Ancient math application problems, what are the ancient Chinese math problems

1. Two rats through the wall.

There is a problem of two rats penetrating the wall in Chapter 7 of the ancient mathematical classic "Nine Chapters of Arithmetic": there is a wall five feet thick, two rats wear through each other, rats are one foot a day, and mice are also one foot a day. Rats are doubled daily, and mice are doubled daily. Q: When will we meet, and how much will each wear?

Today's meaning is: there is a 5-foot thick wall, and two rats make holes through the wall from opposite sides of the wall. The rat enters one foot on the first day and doubles every day thereafter; The baby mouse also enters one foot on the first day, and then halves it every day thereafter. Q: A few days later, when the two rats meet, how many feet do they wear?

2. Chickens and rabbits in the same cage.

Chickens and rabbits in the same cage is one of the famous mathematical problems in ancient China. About 1,500 years ago, this interesting question was recorded in the "Sun Tzu's Sutra". The book is described as follows: There are pheasants and rabbits in the same cage today, with thirty-five heads on the top and ninety-four feet on the bottom.

The meaning of these four sentences is: There are several chickens and rabbits in the same cage, counting from above, there are 35 heads, and counting from below, there are 94 legs. Q: How many chickens and rabbits are in each cage?

3. Li Bai fights wine.

Li Bai walked on the street, carrying a pot to drink; Double the store, see the flowers and drink a bucket; Three encounters with shops and flowers, drink up the wine in the pot. How much wine is in the flask? This is a folk arithmetic problem.

The title is: Li Bai walked on the street, holding a wine pot while drinking and drinking, doubling the wine in the pot every time he met the hotel, and drinking a bucket every time he met a flower (bucket is an ancient capacity unit, 1 bucket 10 liters), so that he met the store and saw the flower 3 times, and drank the wine. Q: How much wine was in the pot?

4. There are many things today.

Today, there are things that do not know their number, two of the three or three numbers, three of the five or five numbers, and two of the seven or seven numbers. Ask about the geometry of things? The meaning of the title is:

There are some items, I don't know how many, I just know that if you count them in three or three, there will be two left; If you count five or five lands, there will be 3 left; Seven and seven lands, there will be two left. What is the minimum number of these items?

5. Timely pear fruit.

In 1303, the Yuan Dynasty mathematician Zhu Shijie compiled the "Four Yuan Jade Jian", which has such a topic: 999 yuan, buy 1,000 pears in time, 9 pears for 11 yuan, and 4 yuan for seven fruits. Q:

How much does a pear fruit cost? The meaning of this question is: buy a total of 1,000 pears and fruits with 999 yuan, buy 9 pears for 11 yuan, and buy 7 fruits for 4 yuan.

Q: How many pears and fruits do you buy, and how much do you pay for each?

1. Fang Tian: It mainly describes the calculation method of the area of plane geometry. It includes eight methods for calculating the area of rectangles, isosceles triangles, right-angle trapezoids, isosceles trapezoids, circles, fans, bows, and rings.

In addition, the four rules of operation of fractions and the methods of finding the greatest common divisor of the numerator and denominator are systematically described.

2. Corn: proportional exchange of cereals; Proposed a proportional algorithm, known as the present technique; The decay chapter proposes the law of proportional distribution, which is called decay.

3. Shaoguang: know the area and volume, and find the length of one side and the length of the diameter; The method of opening the square and opening the square is introduced.

4. Commercial work: earth and stone engineering, volume calculation; In addition to the various three-dimensional volume formulas given, there are also engineering allocation methods.

5. Equal loss: reasonable apportionment of taxes; Solve the problem of reasonable burden of enslavement with attenuation. Today's techniques, decay techniques and their application methods constitute a complete set of proportional theories including today's positive and negative proportions, proportional distribution, complex proportions, and linkage proportions.

It was not until the end of the 15th century that a similar set of methods was developed in the West.

6. Insufficient surplus: that is, the problem of double management; Three types of profit and loss problems are proposed, namely, surplus deficiency, surplus and shortfall, two surplus and two deficit, as well as some solutions to some general problems that can be turned into surplus and deficit problems by two assumptions. This is also the result of being a world leader, and after it spread to the West, it has had a great impact.

1. Hundred chickens.

Today, there are chickens who have been paying five, chicken mothers have three straight money, and chickens have three straight money and one money. Where 100 dollars buy 100 chickens. Ask the chickens and the hens how much they are."

Translation: 5 yuan for a rooster, 3 money for a hen, 1 money for 3 chicks, 100 chickens for 100 yuan, a few roosters and hens and chicks.

2. Surplus deficiency surgery.

Today, there are (people) who buy things together, (each) people give eight (money), and the surplus (surplus) is three money; People pay seven (money), less than four (money), ask the number of people, the price of each geometry."

Translation: Someone buys something, each person pays 8 yuan, 3 yuan is over, each person pays 7 yuan, and there is a shortage of 4 yuan, and asks how many people there are and how much the price is.

3. Methods for calculating the area of rectilinear shapes and circles.

Today there are fifteen steps from the field wide, sixteen steps from (sound zong). Ask for Tian geometry. ”

Translation: There is a field that is 15 steps long and 16 steps wide, and I ask how much the area of the field is.

4. Chickens and rabbits in the same cage.

Now there are pheasants and rabbits in the same cage, with thirty-five heads on it and ninety-four feet on the bottom. Q: What are the geometries of pheasants and rabbits? ”

Translation: There are chickens and rabbits in the same cage, with 35 heads and 94 legs, ask how many chickens and rabbits there are.

5. Gravity difference theory.

Now it is expected that the island, set up two tables, three feet high, a thousand steps before and after, so that the back table and the front table are straight. In the past, the table walked one hundred and twenty-three steps, and people looked at the peak of the island with their eyes, and joined the end of the table. One hundred and twenty-seven steps from the back table, people look at the island peak, and also join the end of the table.

Q: What is the height of the island and how much does it go to the table? Answer: The island is four miles high and fifty-five steps; One hundred and twenty miles and one hundred and fifty steps to the table.

Translation: Suppose you measure the island, the height of the two tables is 3 zhang, and the distance between the front and back is 1000 steps, so that the back table and the front table are in the same straight line, and the front table retreats 123 steps, and the person observes the island peak with his eyes, and the person observes the island peak with his eyes on the ground, and asks how high the island is? How far is the island from the previous table?

The surplus-deficit technique is a unique creation in the history of Chinese mathematics to solve application problems, and it occupies a very important position in ancient Chinese algorithms.

It also spread westward through the Silk Road to the Arab countries of Central Asia, where it received special attention and was called the "Khitan algorithm", and later introduced to Europe, where the "double method" ruled their mathematical kingdom for a long time during the Middle Ages.

Like a hundred steamed buns and a hundred monks.

The three monks are even more indisputable.

The three monks are divided into one.

How many dings are there for the big and small monks? "

There are also Han Xin's soldiers, hook three strands, four hyun five, and so on.

The remaining two of the three and three Chang limb imitation numbers, the remaining three of the five and five numbers, and the remaining two of the seven and seven numbers, ask the geometry of things.

Translated as: divide by 3 and remainder 2, divide by 5 remainder 3, divide by 7 remainder 2 This is a hungry brother problem in Sun Tzu's arithmetic, and the simplest algorithm is to use enumeration:

Divided by 3 and the remainder of 2 are: 2, 5, 8, 11, 14, 17, 23, 26 、、、 fiber.

Divide by 5 and 3 are: 3, 8, 13, 18, 23, 28、、、 divided by 7 and 2 are: 2, 9, 16, 23, 31、、、 so this number is 23

Emperor Qianlong of the Qing Dynasty.

Set up a banquet for thousands of people, and when he learned the age of the oldest old man, Emperor Qianlong molded out the upper link: the sixtieth armor was reopened, and three sevens were added.

Years. Ji Xiaolan.

Sixtieth replay: 60*2=120 Sanqi years: 3*7=21 120+21=141

Ancient Shuangqing: 70 * 2 = 140 One more fight for winter and autumn: 1 140 + 1 = 141

So it should be 141 years old.

During the 50th anniversary of Qianlong, he was in the Qianqing Palace.

A thousand feasts are held. The participant was a 141-year-old man. Qianlong made a sentence with the title of his year:

The sixtieth nail is reopened, plus thirty-seven years;

Ji Xiaolan said:

Ancient and rare double celebrations, and one more spring and autumn.

Sixtieth year, refers to sixty years old. Re-opening, refers to two sixtieth birthdays, one hundred and twenty years old. Sanqi is twenty-one years old.

The total number of Shanglian is one hundred and forty-one years old. Ancient rare, refers to the seventy years old. Shuangqing, refers to two ancient rarities, one hundred and forty years old.

A spring and autumn, that is, one year old. The sum of the lower links is also one hundred and forty-one years old. The characteristic of the coupling is that it is clever in the use of numbers.

60 for one"Sixtieth Nails","Sixtieth Nails"The reunion is 120, 3721, and 37, plus 37, that is, 141 years old.

70 for one"Ancient rarity", Gu Xi Shuangqing is 140, plus one, that is, 141 years old.

Judging from the fact that there are five of them lying on each tree, and one tree is empty.

The number of turtledoves is a multiple of 5.

So there may be 5, 10, 15 turtledoves. Wait a minute.

Suppose there are 5 turtle doves, according to each tree lying five and one tree is empty, then the tree has 2 trees, but 5 turtle doves and 2 trees cannot meet the requirements of three per tree, and there are five turtle doves that have nowhere to go, so it is not 5 turtle doves.

Suppose there are 10 turtle doves, according to five lying on each tree, and one tree is empty, then there are 3 trees, but 10 turtle doves and 3 trees cannot meet the requirements of three per tree, and there are five that have nowhere to go, so it is not 10 turtle doves.

Suppose there are 15 turtle doves, according to five per tree, one tree is empty, then there are 4 trees, but 15 turtle doves and 4 trees cannot meet the requirements of three per tree, and there are five that have nowhere to go, so it is not 15 turtle doves.

Suppose there are 20 turtle doves, according to five lying on each tree, one tree is empty, then the tree has it, 5 trees, 20 turtle doves and 5 trees are satisfied with three lying on each tree, and there are five that have nowhere to go, so it is 20 turtle doves and 5 trees.

The answer is 20 turtle doves, 4 trees.

The following is to help you give you the method of column equations.

There are x turtle doves and y trees.

Then 3y + 5 = x

5(y - 1) = x

Get x = 20 y = 5

In fact, it is not impossible to calculate the solution, but it is difficult to understand, let's look at the equation.

3y + 5 = 5(y - 1)

5y - 3y = 5 + 5

y = (5 + 5)/(5 - 3) = 5

x = 3y + 5 = 3 * 5 + 5 = 20

Then the arithmetic method is.

Find out how many trees there are: (5 + 5) (5 - 3) = 5

Then the tree is found, and the number of turtledoves can be found: 3 * 5 + 5 = 20 complete.

Tree Lesson 5 20 doves, take my answer.

5+5=10

10 divided by 2 = 5

3 times 5 = 15

5 trees, 15 doves, zinc you count.

One solution: come.

Source 1= 1 5 bai1 6 1 8 1 9 1 10 1 12 1 15 1 18 1 20 1 24 24 2 2 2 2 3+1 3-1 4+1 4-1 5+1 5-1 6+1 6-1 6-1 7+1 7-1 8+1 8-1 9+1 9-1 10=1-1 10 So: zhi 1 2+1 6+1 12+1 20+1 30+1 42+1 56+1 72+1 90+1 10=1 i.e.:

1 2-1 6-1 12-1 20-1 30-1 42-1 56-1 72-1 90-1 10=-1 References.

1. Reputation is simple.

Rent for the first year + number of houses 500 = rent for the second year.

Question: How many rooms are there? How much is the rent per house in the first year? How much is the rent per house in the second year? How much is the rent of a house in year x?

First find out the number of houses and the number of stoves:

I just said: the rent of the first year + the number of houses in the celebration book 500 = the rent of the second year.

Therefore, the number of houses = 12 (note the difference between the units of "10,000 yuan" and "yuan") so the first year: rent per house = total rent and number of houses = 8,000 yuan.

The second year: rent per house = rent per house in the first year + 500 = 8,500 yuan.