(1) Question review: To clarify what is known, what is unknown and their relationship, use X to represent a reasonable unknown in the stem of the question.

(2) According to the meaning of the question, find an equivalent relationship that can express all the meanings of the application question. (key step)

(3) According to the equality relation, correctly list the equations, that is, the listed equations should satisfy that the quantities on both sides of the equal sign should be equal; The units of algebraic expressions on both sides of the equation should be the same.

(4) Solving the equation: finding the value of the unknown quantity.

(5) Write the answers clearly and completely after the exam. The test should be that the solution obtained from the test can not only make the equation effective, but also make the application problem meaningful.

2. The types of application problems and the basic quantitative relationship used by each type:

(1) The basic relation of equal product application problem: volume before deformation = volume after deformation.

(2) The characteristics of application problems in deployment are: the quantity relationship before deployment and the new quantity relationship after deployment.

(3) Basic relationship of interest application questions: principal × interest rate = interest, principal+interest = principal and interest.

(4) Commodity profit rate: Commodity profit rate, commodity profit = commodity price-commodity purchase price.

(5) The workload in engineering application problems is not a specific quantity, so the total workload is often regarded as a whole 1, where work efficiency = total workload ÷ working time.

(6) Basic relation of travel application problem: distance = speed × time.

Meeting problem: If A and B go in opposite directions, then: the distance A walks+the distance B walks = the total distance.

Follow-up question: If Party A and Party B are in different places in the same direction, then the distance traveled by the tracker = the distance traveled by the former+the distance between the two places.

Circular runway problem:

① Party A and Party B set off at the same time in the same place and direction on the circular runway: the fast runner must run one more lap to catch up with the slow runner.

(2) A and B set off in opposite directions at the same time on the circular runway: the total distance when they meet is the length of one circle of the circular runway.

Flight problem, basic equivalence relation;

① Downwind speed = windless speed+wind speed

② Upwind speed = windless speed-wind speed

Navigation problem, basic equivalence relation:

(1) downstream speed = still water speed+current speed

② Backward speed = still water speed-current speed

(7) Proportion application: If the ratio of A to B is 2: 3, A can be set to 2x and B to 3x.

(8) Basic relationship of digital application problems: If a three-digit number, the hundredth digit is A, the tenth digit is B and the unit digit is C, then these three digits are:

1 The school organized tree planting activities. It is known that A has 27 people planting trees and B has 18 people planting trees. If A has twice as many trees as B, how many people need to be transferred from B to A?

A and B.

Original number 27 18

The current population is over 27 years old.

18-

relation of equality

The school organized tree planting activities. It is known that A has 23 people planting trees, and B has 17 people planting trees. Now, 20 people are deployed to support it, so that A has twice as many people planting trees as B. How many people should A and B deploy?

The analysis should be transferred to one person, and the number of people involved in the topic and their relationship can be expressed in the following table:

A and B.

Original number 27 18

Increase the number of people

20-

The current population is over 27 years old.

18+20-

Equivalence relation +2

A middle school organizes students to have a spring outing. If each car has 45 seats, 15 people have no seats. If there are 60 people in each car seat, then one car is empty and the other cars are just full. How many cars and students are there?

There are 59 workers in a workshop. It is known that each worker can process 15 type A parts, 12 type B parts or 8 type C parts on average every day. How to arrange daily production to make daily products complete? (3 parts A, 2 parts B, 1 part C as a set)

A square table consists of a table top and four legs. As we all know, one cubic meter of wood can make 50 desktops or 300 legs. Now, how many tables can be made with five cubic meters of wood?

There are 50 students in a class. In a math exam, the pass rate of girls is 80%, boys are 75%, and the whole class is 78%. How many boys and girls are there in this class?

There are 25 questions in a test paper, and each question gives four answers, among which there is only one correct answer. If each question is selected correctly, it will get 4 points, and if it is not selected or selected incorrectly, it will get 1 point. If a student gets 90 points, how many questions does he get right?

Pythagoras was asked how many students there were in his school. He replied, "Half of the students study math, a quarter study music, a seventh are resting, and there are three female students left." Ask how many students there are in Pythagoras' school.

9 Some cards are marked with 5 10, 15, 20, 25 ... respectively. The number on the last card is 5 larger than that on the previous card. Xiaoming got three adjacent cards, and the sum of the numbers on these cards is 240.

(1) What three cards did Xiao Ming get?

(2) Can you get three adjacent cards so that the sum of the numbers on these cards is 63?

The sum of 10 consecutive integers is 72, so these three numbers are respectively

1 1. (Xiao Yong's tuition fee for going to college after six years is 5,000 yuan, and his parents are now involved in education savings. There are two ways to save.

(1) Direct deposit for 6 years with annual interest rate of 2.88%; (2) Deposit for 3 years first, and after 3 years, the principal and interest will be automatically transferred to 3 years. The annual interest rate for three years is 2.7%. What kind of savings do you think will start with less principal? Analysis: To solve the problem of "which savings method has less initial principal", we only need to find out how much the initial amount of the two savings methods is, and then compare them. Let's start saving X yuan. If you follow the first saving method, then the equation is:

X × (112.88% × 6) = 5000, x≈4263 (yuan) If the second storage method is adopted, students can be encouraged to fill out the form themselves, and be guided at an appropriate time to review the students who have difficulties: total principal = principal ten interest: principal x interest rate x number of periods. The equivalence relation is: the sum of principal and interest in the last three afternoons =

So the column equation is 1.0 1x? (1 12.7% × 3) = 5000, and x≈4279 is obtained. That is to say, it is about 4280 yuan. After three years, the principal and interest will be saved for another three years, and after six years, the principal and interest will reach 5000 yuan. Therefore, the first way of saving

12 Answer the following question: (1) According to the report of Beijing Daily in May, 2000 16, the per capita water resource in Beijing is 300 cubic meters, which is only the national average and the world average. What is the national average? How many cubic meters of water resources is per capita in the world?

(2) One year's water leakage in Beijing is equivalent to building a new waterworks. According to incomplete statistics, there are at least 6×l05 faucets and 2×l05 toilets leaking in the city. If a faucet that is not closed tightly can leak one cubic meter of water a month, and a leaky toilet can leak b cubic meters of water a month, how many cubic meters of water will be lost a month? (represented by an algebraic expression containing a and b)

(3) The overdraft of water resources is worrying, and saving water is imminent. In view of the waste of water for residents, Beijing will formulate the water consumption standard for residents, stipulating the monthly standard water consumption for buildings of a family of three, and charging for those exceeding the standard. Assuming that the water consumption per cubic meter of the part that does not exceed the standard is 1.3 yuan, the water consumption per cubic meter of the part that exceeds the standard is 2.9 yuan, the monthly water consumption of a family of three living in a building is 12 cubic meter, and the water fee is 22 yuan, please calculate Beijing by equation.

13 The logging team should cut down 48m3 of timber every day as planned, so it finished the task three days ahead of schedule and cut down more timber than originally planned. How much wood should be cut down according to the original plan?

14 a city collects monthly water charges according to the following regulations: if the monthly water consumption of each household does not exceed 20 cubic meters, the water price per cubic meter will be charged at 1.2 yuan; More than 20 cubic meters, the excess is charged by 2 yuan per cubic meter. If the average water price paid by a resident in a certain month is per cubic meter 1.5 yuan, then he has used _ _ _ _ _ _ _ _ _ _ _ cubic meters of water this month.

15 the calculation method of tax payment for articles and books published by individuals stipulated by the state is as follows: (1) No tax is paid if the remuneration is not higher than that of 800 yuan; (2) If the contribution fee is not higher than 4,000 yuan in 800 yuan, the part exceeding 800 yuan shall be taxed at 14% of the contribution fee; (3) If the contribution fee is more than 4,000 yuan, the tax of 1 1% of the total contribution fee shall be paid. Today, I learned that Mr. Ding received a sum of money and paid personal income tax. 420 yuan asked Teacher Ding that the signature was _ _ _ _ _ _ _ _ _ _ _ _.

16 the worker's master made a cuboid box with a volume of 6cm and a height of 6 cm. It is known that the length of the bottom of the box is 5cm longer than the width, so the width of the bottom of the box is _ _ _ _ _ _ _ _ _ _ _ cm.

17, b two teams of students greening the campus, if the two teams cooperate, it can be completed in 6 days; If going it alone, team B will spend five days more than team A, then how many days will it take for each team to go it alone?

18. The purchase price of a commodity is 200 yuan, and the price is 300 yuan. When it is sold at a discount, the profit is 5%. At what discount is this commodity sold?

19 It takes 40 hours for one person to manage a batch of books. Now let's see, some people work for four hours first, and then two more people work together for eight hours to finish the work. Assuming that these people are equally efficient, how many people will be arranged to work first?

After two consecutive price reductions of 10%, the price of 20 commodities is 486 yuan, and the price before the price reduction is _ _ _ _ _.

The cost price of a certain garment in 2 1 store is 50 yuan, and it is sold at 20% off the list price (that is, 20% off the list price). (1), if each piece is still profitable 14 yuan, what is the list price of this kind of clothing?

(2) If the profit margin is 20%, what is the price tag of this kind of clothing? The shopping mall will increase the cost of a jacket with a price of 100 yuan by 50%, then price it as 150 yuan, and then sell it to a customer at a 20% discount. Please calculate, did the merchant make a profit in this transaction?

The store has a backlog of 65,438+000 items. In order to get rid of this batch of goods as soon as possible, the store adopted the following sales plan, which increased the price to 2.5 times, and then reduced the price three times: the first time, the price was reduced by 30%, marked as "loss price"; Reduce the price by 30% for the second time and mark "Bankruptcy Price"; Reduce the price by 30% for the third time, and mark "jumping price". The sales results of the three price reduction treatments are as follows:

Valence number

The price is several thousand, and it's gone.

(1) What percentage of the original price is the sale price?

(2) The goods are sold according to the new sales plan, which plan is more profitable than the original price?

If the goods are sold at a fixed price, everyone can make a profit. Now, 45 yuan can get the same profit as selling 65,438+02 units in 35 yuan by reducing the fixed price by 8.5%. What is the purchase price and the selling price of this commodity?

24. The distance between B and B is 6 kilometers. If they start at the same time and go in the same direction, A can catch up with B within 3 hours. Going in the opposite direction, 1 hour meets, what is the average speed of the two?

Two people at the age of 25 set out from two places at the same time, which are 18km apart, face to face, and met at 1 hour 48 minutes. If A leaves 40 minutes earlier than B, they will meet at B at 1 hour 30 to find their own speed.

From a to b, go down the mountain first and then take the flat road. Someone rode a bicycle down the mountain from A at a speed of 12km/h and reached B through a flat road at a speed of 9km/h, which lasted for 55 minutes. After returning, he crossed the flat road at a speed of 8 kilometers per hour and went up the mountain at a speed of 4 kilometers per hour. It took him several hours to go back to a place to find the distance between A and B.

Two people, A and B, are walking on the circular runway with a circumference of 400 meters. If they go in opposite directions from the same place at the same time, they will meet in 2 minutes. If they walk in the same direction from the same place at the same time, they will meet in 20 minutes. Knowing that A is faster, ask them to walk faster. (Only list equations, not solve them)

28 people rode bicycles around the 800-meter-long circular runway. They started from the same place. If the directions are opposite, they will meet every 1 minute for 20 seconds. If the directions are the same, they will meet every 13 minutes and 20 seconds. Seek everyone's speed.

A railway bridge is 1000 meters long. There is a train crossing the bridge. It takes 1 minute for the train to cross the bridge, and it takes 40 seconds for the whole train to cross the bridge completely. Find the train speed.

30 places are 280 kilometers apart, and a ship is sailing in them. It takes 14 hour downstream and 20 hours upstream. Find the speed and current speed of the ship in still water.

3 1 A and b walk in opposite directions, with a distance of 36 kilometers. If A leaves before B at 2 o'clock, they will meet after leaving at 2.5 o'clock. If B leaves at 2 o'clock before A, then they meet after A 3. How many kilometers do A and B walk per hour?

The first chapter is a rich graphic world.

1. Common geometric bodies in life: cylinder, cube, cuboid and sphere.

2. Classification of common geometric bodies: sphere, cylinder (cylinder, prism, cube and cuboid) and cone (cone and pyramid).

3. When a plane figure is folded into a solid figure, it should be noted that the number of sides is equal to the number of sides of the base map.

4. The side of the cylinder is unfolded into a rectangle; The total diffusion on the surface is two and one; The total expansion diagram of the cone is one and one; The unfolding diagram of the cube surface is one and two small squares; The expansion diagram of a rectangle is a big two.

5, special three-dimensional graphics section graphics:

(1) The cross sections of cuboids and squares are: triangle, quadrilateral (rectangle, square, trapezoid, parallelogram), pentagon,.

(2) The cross section of a cylinder is circular.

(3) The cross section of the cone is triangular.

(4) The cross section of the ball is:

6. We often refer to the figure seen from the front view, the figure seen from the left view and the figure seen from the top view as.

7. Top view of common three-dimensional graphics

geometry

Cubic

cube

circular cone

column

ball

front view

square

rectangle

planform

rectangle

circle

circle

Left facade

rectangle

square

8, point in, line in, face in.

Chapter II Rational Numbers

1, positive and negative numbers

A number preceded by a minus sign "-"is called a negative number.

It has the opposite meaning to negative number, that is, I learned that numbers other than 0 are called positive numbers (sometimes "+"is added before positive numbers as needed).

2. Rational number

(1) positive integer, 0, negative integer, positive fraction, negative fraction.

Integer and fraction are collectively referred to. 0 is neither a number nor a number.

(2) Numbers are usually represented by points on a straight line, which is called the number axis.

Three elements of number axis: origin and unit length.

Take any point on a straight line to represent the number 0. This point is called.

(3) Numbers with only two different signs are called reciprocal.

Example: the reciprocal of 2 is; The reciprocal of -2 is; The reciprocal of 0 is

(4) The distance between the point representing the number A on the number axis and the origin is called the absolute value of the number A, which is denoted as |a|.

The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal; The absolute value of 0 is 0. Two negative numbers, the larger one has the smaller absolute value.

3. Addition and subtraction of rational numbers

(1) rational number addition rule:

① Add two numbers with the same symbol, take the same, and add the absolute values.

② Add two numbers with different symbols whose absolute values are not equal, take the symbol, and subtract the smaller absolute value.

The sum of two opposite numbers is 0.

③ A number with 0 still gets this number.

(2) Rational number subtraction rule: subtracting a number is equal to adding the reciprocal of this number.

4. Multiplication and division of rational numbers

(1) rational number multiplication rule: two numbers are multiplied, the same sign is positive, the different sign is negative, and the absolute value is multiplied. Any number multiplied by 0 is 0.

(2) Two numbers whose product is 1 are reciprocal. For example, the reciprocal of-is; The absolute value is; The opposite number is.

(3) Rational number division rule 1: dividing by a number that is not equal to 0 is equal to multiplying the reciprocal of this number.

Rule 2 of rational number division: divide two numbers, those with the same sign and those with different signs. Divide 0 by any number that is not equal to 0 to get 0.

(4) The operation of finding the product of n identical factors is called power, and the result of power is called power. In the n power of a, a is the base and n is the exponent.

The odd power of a negative number is negative, and the even power of a negative number is. Any power of a positive number is a positive number, and any power of 0 is 0. The odd power of-1 is; The even power of-1 is.

Chapter three, letters representing numbers

1, and the letters formed by connecting numbers and letters representing numbers with operation symbols are called algebraic expressions.

Note: Single number and single number are also available.

2. Pay attention to the algebraic value: the value of letters must ensure that the algebraic value is meaningful; The value of a letter should ensure that the number it represents is meaningful.

3, algebraic coefficient should include the symbol before this item; If an algebraic term contains only one letter factor, its coefficient is 1 or-1, not 0.

4, similar items contain the same; The same letters are the same.

Note: Similar items have nothing to do with coefficient and alphabetical order; Several constant terms are similar.

5. Rules for merging similar items: When merging similar items, add the coefficients of similar items and keep them unchanged.

6. Rules for removing brackets:

(1) is preceded by a "+". After removing the brackets and the preceding "+"sign, the original brackets

(2) brackets before "-",remove the brackets and the "-"before it, the original brackets.

Chapter IV Plane Figure and Position Relationship

1, line, ray, line segment

(1) The difference between straight line, ray and line segment: straight line endpoint: ray endpoint: line segment has endpoint.

(2) Axiom of line segment: Of all the connecting lines between two points, the line segment is the shortest.

The length of a line segment connecting two points is called.

(3) Line segment comparison method: superposition method and measurement method.

(4) the midpoint of the line segment: if m is the midpoint of AB, then; On the other hand, if the point m is at

On the line segment AB, and there is (AB = BM), then point M is the midpoint of AB.

Example: C is the midpoint of line segment AB, so we can get AC= =, or 2AC= =AB.

AC+ =AB，BC=AB- .

2. Angle measurement and representation

(1) 1 degree =; 1 =; 1 fillet = degree; 1 flat angle = degree = fillet

(2) Three representations of angles: three uppercase English letters or one uppercase English letter (such as < < ABC, < a; Expressed in Greek letters (e.g.