道日数而一，认为衰退。甲减少一百二十五，乙、丙衰各九十五，丁已六十一，副都为法。以征收粮食、几辆车还没有统一的，各自为实。确实如法得到一辆车。有分的，上下等人的。以二十五斛乘车数，即粮食数量。

Now there is an equal supply of grain: 10000 households in County A, traveling for eight days; Nine thousand five hundred households in Yi County, walking for ten days; Bingxian passes through 12350 households for 13 days on the road; Twelve thousand and two hundred households in Ding County, on the road for twenty days, each arriving at the transportation station. There are four counties in total, with a transportation capacity of 250000 mu and a passenger car of 10000. I want to know the distance of the road, the number of household registrations, and the number of people passing through it. How much is it for grain and cars respectively? He replied: Jia County has ten thousand mu and eight dried grains, and four chariots of 3320. The grain in Yixian County is 60000 mu, 3155 mu, and seven vehicles are 2520 mu. Bingxian has a grain of 60000 mu and 315 mu, with seven vehicles and 2520 mu. Ding County has 4550 grains and two carts of 1620. According to the theory of equal loss, the number of households in each county should be consistent with the number of days on the road, indicating a decline. A has decreased by 125, B and C have declined by 95 each, D has decreased by 61, and both sides are affected by the law. The collection of grain and the number of vehicles that have not yet been unified shall be considered as actual. It is indeed impossible to obtain a car. Divided, upper and lower class. The quantity of grain is calculated based on the number of vehicles carried by 25 dendrobiums.

假设现在有均输卒：甲县一千二百人，守卫边塞；乙县一千五百五十人，行程一天；丙县一千二百八十人，行程两天；丁县九百九十人，行程三天；戊县一千七百五十人，行程五天。总共五个县，每月输送兵员一千二百人。要根据距离的远近、户数的比例，按多少递减出兵员。问每个县各应出多少兵员？
答案是： 甲县应出二百二十九人。 乙县应出二百八十六人。 丙县应出二百二十八人。丁县应出一百七十一人。 戊县应出二百八十六人。 计算方法如下：让各县兵员，根据他们居住的地方以及行程日数的总和作为一个单位，以此作为递减的基准。甲的单位是四，乙的单位是五，丙的单位是四，丁的单位是三，戊的单位是五，将这些单位的副并起来作为标准法则。以人数乘以未合并的各自作为实际数值。实际数值除以法则。如果有分数，就上下调整。

Suppose there are transport soldiers currently: County A with 1,200 people, guarding the border; County B with 1,550 people, a one-day journey; County C with 1,280 people, a two-day journey; County D with 990 people, a three-day journey; County E with 1,750 people, a five-day journey. In total, these five counties contribute 1,200 soldiers per month. The contribution should be adjusted according to the distance, household rates, and in descending order of magnitude. How many soldiers should each county contribute? The answer is: County A should contribute 229 people. County B should contribute 286 people. County C should contribute 228 people. County D should contribute 171 people. County E should contribute 286 people. The method is as follows: Let the number of soldiers from each county be divided by the sum of their residence and travel days as a unit, which serves as the basis for the decline. The unit for A is four, for B is five, for C is four, for D is three, for E is five. Combine these units as a standard rule. Multiply the number of people by the respective uncombined units as the actual value. Divide the actual value by the rule. If there is a fraction, adjust it up or down accordingly.

中文翻译： 假设现在有均输卒：甲县一千二百人，守卫边塞；乙县一千五百五十人，行程一天；丙县一千二百八十人，行程两天；丁县九百九十人，行程三天；戊县一千七百五十人，行程五天。总共五个县，每月输送兵员一千二百人。要根据距离的远近、户数的比例，按多少递减出兵员。问每个县各应出多少兵员？
答案是： 甲县应出二百二十九人。 乙县应出二百八十六人。 丙县应出二百二十八人。丁县应出一百七十一人。 戊县应出二百八十六人。 计算方法如下：让各县兵员，根据他们居住的地方以及行程日数的总和作为一个单位，以此作为递减的基准。甲的单位是四，乙的单位是五，丙的单位是四，丁的单位是三，戊的单位是五，将这些单位的副并起来作为标准法则。以人数乘以未合并的各自作为实际数值。实际数值除以法则。如果有分数，就上下调整。

Suppose there is a balanced tax grain distribution: County A has 20,520 households, with grain priced at 20 coins per hu (a unit of volume), self-transported to the county; County B has 12,312 households, with grain priced at 10 coins per hu, with a transport distance of 200 li (a unit of distance); County C has 7,182 households, with grain priced at 12 coins per hu, with a transport distance of 150 li; County D has 13,338 households, with grain priced at 17 coins per hu, with a transport distance of 250 li; County E has 5,130 households, with grain priced at 13 coins per hu, with a transport distance of 150 li. In total, these five counties are to contribute 10,000 hu of grain. One cart can carry 25 hu, and the cost to rent a cart is 1 coin per li. The grain should be transported according to the number of households in each county, so that the costs and labor are equal. How much grain should each county contribute? The answer is: County A should contribute 3,571 hu and 517/2,873 of a hu. County B should contribute 2,380 hu and 2,260/2,873 of a hu. County C should contribute 1,388 hu and 2,276/2,873 of a hu. County D should contribute 1,719 hu and 1,313/2,873 of a hu. County E should contribute 939 hu and 2,253/2,873 of a hu. The method is as follows: Multiply the cost of renting a cart per li by the distance to the delivery location, divide by the amount one cart can carry (25 hu), then add the price of one hu of grain to get the cost of transporting one hu of grain. Then, reduce this according to the number of households in each county to create a decreasing standard. The unit for A is 1,026, for B is 684, for C is 399, for D is 494, for E is 270. Combine these units as a standard rule. Multiply the amount of grain to be contributed by the respective uncombined values to get the actual amount to be contributed. Divide the actual amount by the rule to get the result.

假设现在有平衡赋税的粮食：甲县有二万五百二十户，每斛粟售价二十钱，自行输送到本县；乙县有一万二千三百一十二户，每斛粟售价十钱，输送地点距离二百里；丙县有七千一百八十二户，每斛粟售价十二钱，输送地点距离一百五十里；丁县有一万三千三百三十八户，每斛粟售价十七钱，输送地点距离二百五十里；戊县有五千一百三十户，每斛粟售价十三钱，输送地点距离一百五十里。总共五个县要缴纳的粮食是一万斛。一辆车可以装载二十五斛，且租用车辆每里需要支付一钱。要根据各县的户数来输送粮食，使得费用和劳力等同。问每个县各应缴纳多少粮食？
答案是： 甲县应缴纳三千五百七十一斛又二千八百七十三分之五百一十七。 乙县应缴纳二千三百八十斛又二千八百七十三分之二千二百六十。 丙县应缴纳一千三百八十八斛又二千八百七十三分之二千二百七十六。 丁县应缴纳一千七百一十九斛又二千八百七十三分之一千三百一十三。 戊县应缴纳九百三十九斛又二千八百七十三分之二千二百五十三。 计算方法如下：以每里的租车价格，乘以至输送地点的距离，除以一车能装载的二十五斛的数量，再加上一斛粟的售价，得出运送一斛的费用。然后根据各县的户数进行约简，作为递减的标准。甲的单位是一千零二十六，乙的单位是六百八十四，丙的单位是三百九十九，丁的单位是四百九十四，戊的单位是二百七十，将这些单位的副并起来作为标准法则。用所要缴纳的粮食数量乘以未合并的各自数值，得到实际应缴纳的数量。实际数量除以法则得出结果。

Suppose there is a balanced tax grain distribution: County A has 20,520 households, with grain priced at 20 coins per hu (a unit of volume), self-transported to the county; County B has 12,312 households, with grain priced at 10 coins per hu, with a transport distance of 200 li (a unit of distance); County C has 7,182 households, with grain priced at 12 coins per hu, with a transport distance of 150 li; County D has 13,338 households, with grain priced at 17 coins per hu, with a transport distance of 250 li; County E has 5,130 households, with grain priced at 13 coins per hu, with a transport distance of 150 li. In total, these five counties are to contribute 10,000 hu of grain. One cart can carry 25 hu, and the cost to rent a cart is 1 coin per li. The grain should be transported according to the number of households in each county, so that the costs and labor are equal. How much grain should each county contribute? The answer is: County A should contribute 3,571 hu and 517/2,873 of a hu. County B should contribute 2,380 hu and 2,260/2,873 of a hu. County C should contribute 1,388 hu and 2,276/2,873 of a hu. County D should contribute 1,719 hu and 1,313/2,873 of a hu. County E should contribute 939 hu and 2,253/2,873 of a hu. The method is as follows: Multiply the cost of renting a cart per li by the distance to the delivery location, divide by the amount one cart can carry (25 hu), then add the price of one hu of grain to get the cost of transporting one hu of grain. Then, reduce this according to the number of households in each county to create a decreasing standard. The unit for A is 1,026, for B is 684, for C is 399, for D is 494, for E is 270. Combine these units as a standard rule. Multiply the amount of grain to be contributed by the respective uncombined values to get the actual amount to be contributed. Divide the actual amount by the rule to get the result.

假设现在有7斗谷子，三个人分别将其舂成不同等级的米，第一个人舂成粗米，第二个人舂成较细的米，第三个人舂成最细的米，要求每种米的量相等。问每个人应该取多少谷子来舂？
答案是： 粗米取2斗加上121分之10。 较细的米取2斗加上121分之38。 最细的米取2斗加上121分之73。 这样每个人舂出的米都是1斗加上605分之151。计算方法是：列出粗米30份，较细的米27份，最细的米24份，然后按反比递减，将副产品合并起来作为标准。用7斗乘以未合并的数值，各自得出取谷子的实际数量。实际数量除以标准得到1斗。如果要求米的量相等，就用原来的比例分别乘以确定取出的谷子作为实际数值，以谷子的比率50作为标准，实际数值除以标准得到1斗。

Suppose there are seven pecks of millet, which are to be divided among three people for milling. One person mills it into coarse rice, another into medium rice, and the third into fine rice, such that the quantity of each type of rice is equal. How much millet should each person take to mill? The answer is: Coarse rice takes two pecks plus ten out of one hundred and twenty-one. Medium rice takes two pecks plus thirty-eight out of one hundred and twenty-one. Fine rice takes two pecks plus seventy-three out of one hundred and twenty-one. In this way, each person mills one peck plus one hundred and fifty-one out of six hundred and five. The method is as follows: list thirty parts for coarse rice, twenty-seven parts for medium rice, and twenty-four parts for fine rice, then decrease inversely proportionally, combining the by-products as a standard. Multiply the uncombined numbers by seven pecks to get the actual amount of millet taken. The actual amount divided by the standard gives one peck. If the amount of rice is to be equal, multiply the determined amount of millet taken by the original ratio as the actual value, with the millet rate of fifty as the standard, and the actual value divided by the standard gives one peck.

现在有一个人应该领取两斛（古代容量单位，一斛约等于50升）的粟米。但是仓库里没有粟米了，想要给他一升米和两升豆子来代替应该领取的粟米。问应该如何换算？
回答是： 米五斗一升，也就是七分之三升。豆子一斛二升，也就是七分之六升。 计算方法：将一升米和两升豆子折算成粟米的数量。合并起来得到三分之八，作为换算的标准。同样地，也将一升米和两升豆子，用两斛粟米的数量去乘，分别得到实际的折算数量。实际的折算数量除以标准得到一斛。

Now there is a person who should receive two pecks of millet. There is no millet in the storehouse, and they want to give him one liter of rice and two liters of beans instead of the millet he should receive. How should it be converted? The answer is: Rice: five bushels and one liter, which is three-sevenths of a liter. Beans: one peck and two liters, which is six-sevenths of a liter. Calculation method: Convert one liter of rice and two liters of beans into the quantity of millet. Combined, this gives eight-thirds, which serves as the conversion standard. Similarly, also take one liter of rice and two liters of beans, multiply by the quantity of two pecks of millet, to get the actual conversion amounts. The actual conversion amounts divided by the standard give one peck.

假设现在有人被雇来背着两斛（古代量度单位，一斛等于五斗）盐走一百里路，他会得到四十文钱的报酬。现在这个人背了一点七斗三升（比一斛少半升）的盐走了八十里路。问他应该得到多少钱？
答案是：应该给他二十七文钱加上十五分之十一文钱。 计算方法如下：将两斛盐换算成升数，乘以一百里作为标准。用四十文钱乘以现在背的盐的升数，再乘以八十里，得出实际应给的钱数。实际应给的钱数除以标准得到每一文钱的价值。

Suppose someone is hired to carry two hu (an ancient unit of measure, one hu equals five dou) of salt for a hundred miles, and they are paid forty wen (ancient Chinese currency). Now this person has carried a bit over one dou and three sheng (a sheng being half a liter) of salt, which is slightly less than one hu by half a sheng, for eighty miles. How much should they be paid? The answer is: They should be paid twenty-seven wen and eleven parts out of fifteen wen. The method is as follows: Convert the two hu of salt into sheng, and multiply by a hundred miles to set the standard. Multiply the number of sheng of salt currently carried by forty wen, and then multiply again by eighty miles to find the actual payment. The actual payment divided by the standard gives the value of each wen.

现在有一个背负笼子的人，笼子重1石17斤，走了76步，来回走了50次。现在这个人背的笼子重1石，走了100步，问可以来回走多少次？
答案是：43次来回，以及60分之23次来回。 计算方法：用现在走的步数乘以现在笼子的重量作为标准，原来的笼子重量乘以原来的步数，再乘以来回的次数，作为实际数值。实际数值除以标准得到一次来回。

There is a person carrying a cage, the weight of the cage is 1 shi and 17 jin (ancient Chinese units of weight, with 1 shi = 100 jin), who walks 76 steps and makes 50 round trips. Now the weight of the cage this person carries is 1 shi, and he walks 100 steps. How many round trips can be made? The answer is: 43 round trips, and 23/60 of a round trip. Method of calculation: Multiply the number of steps walked now by the weight of the cage now to create a standard; multiply the original weight of the cage by the original number of steps, then multiply by the number of round trips to get the actual value. Divide the actual value by the standard to get one round trip.

假设现在有一个运输任务，空车每天能行驶70里，重车每天能行驶50里。现在有一辆车装载着太仓的粟米运往上林，五天内往返了三次。问太仓到上林的距离是多少？
答案是：48里加上18分之11里。 计算方法如下：将空车和重车的里程数相加，然后乘以三次往返作为标准。让空车和重车的里程数相乘，再乘以五天，得出实际里程数。实际里程数除以标准得到单程距离。

Suppose there is a transportation task, with an empty cart traveling 70 miles per day and a loaded cart traveling 50 miles per day. Now, a cart loaded with grain from the Tai Granary is transported to the Shanglin Park, making three round trips in five days. How far is it from the Tai Granary to Shanglin Park? The answer is: 48 miles plus 11/18 of a mile. The method is as follows: Add up the mileage for both empty and loaded carts, then multiply by the number of round trips to establish a standard. Multiply the mileage for empty and loaded carts together, and then multiply again by the number of days to get the actual mileage. The actual mileage divided by the standard gives the distance for one trip.

现在有络丝一斤，可以练成练丝十二两；练丝一斤，可以得到青丝一斤十二铢。现在有青丝一斤，问原本的络丝有多少？
答案是：一斤四两十六铢，再加上全部的三分之一十六分之一。 计算方法：用练丝十二两乘以青丝一斤十二铢作为标准。用青丝一斤的铢数乘以练丝一斤的两数，再乘以络丝一斤的数量，得到实际数值。实际数值除以标准得到一斤络丝。

Now, one catty of raw silk can be processed into twelve taels of refined silk; one catty of refined silk can produce one catty and twelve zu of green silk. Now, with one catty of green silk, how much raw silk was originally required? The answer is: one catty, four taels, sixteen zu, plus one-sixteenth of the total. Method of calculation: Multiply twelve taels of refined silk by twelve zu of green silk to get the standard. Multiply the number of zu in one catty of green silk by the number of taels in one catty of refined silk, then multiply by the quantity of one catty of raw silk to get the actual value. The actual value divided by the standard gives one catty of raw silk.

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.

假设现在有个人走路非常快，能走100步，而另一个人走得比较慢，只能走60步。如果走得慢的那个人先走了100步，走得快的那个人追赶他，问他需要多少步才能追上？
答案是：250步。 计算方法如下：将走得快的人的100步减去走得慢的人的60步，剩下40步作为标准。用走得快的人的100步乘以走得慢的人先走的100步得出实际步数。实际步数除以标准得出每一步的距离。

Suppose there is a fast walker who can take 100 steps, and a slow walker who can only take 60 steps. If the slow walker starts with a 100-step lead, how many steps does the fast walker need to catch up? The answer is: 250 steps. The method is as follows: Subtract the slow walker's 60 steps from the fast walker's 100 steps, leaving 40 steps as the standard. Multiply the fast walker's 100 steps by the slow walker's initial 100 steps to find the actual number of steps. The actual number of steps divided by the standard gives the distance per step.

现有不善行者先行10里，善行者追赶100里，比不善行者先到达20里。问善行者在多少里后能追上不善行者？
答案是：33里又1/3里。 解法是：将不善行者先行的10里加上善行者先到达的20里，作为除数。将不善行者先行的10里乘以善行者的100里，作为被除数。用被除数除以除数得到追及的距离。

Suppose there is a slower traveler who starts with a 10-li lead, and a faster traveler chases over 100 li, reaching the destination 20 li ahead of the slower traveler. How many li does the faster traveler need to catch up with the slower traveler? The answer is: 33 li and 1/3 of a li. The method is as follows: Take the 10 li that the slower traveler started with and add the 20 li that the faster traveler arrived ahead of, to use as the divisor. Multiply the 10 li that the slower traveler started with by the 100 li that the faster traveler covers, to use as the dividend. Dividing the dividend by the divisor gives the distance of pursuit.

假设现在有一只兔子先走了100步，然后一只狗开始追赶，追了250步后，还差30步就能追上兔子。问如果狗不停下来，继续走，还需要多少步才能追上兔子？
答案是：107步加上七分之一（1/7）步。 计算方法如下：将兔子先走的100步减去狗追赶时差的30步，剩下的作为标准。用这个差值乘以狗追赶的步数得出实际步数。实际步数除以标准得出每一步的距离。

Suppose a rabbit has a head start of 100 steps, and then a dog starts to chase. After running 250 steps, the dog is still 30 steps behind the rabbit. If the dog doesn't stop, how many more steps does it need to catch up with the rabbit? The answer is: 107 steps plus one seventh (1/7) of a step. The method is as follows: Subtract the 30 steps that the dog is short from the rabbit's initial 100 steps, and the remainder becomes the standard. Multiply this difference by the number of steps the dog has chased to get the actual number of steps. The actual number of steps divided by the standard gives the distance per step.

假设现在有人拿着12斤金出关，关税是金子重量的十分之一。现在关税收了2斤金，并补偿了5000钱。问一斤金值多少钱？
答案是：6250钱。 计算方法如下：用10乘以2斤金，然后从12斤金中减去这个数值，剩下的作为标准。再用10乘以5000钱得出实际的钱数。实际的钱数除以标准得出每斤金的价值。

Suppose someone is carrying 12 catties of gold when leaving the customs. The customs tax is one-tenth of the weight in gold. Now, the customs took 2 catties of gold and compensated 5000 coins. How much is one catty of gold worth? The answer is: 6250 coins. The method is as follows: Multiply 2 catties by 10, then subtract this value from 12 catties; the remainder serves as the standard. Multiply 5000 coins by 10 to get the actual amount of money. The actual amount of money divided by the standard gives the value per catty of gold.

假设有一个客人的马每天能走300里。客人离开时忘记带衣服，走了一天的三分之一后，主人才发现。主人拿着衣服追赶上客人并一起返回，到家时太阳已经过去了四分之三天。问主人的马如果不休息，每天能走多少里？
答案是：780里。解题方法是：取四分之三天的三分之一，再除以三分之第一天的一半，作为除数。再将这个除数增加三分之第一天的数量，然后乘以300里，得到实际距离。实际距离除以除数，就得到主人的马每天行走的距离。

Suppose a guest's horse can travel 300 li per day. The guest forgot to take his clothes when he left, and the host realized this after one-third of the day had passed. The host then chased after the guest with the clothes and returned together, arriving home by three-quarters of the day. How many li can the host's horse travel in a day without rest? The answer is: 780 li. The method is as follows: Take one-third of three-quarters of the day, divide it by one-half of one-third of the day, to use as the divisor. Then increase the divisor by one-third of the day, multiply by 300 li, to get the actual distance. Divide the actual distance by the divisor to get the daily travel distance of the host's horse.

假设现在有一把金锤，长5尺。从头部斩下一尺重4斤，从尾部斩下一尺重2斤。问接下来的每一尺各重多少？
答案是：最后一尺重2斤； 次一尺重2斤8两； 次一尺重3斤； 次一尺重3斤8两； 次一尺重4斤。 计算方法如下：让尾部的重量减去头部的重量，余数即为差率。再将头部的重量乘以4，得到最下面的第一个衰（序列）。副置，用差率减去它，每尺各自为衰。副置下第一衰作为标准，用头部的4斤重量乘以列衰，各自得到实际重量。实际重量除以标准得到每一斤的重量。

Suppose there is a gold hammer, 5 feet long. If the first foot cut from the top weighs 4 catties, and the first foot cut from the bottom weighs 2 catties, how much does each subsequent foot weigh? The answer is: The last foot weighs 2 catties; The second-to-last foot weighs 2 catties and 8 taels (2.8 pounds); The third-to-last foot weighs 3 catties; The fourth-to-last foot weighs 3 catties and 8 taels (3.8 pounds); The fifth-to-last foot weighs 4 catties. The method is as follows: Subtract the weight of the bottom part from the top part to get the difference ratio. Then multiply the weight of the top part by 4 to get the first sequence (decay). Place this aside, subtract the difference ratio from it, and each foot will have its own decay. Place the first decay as the standard, multiply it by the top weight of 4 catties, and get the actual weights for each foot. The actual weights divided by the standard give the weight per catty.

假设现在有5个人分5钱，要求上面2个人所得的钱和下面3个人的钱相等。问每个人各得多少钱？
答案是：甲得1钱又2/6钱，乙得1钱又1/6钱， 丙得1钱，丁得5/6钱， 戊得4/6钱。 计算方法如下：将钱按照锥形排列，上面2个人的总和为9，下面3个人的总和为6。6比9少3。用3平均加到每个人头上，然后加起来作为标准。用所分的钱乘以未合并的数各自得到实际的钱数。实际的钱数除以标准得到每一钱的价值。

Suppose there are 5 people sharing 5 coins, with the condition that the amount received by the top 2 people should be equal to that of the bottom 3 people. How much does each person get? The answer is: The first person gets 1 coin and 2/6 of a coin, the second person gets 1 coin and 1/6 of a coin, The third person gets 1 coin, the fourth person gets 5/6 of a coin, The fifth person gets 4/6 of a coin. The method is as follows: Arrange the coins in a tapering row, with the sum of the top two people being 9, and the sum of the bottom three people being 6. 6 is less than 9 by 3. Add 3 evenly to each person, then sum them up to use as the standard. Multiply the distributed coins by the unmerged numbers to get the actual amount of money for each person. The actual amount divided by the standard gives the value of each coin.

现有竹子共9节，下面3节的容量总共是4升，上面4节的容量总共是3升。问中间的2节每节各容纳多少升？
答案是：最下面的一节是1升又66分之29升， 其次是1升又66分之22升， 再次是1升又66分之15升， 然后是1升又66分之8升， 接着是1升又66分之1升， 然后是66分之60升， 再接着是66分之53升， 然后是66分之46升， 最后是66分之39升。 解题方法是：将下面3节的总容量4升作为下率，上面4节的总容量3升作为上率。用上率减去下率，得到的差值作为实际值。将4节和3节各自一半后相减，从9节中减去，余下的作为除数。实际值除以除数得到1升，这就是每节之间的差距。下率中，比1升少的部分，就是下面第二节的容量。

We have a bamboo with 9 segments; the bottom three segments hold a total of 4 liters, while the top four segments hold a total of 3 liters. What is the capacity of each of the middle two segments? The answer is: The very bottom segment holds 1 liter and 66/29 liters, followed by one holding 1 liter and 66/22 liters, then one with 1 liter and 66/15 liters, then one with 1 liter and 66/8 liters, then one with 1 liter and 66/1 liters, then one with 66/60 liters, then one with 66/53 liters, then one with 66/46 liters, and finally one with 66/39 liters. The method for solving this is as follows: Take the total capacity of the bottom three segments, which is 4 liters, as the lower rate, and the total capacity of the top four segments, which is 3 liters, as the upper rate. Subtract the lower rate from the upper rate to get the actual value. Halve the number of segments for the top four and bottom three, then subtract that from the nine segments to get the remainder, which serves as the divisor. Dividing the actual value by the divisor yields 1 liter, which represents the difference between each segment. The part of the lower rate that is less than 1 liter corresponds to the capacity of the second segment from the bottom.

假设现在有一只野鸭从南海起飞，7天后到达北海；一只雁从北海起飞，9天后到达南海。现在野鸭和雁同时起飞，问它们在哪一天相遇？
答案是：第3天加上16分之15。 计算方法如下：将天数相加作为标准，天数相乘得到实际天数，实际天数除以标准得到相遇的天数。

Suppose there is a wild duck that starts flying from the South Sea and reaches the North Sea in 7 days; a goose starts from the North Sea and arrives at the South Sea in 9 days. Now, if the wild duck and the goose take off at the same time, on which day will they meet? The answer is: on the third day plus fifteen-sixteenths of a day. The method is as follows: Add the number of days together to get the standard, multiply the number of days to get the actual number of days, and divide the actual number of days by the standard to get the day of meeting.

现有甲从长安出发，五天到达齐地；乙从齐地出发，七天到达长安。现在乙已经提前两天出发了，甲才从长安出发。问他们多少天后会相遇？
答案是：两天又十二分之一。 解题方法是：将五日和七日相加作为除数。用乙提前出发的两日减去七日，剩下的天数乘以甲的日数作为被除数。被除数除以除数得到一日。

Suppose there is a person A departing from Chang'an, reaching Qi in five days; another person B departs from Qi and arrives at Chang'an in seven days. Now, B has already started two days earlier, and A just departed from Chang'an. How many days will it take for them to meet? The answer is: two days and one twelfth of a day. The method is as follows: Add the five days and seven days together to use as the divisor. Subtract the two days that B started early from the seven days, and multiply the remaining days by the number of days A travels to get the dividend. The dividend divided by the divisor yields one day.

假设现在有一个人一天能做38枚公瓦，另一个人一天能做76枚母瓦。现在让一个人一天做的瓦片数量是公瓦和母瓦各半，问他一天能做成多少瓦？
答案是：25枚又1/2枚。 计算方法如下：将公瓦和母瓦的数量相加作为标准，公瓦和母瓦的数量相乘得到实际瓦数，实际瓦数除以标准得到每枚瓦的数量。

Suppose there is a person who can make 38 male tiles in one day, and another person can make 76 female tiles in one day. Now, if a person is asked to make an equal number of male and female tiles in one day, how many tiles can he complete? The answer is: 25 tiles and a half. The method is as follows: Add the number of male and female tiles together to get the standard, multiply the number of male and female tiles to get the actual number of tiles, and divide the actual number of tiles by the standard to get the number of tiles per tile.

假设现在有一个人，一天可以矫正箭矢50支，另一个人一天可以给箭矢装上羽毛30支，还有一个人一天可以装上箭尾15支。如果让一个人在一天内完成矫正、装羽毛和装箭尾的整个过程，问最终能完成多少支箭？
答案是：8支又1/3支箭。 解题方法如下：制作50支矫正箭需要1个工人；制作50支羽箭需要1个工人加上2/3个工人；制作50支装有箭尾的箭需要3个工人加上1/3个工人。将这些人合并起来，得到6个工人的工作量，以此作为标准。以制作50支箭为实际工作量。按照这个工作量与标准的比例，我们可以计算出一个工人一天能完成的箭矢数量。

Suppose there is a person who can straighten 50 arrows in one day, another person who can feather 30 arrows in one day, and another person who can attach the arrowheads to 15 arrows in one day. If one person is asked to complete the entire process of straightening, feathering, and attaching arrowheads in one day, how many arrows can ultimately be completed? The answer is: 8 and 1/3 arrows. The method for solving this is as follows: To make 50 straightened arrows requires 1 worker; to make 50 feathered arrows requires 1 worker plus 2/3 of a worker; to make 50 arrows with attached heads requires 3 workers plus 1/3 of a worker. Combining these workers together results in the workload of 6 workers, which is taken as the standard. With the actual workload being the production of 50 arrows, we can calculate the number of arrows one worker can complete in a day based on the ratio of the actual workload to the standard.

假设现在有租田，第一年每3亩地租1钱，第二年每4亩地租1钱，第三年每5亩地租1钱。总共3年得到100钱，问共有多少亩田？
答案是：1顷又27亩加上47分之31亩。 计算方法如下：将亩数和钱数放在一起，让亩数相互乘以钱数，总和作为标准。亩数相乘，又用100钱相乘，得到实际的亩数。实际的亩数除以标准得到每亩的价值。

Suppose there is some leased land, where the rent is 1 coin for every 3 mu of land in the first year, 1 coin for every 4 mu in the second year, and 1 coin for every 5 mu in the third year. A total of 100 coins are collected over three years. How many mu of land are there? The answer is: 1 qing (a unit of area equal to 100 mu), plus 27 mu and an additional 31/47 mu. The method is as follows: Place the number of mu and the number of coins together, have the number of mu multiply the number of coins, and add them together to use as the standard. Multiply the number of mu by itself, then multiply by 100 coins to get the actual number of mu. The actual number of mu divided by the standard gives the value per mu.

假设现在有一个耕作的过程，其中一个人一天可以翻土7亩，另一个人一天可以耕地3亩，还有一个人一天可以播种5亩。如果现在让一个人在一天内完成翻土、耕地和播种的全部过程，问他能处理多少亩田地？
答案是：1亩加上114步，以及66/71步。 解题方法如下：将翻土、耕地和播种的亩数分别乘以对应的人数，然后将这些乘积相加，得到的结果作为标准（法）。将翻土、耕地和播种的亩数相互乘积，得到的结果是实际完成的工作量（实）。用实际工作量除以标准，就可以得到一天内一个人能完成的亩数。

Suppose there is a farming process where one person can plow 7 mu of land in one day, another person can till 3 mu, and another person can sow seeds over 5 mu in a day. If now one person is asked to complete the entire process of plowing, tilling, and sowing in one day, how many mu of land can he handle? The answer is: 1 mu plus 114 steps, and an additional 66/71 steps. The method for solving this is as follows: Multiply the number of mu for plowing, tilling, and sowing by the respective number of people, and add these products together to get the standard (fa). The actual work completed (shi) is obtained by multiplying the number of mu for plowing, tilling, and sowing together. Dividing the actual work by the standard gives the number of mu that one person can complete in a day.

假设现在有一个池塘，由五个渠道供水。第一个渠道开放，需要半天才能填满；第二个渠道，需要1天才能填满；第三个渠道，需要2.5天才能填满；第四个渠道，需要3天才能填满；第五个渠道，需要5天才能填满。现在这五个渠道都打开了，问池塘多久能被填满？
答案是：74又15分之1天。 计算方法如下：分别设置每个渠道一天内能填满池塘的比例，将它们加起来作为标准。以1天为实际值。实际值除以标准得到池塘被填满的天数。另一种方法是，列出日数和填满的次数，让日数相互乘以填满的次数，总和作为标准，日数相乘得到实际值，实际值除以标准得到池塘被填满的天数。

Suppose there is a pond, which is fed by five channels. The first channel can fill the pond in half a day; the second one takes one day to fill it; the third one takes two and a half days; the fourth one takes three days; the fifth one takes five days. Now all these channels are opened, how long will it take to fill the pond? The answer is: 74 days and 1/15 of a day. The method is as follows: Set up the number of days each channel can fill the pond on its own, add them together to use as the standard. Use one day as the actual value. The actual value divided by the standard gives the number of days it takes to fill the pond. Another method is to list the number of days and the number of times filled, multiply the days by the times filled, add them together to use as the standard, multiply the numbers of days to get the actual value, and the actual value divided by the standard gives the number of days to fill the pond.

假设现在有人带着米通过三个关卡，外关抽税三分之一，中关抽税五分之一，内关抽税七分之一，最后剩下5斗米。问他原本带了多少米？
答案是：10斗又9升加上8分之3升。 计算方法如下：将剩余的5斗米分别乘以3、5、7作为实际值。将未被抽税的部分2、4、6相乘作为标准。实际值除以标准得到每斗米的数值。

Suppose there is a person carrying rice through three checkpoints. At the outer checkpoint, one-third of the rice is taxed; at the middle checkpoint, one-fifth is taxed; at the inner checkpoint, one-seventh is taxed. After passing through all the checkpoints, there are 5 dou of rice left. How much rice was originally carried? The answer is: 10 dou and 9 sheng plus 3/8 of a sheng. The method is as follows: Take the remaining 5 dou of rice. Multiply it by three, five, and seven to get the actual amount. Multiply the untaxed parts, which are two, four, and six, to get the standard. The actual amount divided by the standard gives one dou of rice.

假设现在有人拿着金子经过五个关口，第一个关口收税金的二分之一，第二个关口收税金的三分之一，第三个关口收税金的四分之一，第四个关口收税金的五分之一，第五个关口收税金的六分之一。这五个关口总共收取的税正好是一斤（金）。问这个人最初拿了多少金子？
答案是：一斤三两四铢，再加上五铢的四分之五。 解题方法如下：设初始金子的重量为一斤，将通过每个关口所需缴纳的税额相加作为实际支付的税金（实）。同时，将每个关口未缴税部分相减，得到的差值为法。用实际支付的税金除以法，就可以得到最初持有的金子重量。

Suppose there is a person carrying gold through five checkpoints. The first checkpoint taxes one-half of the gold, the second checkpoint taxes one-third, the third checkpoint taxes one-fourth, the fourth checkpoint taxes one-fifth, and the fifth checkpoint taxes one-sixth. The total tax collected from all five checkpoints amounts to exactly one jin (of gold). How much gold did this person originally have? The answer is: one jin, three liang, four zhu, and four-fifths of five zhu. The method for solving this is as follows: Assume the initial weight of the gold is one jin. Add up the tax rates at each checkpoint to calculate the actual tax paid (shi). Also, subtract the untaxed portions at each checkpoint from the total, and the remainder becomes the standard (fa). Dividing the actual tax paid by the standard gives the initial amount of gold the person had.