999 pence for sweet fruits and bitter fruits, a bunch of sweet fruits, nine eleven pence for bitter fruits and seven pence for four pence. How many sweet fruits and bitter fruits are there? How much is each?

Suppose the sweet one bought X and the bitter one bought Y.

X+Y= 1000

X/9* 1 1+Y/7*4=999

Solution: X=657

Y=343

A: I bought 657 sweet ones and 343 bitter ones.

2. A square table consists of 1 desktop and 4 legs. If 1 m3 of wood can be used to make 50 square tables or 300 legs, and there are 5 m3 of wood, how many m3 of wood can be used to make a desktop and how many m3 of legs can be used to make a square table? How many square tables can you make?

Set x cubic meters as the desktop and y cubic meters as the legs. You can make 50 times the desktop and 300 legs.

There are 5 cubic meters of wood, and the equation1:x+y = 5 is obtained;

According to the table top and legs, the square table can be made into equation 2: 50x:300y= 1:4.

Solution: x = 3, y = 2.

So use 3 cubic meters to make the table top and 2 cubic meters to make the table legs.

A total of 50× 3 = 150 tables.

These two questions are both classic.

Classical problems of linear equations

1. The solution of the unknown equation is ()

A.B. C. D。

2. To make the sum reciprocal, then the value is ().

A.B. C. D。

As we all know, this is a linear equation, then

4. If and are similar projects, then

5. If it is about the solution of the equation, then

6. If the equation about is a linear equation, the solution of this equation is.

6. It is known that if there is a maximum, the solution of the equation is.

7. The equation is represented by an algebraic expression containing X, and X is represented by an algebraic expression containing Y. ..

3. When solving the equation, turn the denominator into an integer.

2. the solution of the equation and the solution of the equation about x are reciprocal, and the value of k is found.

7.

6.3. 1 From practical problems to equations

First, please manage the key points of this lesson.

The general steps to solve application problems with column equations are:

(1) "Find": See the meaning of the question clearly, analyze the problem and its relationship, and find out the _ _ _ _ _ _ _ _ _ _ used to list the equations;

(2) "Hypothesis": use letters (such as X) to represent the _ _ _ _ _ of the question;

(3) "Column": use algebraic expressions of letters to express relevant quantities, and list equations according to _ _ _ _ _ _ _;

(4) "solution": solving the equation;

(5) "Test": check whether the obtained value is correct and conforms to the actual situation, and write the answer;

(6) "Answer": Answer the questions raised in the title.

Second, the basic questions, please do.

1. It is known that the circumference of a rectangle is 20cm, and if its length is x cm, its width is ().

A.20 x b . 10-x c . 10-2x d . 20-2x

2. Student A, with every 10 person as a group, has two groups, each with 1 person, so there are () groups of students.

A. 10a-2 b . 10-2a c . 10-(2-a)d .( 10+2)/a

Third, the comprehensive questions, please have a try.

1. In extracurricular activities, Teacher Zhang found that most of the students were 13 years old. He asked his classmates: "I am 45 years old this year. In a few years, your age will be one-third of mine? "

Xiaoming's father saved 3000 yuan for Xiaoming's education three years ago. Due this year, the total principal and interest is 3243 yuan. Please help Xiao Ming calculate the annual interest rate of this savings.

3. Xiao Zhao went to the store to buy exercise books. When he came back, he asked his classmates, "The shopkeeper told me that if I bought more, I would get a 20% discount. I bought 20 copies and it turned out to be cheaper 1.60 yuan. " Can you list the equations?

Fourth, it is easy to make mistakes. Please think about it.

1. When pouring cement column, the construction personnel should bend the steel bar into a square. If the area of each square is 400 square centimeters, which steel bar should be selected in the table below?

Model A B C D

Length (cm) 90 70 82 95

Idea: The solved equation has two values, so it is necessary to check whether the obtained values are correct and conform to the actual situation. Because the reinforcement length is positive, x=80 is selected, so C-shaped reinforcement should be selected.

2. Do you make mistakes in your homework? Please record it and analyze the cause of the error.

6.3.2 Travel issues

First, please manage the key points of this lesson.

1. Basic relationship: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _;

2. Basic types: problems encountered; Distance problem; ____________ ;

3. Basic analysis method: draw a schematic diagram to analyze the meaning of the question, distinguish between speed and time, and find the equivalence relationship (the distance is divided into several parts).

4. The quantitative relationship of navigation problems:

(1) downstream (wind) sailing distance = downstream (wind) sailing distance

(2) Downstream (wind) speed = _ _ _ _ _ _ _ _ _ _ _ _ _ _

Current (wind) speed = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

Second, the basic questions, please do.

1, A's speed is 4 kilometers per hour, so he drives () kilometers per hour.

2. If B walks X kilometers in 3 hours, his speed is ().

3. Party A travels 4 kilometers per hour and Party B travels 5 kilometers per hour, so Party A and Otsuichi travel () kilometers and () kilometers per hour.

4. A certain distance is X kilometers. If the train runs at a speed of 49 kilometers per hour, it will take () hours to complete the train journey.

Third, the comprehensive questions, please have a try.

1.The distance between a and b is180km. One person rides a bike from A15km, and the other person rides a motorcycle from B.. As we all know, the speed of a motorcycle is three times that of a bicycle. If two people start at the same time and go in opposite directions, how long will it take to meet?

2. the distance between a and b is 180km. One person rides a bike every hour from A 15km, and the other person rides a motorcycle from B. As we all know, the speed of a motorcycle is three times that of a bicycle. If two people travel in the same direction, ride a bike first and start for 2 hours. How long will it take for the motorcycle to catch up with the bike?

3. A helicopter flies between two cities, A and B. It takes 4 hours to fly with the wind and 5 hours to fly against the wind. If the wind speed is 30 km/h, find the distance between cities A and B. 。

Fourth, it is easy to make mistakes. Please think about it.

1. Both Party A and Party B are running at a constant speed on the 400-meter circular track. They both start from the same place and walk in the same direction at the same time. The speed of Party A is 100 m/min, which is 3/2 times that of Party A. Q (1) How long did it take them to meet for the first time? (2) What happened the second time?

Thinking: This question is about the types of travel in the same direction in the travel problem. As can be seen from the title, when A and B met for the first time, B walked one more lap than A; When we met for the second time, the distance between them was two laps. So we met for the first time after 8 minutes, 16 minutes later for the second time.

2. Do you make mistakes in your homework? Please record it and analyze the cause of the error.

6.3.3 Distribution problem

First, please manage the key points of this lesson.

Initially learn to use equations to solve various application problems of allocation problems; The basic methods and key points of analyzing a class of application problems with total amount equal to _ _ _ _ _ _.

Second, the basic questions, please do.

1. Someone made 330 parts in three days, and it is known that there are three more parts on the second day than on the first day, and two times less on the third day. How many parts did he make on the first day?

Solution: If he makes X parts on the first day, he will make _ _ _ _ _ _ parts on the second day.

The third day, _ _ _ _ _ _ _ _ _ _ the third day, _ _ _ _ _ _ _ the third day.

List the equations: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

To solve this equation: _ _ _ _ _ _ _ _ _.

On the first day, he made _ _ _ _ _ parts.

2. There are 48 students in Class A and 52 students in Class B. Now 12 students from other schools are inserted into Class A, and the rest are inserted into Class B. After the insertion, there are _ _ _ _ _ _ _ _ _ _ _ _ students in Class A and Class B. It is known that the number of students in Class A is three times more than that in Class B..

Third, the comprehensive questions, please have a try.

1. There are 23 people working in A, and there are 17 people working in B. Now, 20 people are transferred to support, so the number of people working in A is twice as much as that working in B. How many people should be transferred to A and B respectively?

2. In order to encourage water conservation in a certain place, the monthly water fee is charged according to the following regulations: if the monthly water consumption of each household does not exceed 20 tons, it will be charged 1.2 yuan per ton of water; If the monthly water consumption of each household exceeds 20 tons, the excess will be charged at 2 yuan per ton. If a user's water fee in May averages per ton 1.5 yuan, how much should the user pay in May?

3. The unit price of methyl sugar per kilogram is 20 yuan, and the unit price of ethyl sugar per kilogram is 15 yuan. If you want to prepare 200 kilograms of mixed candy with the unit price of 18 yuan per kilogram, and keep the total revenue of selling the two kinds of candy unchanged, how many kilograms do you need for each of the two kinds of candy?

Fourth, it is easy to make mistakes. Please think about it.

1. Prepare a kind of concrete, and the mass ratio of cement, sand, gravel and water is 1: 3: 10: 4. How many kilograms of various raw materials are needed to prepare 360 kilograms of this concrete?

The key to this problem is how to set an unknown number and then solve it according to the relationship between the partial sum and the population, in which cement accounts for 20 kg.

2. Do you make mistakes in your homework? Please record it and analyze the cause of the error.

Engineering problems

First, please manage the key points of this lesson.

1. Basic relations in engineering problems:

Total amount of work = working efficiency × working time

Sum of workload of each part = total workload

Second, the basic questions, please do.

1. To do a certain job, it takes 8 points for Party A to do it alone and Party B to do it alone 12. Q:

① When doing 1, how much of the total workload did A complete? _____ 。

② How much of the total workload did B complete when doing 1? _____ 。

③ How much of the total workload did Party A and Party B complete when doing 1? _____ 。

④ When executing X, how much part of the total workload did A complete? _____ 。

⑤ How many parts of the total workload did Party A and Party B complete when doing X? _____ 。

When A finished 2 first, how much of the total workload did he finish? _____ 。

B What is the amount of work completed in the last 3 hours? _____ 。

How much of the total workload did Party A and Party B jointly do X? _____ 。

What is the proportion of the workload completed three times to the total workload?

In this way, the work is completed and the equations can be listed: _ _ _ _ _ _ _ _

Third, the comprehensive questions, please have a try.

1. For a project, it takes 10 days for Party A to do it alone, and 15 days for Party B to do it alone. After four days, Party B will do the rest alone. How many days will it take?

There are several tons of coal stored in the canteen, which used to burn 4 tons every day. After using 15 tons, the equipment was improved and the coal consumption was reduced to half of the original. As a result, it took 10 to find the original coal storage point.

3. A pool can be filled with water only by opening the water inlet pipe for 3 hours and opening the water outlet pipe for 4 hours. Now open the water inlet pipe for 2 hours, then open the water outlet pipe, so that the water inlet pipe and the water outlet pipe can be opened together. How many hours can the pool be filled?

Fourth, it is easy to make mistakes. Please think about it.

1. For a project, it takes 10 days for Party A to do it alone, 15 days for Party B to do it alone, and 5 days for Party A to do it alone. Then Party A and Party B cooperated to complete the project, and got a total of 1000 yuan. If the remuneration is calculated according to the amount of work completed by each person, how should Party A and Party B distribute it?

Thinking: The concern of this issue is that the basis of salary distribution is their respective workload. So Party A and Party B got 800 yuan and 200 yuan respectively.

2. Do you make mistakes in your homework? Please record it and analyze the cause of the error.

6.3.5 Savings problem

First, please manage the key points of this lesson.

1. Relationship among principal, interest rate, interest and principal and interest:

(1) interest = principal × interest rate

(2) Principal and interest = principal+interest

(3) After-tax interest = interest-interest × interest tax rate

2. Through the process of "problem situation-establishing mathematical model-explanation, application and expansion", we can understand and appreciate the role of mathematical modeling thought in solving practical problems.

Second, the basic questions, please do.

1. If a commodity is sold at a 20% discount and the price is 14.80 yuan, the original price is _ _ _ _ _ _ _.

2. Sheng Chao regularly deposits the lucky money of 1000 yuan given by his parents in the bank. At that time, the annual interest rate of one-year time deposit was 1.98%, and the interest tax rate was 20%. When the withdrawal is due, the interest is _ _ _ _ _ _ _ _.

After-tax interest is _ _ _ _ _ _ _ _, and Xiaoming's principal and net interest are _ _ _ _ _ _ _.

Two kiosks, A and B, sell goods at the same price. A week later, A lowered the price by 10%, and another week it increased by 20%. B After two weeks, the price will be raised 10%, and after two weeks, the price of _ _ _ _ kiosk will be low.

4. A clothing vendor sells two sets of clothing at the same time, and each set costs 168 yuan. If one set earns 20% and the other set loses 20%, then the seller will sell _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _.

Third, the comprehensive questions, please have a try.

1. Xiaoming's father deposited a two-year time deposit the year before last, with an annual interest rate of 2.43%. After the expiration of this year, after deducting interest tax, the interest tax rate is 20%. The interest earned just bought Xiaoming a calculator worth 48.60 yuan. How much did Xiaoming's father save the year before last?

Qingqing's mother spent 4500 yuan the year before last to buy a two-year bond of a company, which will expire this year. After deducting interest tax, she will get a total of about 4700 yuan of principal and interest, and the interest tax rate is 20%. What is the annual interest rate of this bond? (accurate to 0.0 1%)

3. A store increased the original price of a color TV by 40%, and then wrote in the advertisement "Big reward, 20% discount". After the customer complains, the law enforcement department will impose a fine of 2700 yuan per set according to the illegal income 10, and seek the original price of each color TV?

Fourth, it is easy to make mistakes. Please think about it.

1. The unit price of a commodity is 1500 yuan. If the gross profit of selling a commodity is 65438+ 05% of the selling unit price, what should the selling unit price be? (accurate to 1 yuan)

Thinking: From "profit = selling price-buying price", we can know that the unit price of this commodity should be set at 2000 yuan.

2. Do you make mistakes in your homework? Please record it and analyze the cause of the error.

1. Several branches grow on the trunk of a plant, and the same number of twigs grow on each branch. There are 73 trunks, branches and twigs. How many twigs are there on each branch?

Let each branch grow x twigs.

1+x+x^2=73，

x^2+x-72=0，

(x+9)(x-8)=0，

X=8, x=-9 (truncation).

A: Each branch grows 8 twigs.

2. The sum of two numbers is 8, and the product is 9.75. Find these two numbers.

Let one of the numbers be x and the other be 8-x.

(8-X)* X = 9.75；

x = 1.5；

8- 1.5=6.5

3. A two-digit number, the sum of which is 5, is inverted with the ten-digit number, and the product of the obtained two-digit number and the original two-digit number is 736, thus finding the original two-digit number.

If the unit number is x and the decimal number is (5-X), it is 10(5-X)+X, and the reverse is 10X+(5-X), then

[ 10(5-X)+X]*[ 10X+(5-X)]= 736

The solution is X=2 or 3.

This number is 23 or 32.

3. A bookstore owner went to the wholesale market to buy a book. The first time he bought a book, he spent 100 yuan. According to the book's pricing, he sold it in 2.8 yuan and sold it out quickly. Because the books sold well, when he bought them for the second time, the wholesale price of each book was higher than that of the first time. It cost 150 yuan, and he bought more books than the first time 10. When buying this batch, how much will you lose if you drop the line? If you make money, how much?

Solution: Buy X books for the second time and (x- 10) books for the first time.

From the meaning of the title, [100/(x-10)]+0.5 =150/x.

Organize x? -1 10x+3000=0, the solution is x 1=50, x2=60.

It is verified that X 1 = 50 and X2 = 60 are the roots of the original equation.

When x=50, the wholesale price of each book is 150÷50=3 (yuan), which is higher than the book price and irrelevant.

When x=60, the wholesale price of each book is 150÷60=2.5 (yuan), which is lower than the book's pricing and meets the question. Therefore, I bought 60 books for the second time.

[60× (4/5 )× 2.8+60× ( 1/5 )× 2.8× ( 1/2)]- 150 = 15 1.2- 158.

A: The boss made 1.2 yuan money by selling books for the second time.