1, and different times

2. Three basic characteristics of age problems:

1 The age difference between two people is constant;

2 The age of two people is increasing or at the same time;

3 The multiple of the ages of the two is changing;

3. Basic Features of a question:

There is a constant amount in the question, which is generally the “single amount”, the topic is generally expressed in the “speed” …

Key Problem: Determine and find a single amount according to the conditions in the topic;

4. Tree problem

5. Chicken rabbit

Basic concept: Chicken rabbit is also known as

Replacement problem, assumption problem,

that is

Replace the part of the hypothesis;

The basic idea:

1 Suppose, it is assumed that some phenomenon exists (the same as B or B and A):

2 After the assumption, the difference between the topic has different conditions, it is to find out how much this difference is

3 The difference caused by each thing is fixed, thus finding this difference;

4 Further adjustments according to these two differences, eliminating the difference in emergence.

Basic formula:

1 put all the chicken vacation into a rabbit: the number of chickens = (rabbit foot × total header – total feet) ÷ (number of rabbits – number of chicken feet)

2 put all the rabbits to the chicken: the number of rabbits = (total number of pieces of a chicken is in total number of heads) ÷ (the number of rabbit is a chicken)

Key Problem: Find out the difference between the total amount and the amount of unit.

6. Profit and loss problem

Basic concept: A certain amount of object, in accordance with a certain standard group, producing a result: according to another standard group, producing a result, due to the difference between the group, resulting in the difference in results, and the object grouping The number of groups or the total amount of the object.

Basic ideas: Two allocation schemes are first compared, and the total number of participated in allocation is determined based on this relationship, and then the total amount of the object is obtained according to the question.

Basic title:

1 There are remaining numbers at a time, and the other is insufficient;

Basic formula: total total number = (number of remaining + less than count) ÷ differences in each part

2 When there are balances twice;

Basic formula: total number = (larger than a small amount) ÷ differences in each part

3 When it is not enough twice;

Basic formula: total number = (larger less than a small number of less than count) ÷ differences in each part

Basic Features: The total number of objects and total groups are constant

.

Key Problem: Determine the total amount of objects and the total number of groups.

7. Cow to graze problem

Basic ideas: Suppose the speed of each cow grazing is “1”, according to two different ways, seeking the difference in total grass; finding the cause of this difference, you can determine the grass Growth speed and total grass.

Basic Features: The growth rate of raft and new grass growth is constant;

Key Problem: Determine two unchanged amounts.

Growth = (longer × long time cattle number – shorter time × short time cattle) ÷ (long – time – short time);

Total grass = longer × long time cattle head – longer × growth amount;

8. Periodic cycle and number table law

Cycle Phenomenon: Things are in the process of motion changes, some features have regular cycles.

Cycle: We put the time called twice in a row.

Key Problem: Determine the cycle cycle.

Leap year: 366 days a year;

1 year can be divided by 4; 2 If the year can be 100, the year must be 400;

Pingnian: 365 days a year.

1 year cannot be divided by 4; 2 If the year can be 100, it cannot be divided by 400;

9. Average

Basic formula: 1 average number = total quantity ÷ total number

Total number = average × total number

Total number = total number ÷ Average

2 average number = reference number + each number and reference number difference and total total

Basic algorithm:

1 The total quantity and total share are obtained, and the basic formula 1 is calculated.

2 Baseline method: Determine a reference number according to the relationship between the numbers; the number of times the number or the number of intermediates is generally selected is a reference number; the number of reference is standard, and all the number of reference to the reference The difference is poor; the average number of these differences; finally, the sum of this difference and the number of references are the average of the results, the specific relationship is found to see the basic formula 2

10. Drawer principle

Drawer Principle One: If the (N + 1) object is placed in N drawers, then there is at least 2 objects in a drawer.

Example: Putting 4 objects in three drawers, that is, decomposing 4 into three integers, then there are four situations:

14 = 4 + 0 + 0 24 = 3 + 1 + 0 34 = 2 + 2 + 0 44 = 2 + 1 + 1

Watch the above four kinds of lands, we will find a common feature: there is 2 or more than 2 objects in a drawer, that is, there must be at least 2 objects in a drawer.

Drawer Principle 2: If the n object is placed in the M drawers, where n> m, then there is at least one drawer:

1k = [n / m] +1 object: When N cannot be divided by M.

2k = n / m object: When N can be divided by M.

Understanding knowledge points: [x] indicates the maximum integer of not exceeding X.

Example [4.351] = 4; [0.321] = 0; [2.9999] = 2;

Key Problem: Constructing objects and drawers. That is to find the amount representing the object and drawer, and then calculates according to the drawer principle.

11. Define new operations

Basic Concept: Define a new calculation symbol, which contains a variety of basic (mixed) operations.

Basic ideas: Strictly follow the newly defined arithmetic rules, translate into the calculation of the addition and subtraction, followed by the basic arithmetic process, law.

Key Problem: Correctly understand the meaning of the defined arithmetic symbol.

Note: 1 New operations do not necessarily meet the rules of operation, pay special attention to the order of operation.

2 Each newly defined arithmetic symbol can only be used in this question.

12. Digital and

Alternative column: in a column number,

Arbitrary adjacent

The difference between the two numbers is certain, such a column, is called an equal number of columns.

Basic concept: First: The first number of different columns, generally use A

1

Express;

Number of items: Number of all numbers of different columns, generally use n to indicate;

Tolerance: The difference between two numbers in any neighboring number of numbers, generally used D;

General item: Represents the formula of each number in the number, generally used A

n

Number of sum: this number of all numbers and general use

SN

Express.

Basic ideas: Part of five quantities in the equivalent column: a

, a

, D, N, S

, In the general formula, four quantities, if you know three, you can find four quantities in the formula, if you know three, you can ask for this fourth.

Basic formula: Welfare formula: a

= a

+ (N-1) D;

General item = first + (item 1) × tolerance;

Digital and formula: s

, = (a

+ a

) × n ÷ 2;

Digital and = (first + end) × item number ÷ 2;

Number of items: n = (a

) ÷ D + 1;

Number of items = (end – first) ÷ tolerance +1;

Tolerance formula: D = (a

-A

)

÷ (N-1);

Tolerance = (End – First) ÷ (item number -1);

Key Problem: Determine known and unknown amounts to determine the formula used;

13. Binary and its application

Decimation: Use 0 ~ 9 ten numbers, all 10 into 1; Different digits represent different meanings, 2 represented 20, 2 represents 200 on the ten digits. So 234 = 200 + 30 + 4 = 2 × 10

2

+ 3 × 10 + 4.

= A

× 10

N-1

+ A

N-2

N-3

N-4

N-5

N-6

N-7

+ … + a

3

0

Note: n

= 1; N

1

= N (where n is any natural number)

Binary: Use 0 ~ 1 two numbers to represent, all 2 into 1; different digits of numbers represent different meanings.

= A

× 2

Note: AN is not 0 is 1.

Ten into binary:

1 According to the characteristics of binary 2 into 1, use 2 to continuously remove this number, until the business is 0, and then written according to the rest of each time it is pressed.

2 First identify the N times of the number of 2, then ask them, then find the N times of 2 N times, no more than this difference, and find the difference between 0, according to the binary launch characteristics Written.

14. Access multiplication principle and geometric count

Addition principle: If a task is completed, there is a n class method, there is M in the first class method.

Different methods, mm in the second type of method

Different methods …, in the Nth method

Different methods, then complete this task: M

+ M

……. + M

Different methods.

Key Issues: Determining the classification method of the work.

Basic Features: Each method can complete the task.

Multiplication Principle: If you need to complete a task, you need to divide n steps, do the first step.

Method, no matter which method is used in step 1, the second step is M

Method … No matter which method is used in front of N-1, the Nth step is always M

Method, then complete this task: M

× M

Key Problem: Determine the completion steps of the work.

Basic Features: Each step can only complete part of the task.

Straight line: A point in a certain direction or in opposite directions at a certain direction or in the opposite direction.

Straight line characteristics: no endpoint, no length.

Line segment: The distance between the two points on the line. These two point are called endpoints.

Direction: There are two endpoints, which have a length.

Ray: Extend the end of one end of the line.

Ray characteristics: there is only one end point; there is no length.

1 Digital section rules: total = 1 + 2 + 3 + … + (point 1);

2 Number regular = 1 + 2 + 3 + … + (ray number 1);

3 count rectangular rules: a number = long line segment × width line segment:

4 rectangular rules: number = 1 × 1 + 2 × 2 + 3 × 3 + … + line number × column number

15. Volume and compliance

Miga: One number except 1 and it itself, there is no other approval, this number is called the number, and it is also known.

Compassion: A number except 1 and itself, there is still another amount, this number is called a compliment.

Cymmine: If a certain number is a number of a few, then this number is called this number of quality.

Decompose the quality factor: expressed a number of rigidities in the form of a number, called the decomposition factor. It is usually used to remove the fracture factor. Any result of any composite division factor is unique.

Standard representation of the decomposition quality factor: n =, where A

… a

Both is the number of syndromes n, and A

Formula of the number of request: P = (r

+1) × (r

+1) × … × (r

+1)

Number of mutual: If the number of two numbers is 1, these two numbers are called mutual numbers.

16. More than the multiple

More than the number of times: If the integer A can be divided by B, a multiple of B is called B is called a approximation.

Number of Covenants: Several counted records, called these number of conventions; one of these, called these number of largest cities.

The nature of the maximum number of conventions:

1, several numbers are divided by their largest number of conventions, the resulting more than the number of mutual numbers.

2, the number of largest civil companies in several numbers is the number of more than the number.

3, the number of conventions of several numbers is the number of the maximum number of conventions.

4, several numbers multiply by a natural number M, the maximum number of conventions obtained, equal to the maximum number of conventions of these numbers multiplied by M.

For example: 12 has a number of 1, 2, 3, 4, 6, 12;

18 of 18: 1, 2, 3, 6, 9, 18;

Then the number of conventions of 12 and 18 are: 1, 2, 3, 6;

Then the number of 12 and 18 the largest conventions are: 6, record (12, 18) = 6;

Seeking the basic method of the biggest convention:

1. Decompose the quality factor: First divide the degrees, and then multiply the same factor.

2, short-range: first find a public approach, then multiply.

3, rolling phase division: Every time the division and the remainder can be used, the remainder that can be removed is the maximum number of conventions.

The number of common multiple: several few public times, called these many common multiple; the smallest one is called the minimum common multiple of these numbers.

12 multiple of: 12, 24, 36, 48 …;

18 The multiple of: 18, 36, 54, 72 …;

So 12 and 18 is: 36, 72, 108 ……;

Then the number of larger variations in 12 and 18 is 36, and [12,18] = 36;

The nature of the least common multiple:

1. Two number of any common multiple of them is the multiple of their least common multiple.

2, the number of two largest courses and the minimum male number is equal to the product of these two numbers.

Ask the least common multiple basic method: 1. Short off the minimum common multiple; 2, method of decomposition quality factor

17. Number

I. Basic concepts and symbols:

1, division: If an integer A, divide a natural number B, get an integer mer C, and there is no remainder, then the A can be tied to b | A, remember B | a.

2. Common Symbol: Take the symbol “|”, can not complete the symbol “”; because the symbol “∵”, the symbol “∴”;

Second, the entire judgment method:

1. Can be removed by 2, 5: the number on the end can be divided by 2, 5.

2. Can be 4,25: the number of numbers of the last two digits is 4,25.

3. Can be 8,125: the number of numbers of the last three digits can be divided by 8, 125.

4. Can be 3,9: the sum of the numbers on each digit is 3, 9.

5. Can be 7 consolidation:

1 The difference between the number of numbers composed of the last three-digit number and the number of previous numbers in the last three digits can be divided by 7.

2 Remove the last digit number and minus the 2 times of the last digit can be removed.

Basic formula:

.

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Express;

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

, a

, a

+ a

)

2

2

2

2

2

2

2

2

2

2

2

× 10

× 10

× 10

× 10

× 10

× 10

× 10

× 10

N-1

N-1

N-1

+ A

+ A

+ A

+ A

+ A

+ A

+ A

+ A

+ A

+ A

+ A

+ A

+ A

N-2

N-2

N-2

N-3

N-3

N-3

N-4

N-4

N-4

N-5

N-6

N-7

+ … + a

3

3

3

3

0

0

× 2

× 2

× 2

× 2

× 2

× 2

× 2

× 2

Different methods.

× M

+1) × (r

6. Can be 11th:

1 The difference between the number of numbers composed of the last three digits and the last three previous numbers were 11th.

2 The numbers and differences between the numbers on the odd digit and the numbers and differences of the number of digits are 11.

3 Remove the last digit number again and subtract the last digit to be 11th.

7. Can be 13 consolidations:

1 The difference between the number of numbers composed of the last three-digit number and the last three digits of the last three times can be removed by 13.

2 Remove the last digit number again and minus 9 times the last digit can be removed.

Third, the nature of the tightening:

1. If A, B can be divided by C, then (A + B) and (A-B) can be divided by C.

2. If a can be divided by B, C is an integer, then A is multiplied by C and can be used by B.

3. If the A can be divided by B, B can be divided by C, then A can also be divided by C.

4. If A can be divided by B, C, then A can also be divided by the least common multiple of B and C.

18. The remainder and its application

Basic concept: For any natural number a, b, q, r, if a ÷ b = Q … r Full business.

The nature of the remainder:

1 The remainder is smaller than the division.

2 If A, B is divided by the remainder of C, C | A-B or C | B-A.

3A and B and divided by the remainder of C, which is equal to the remainder of C in C, divided by the remainder of C with C.

The accumulation of 4A and B is equal to the remainder of C with C by the remainder of C and B divided by C.

19. The remaining number, the same fun and cycle

First, the same definition:

1 If two integers A, B is divided by the remainder of M, it is called A and B for the mold M.

2 It is known to be three integers A, B, M, if M | A-B, saying A, B for the mold M, records AUB (MOD M), reads A with more than the B mode M.

Second, the same in the same nature:

1 self-sex: AUA (MOD M);

2 Symmetry: If AUB (MOD M), BOA (MOD M);

3 Transmissionability: If AUB (MOD M), B≡C (MOD M), A≡ C (MOD M);

4 and differential: if AC (MOD M), C≡D (MOD M), A + C≡B + D (MOD M), A-C≡B-D (MOD M);

5 multiplier: If AT B (MOD M), C≡D (MOD M), A × C≡ B × D (MOD M);

6 Passengeatic: If AUB (MOD M), A

n

≡B

(MOD M);

7 epagism: If AT B (MOD M), the integer C, then A × C≡ B × C (MOD M × C);

Third, about the preparatory knowledge of the passenger:

1 If a = a × b, then M

A

= M

A × B

(M)

a

)

b

2 If b = c + D m

B

C + D

c

× M

di

Fourth, the remaining characteristics of 3, 9, 11:

1 A natural number m, n represents the sum of the numbers of the number of M, then M≡n (MOD 9) or (MOD 3);

2 A natural number M, X represents the sum of the numbers of M, Y represents the sum of the numbers of M’s number, then MOY-X or MT11- (X-Y) (MOD 11);

V. Fahima Little Theorem: If P is the number (number of prime numbers), A is a natural number, and A cannot be divided by P, then A

P-1

1 (MOD P).

20. Application of score and percentage

Basic concepts and properties:

Score: The average unit “1” is divided into several copies, indicating such a number of copies.

The nature of the fraction: the molecule and denominator of the score are simultaneously multiplied or divided by the same number (except), the size of the score is unchanged.

Score unit: divide the unit “1” into several copies, indicating such a number.

Percentage: indicating that one is another number of numbers.

Common methods:

1 Reverse thinking method: Thinking from the reverse direction (or results) of the topic.

2 correspondence method: find out the direct correspondence between the specific amount in the topic and the rate it occupies.

3 Transformation Thinking: Translate a class of application into another type of application to answer. The most common is the conversion into a ratio and converted into multiple relationships; divide different standards (in the fraction is generally a magnification) in the same basis of the same condition. Common processing methods are determined that different criteria are once a quantity.

4 Hypothesis Method: For the convenience of the problem, you can assume that the amount of inconsistency in the topic is equal or assume that some case is established, calculate the corresponding results, then adjust, and final results.

5 quantity unchanged thinking: In various amounts of change, there is always one amount that is constant, regardless of how the other amount changes, and this amount is always fixed. There are three situations: a, the component changes, the total amount is unchanged. B. The total amount changes, but the components therein are constant. C, total amount and components change, but the difference between the components does not change.

6 Replacement of thinking: replacing another amount by one amount, thereby aligning the quantity relationship, and the amount ratio is unclear.

7 Same period: Treated between the total amount and the components.

8 concentration ratio: Generally applied to the total amount and components.

21. Comparison of fractional size

basic method:

1 Points Molecular Law: The same molecules of all scores, according to the relationship between the same molecular fraction and the denominator.

2 Positivity Method: Make the same fractal weight, according to the relationship between the same splitter fraction and the molecule.

3 Benchmark: Determine a standard to compare all the scores.

4 molecules and denominator comparison methods: When the difference in molecules and denominates, the greater the fraction value of the molecule or the denominator.

5 magnification comparison method: When comparing two molecules or denominators, the size of the score is compared, in addition to using the above method, the size of the fraction can be compared using the same magnification. (Specific use of the same magnification change law)

6 Transformation comparison method: Convert all scores into decimal (value of the fraction).

7 multiple comparison method: Use a number to compare with another number, result and 1.

8 Size comparative method: minus another score with one score, and the number and 0 comparison.

⑨ Reverse comparison method: Using the countdown comparison, then determine the size of the original number.

⑩ Baseline comparison method: determine a reference number, each number is compared to the reference number.

3. Full level

Full squared characteristics:

1. The last digit can only be: 0, 1, 4, 5, 6, 9; vice versa.

2. Remove more than 3,0 or after 1;

3. Remove more than 4 0 or less;

4. The number of approximes is odd;

5. The odd square ten digits are even; vice versa.

6. odd square numbers are odd; even square numbers are even.

7. The square of the two levels of the integer cannot have a square number.

Squary fault formula: x

2

-Y

= (X-y) (x + y)

Full square and formula: (x + y)

= X

+ 2XY + Y

Full squared equation: (X-Y)

-2xy + y

24. Comparison and ratio

Comparison: Two numbers are called more than two numbers. The quota before the quotes, the ratio of the ratio of the number of than the number of thanks later.

The ratio: the ratio of the previous item is called the ratio.

Procompared properties: The ratio is multiplied or divided by the same number (zero), the ratio is unchanged.

Proportion: It is called a ratio of two compared to equal style. A: B = C: D or

Proportional properties: two external numbers are equal to two in-internal volumes (cross multiplication), AD = BC.

Proportion: If A is expanded or reduced for several times, B is also expanded or reduced several times (AB’s business is not changed), then A and B are proportional.

Inversion ratio: If A is expanded or reduced, B is reduced or expanded several times (the accumulation of AB is not changed), then A and B are inversely proportional.

Scale: The ratio of the distance from the actual distance is called a scale.

Based on proportion: divide several numbers into several copies, named proportional assignment.

25. Combine

Basic concept: The travel problem is to study object motion, it studies the relationship between objects, time and journey.

Basic formula: journey = speed × time; distance ÷ time = speed; journey ÷ speed = time

Key Problem: Determine the position and direction during exercise.

Message: Speed and × encounter time = encounter away (please write other formula)

Chasing the problem: chasing time = road difference ÷ speed difference (write other formula)

Water problems: Shunshui stroke = (ship speed + water speed) × Shush water time

Backwater Tour = (Ship Speed - Water Speed) × Returning Time

Shooting speed = ship speed + water speed

Back water speed = ship speed – water speed

Static water speed = (smooth water speed + reverse water speed) ÷ 2

Water speed = (smooth water speed – counter water speed) ÷ 2

Water problem: The key is to determine the speed of the motion movement, refer to the above formula.

Passing the bridge question: The key is to determine the distance movement of the object and refer to the above formula.

Main method: draw line diagram method

Basic topic: known journey (encountering distance, chasing distance), time (encounter time, chasing time), speed (speed and speed difference), two quantities, the third amount.

26. Project issues

Basic formula:

1 Total work = Working efficiency × working time

2 Working efficiency = total work time ÷ working time

3 working hours = total workload ÷ work efficiency

The basic idea:

1 Suppose the total work is “1” (independent of total workload);

2 Suppose a convenient number is the total amount of work (generally the least common multiple of the time used in the total work), using the above three basic relationships, can simply represent the work efficiency and working hours.

Key Question: Determine the workload, working hours, and two pairs of relationships between work efficiency.

Experience comment: long-lasting, long-term must.

27. Logic Reasoning

Basic Method Introduction:

1 Conditional Analysis – Hypothesis: Assumptions may be established in the case, then decisions according to this hypothesis, if there is a situation in which the condition is contradictory, indicating that the hypothesis is not established, then the opposite is true with him. . For example, suppose A is an even number set, and there is a contradiction during the judgment process, then A must be odd.

2 Conditional Analysis – List Method: When the topic is much more, you need to make a list to assist the analysis when you need multiple assumptions. The list method is to represent the conditions of the subject in a rectangular table, the list of tables, the columns represent different objects and situations, and observe the topic in the table, and use logic laws to determine.

3 Conditional Analysis – Chart Method: When there is only two relationships between two objects, the relationship between the two objects can be used, and there is a connection of “Yes, there is”, and there is no connection. The line indicates a negative state. For example, there is a relationship or do not know two states between A and B, and there is a connection to understand, and there is no unknown.

4 Logic calculation: In addition to the reasoning of the reason, the corresponding calculation is carried out according to the calculation results, and a new judgment filter is provided in accordance with the results of the calculation.

5 Simple summary and reasoning: According to the features and data provided by the topic, analyze the laws and methods of which are existed, and extend from special circumstances to the general situation, and deliver relevant relationships, resulting in a problem solving.

28.

Geometric area

In the calculation of some area, it is not possible to use the formula, generally need to cut, translate, rotate, turn, break, decomposition, deformation, overlap, etc., make the irregular graphic graphics of the rule of the graphics; In addition, you need to master and remember some of the regular area laws.

Connector line method

2. Use two triangular area equal to the equipotentheve.

3. Bold hypothesis (some point of setting is arbitrary, the question can be set at a special location when solving the problem).

4. Utilize special laws

1 Waist right angle triangle, known or one can be found. (The square of the slope is equal to 4 equals the area of the waist angle triangle)

2 After the ladder diagonal is connected, the two waist portion is equal.

3 The area of the circle accounts for 78.5% of the external square area.

29. Clock problem – quick slow table problem

1. Strove the problem according to the thinking method in the trip problem;

2. Different tables are different moving objects;

3, the unit of the distance is the point (the table is 60 points);

4, time is the time passed by the standard table;

5. Rational use of proportional relationships in stroke issues;

30. Clock problem – clock chasing

Basic Thoughts: Chasing Problems on Closed Curves.

Key Question: 1 Determine the initial position of the minute and the hour hand;

2 Determine the difference between the minute and the hourly needle;

1 Pirated method:

The clock of the clock is evenly divided into 60 small grids, and we are called 1 point. The minute needle walks 60 points per hour, the first week; the needle is only 5 points, so the minute should take 1 point every minute, and the hour should take 1/12 points per minute.

2 degree method:

From the perspective view, the clock round is 360 °, the minute needle is rotation per minute, that is, 6 °, hourly rotation, degree.

31. Concentration and ratio

Experience Summary: There is such a reverse proportional relationship in the process of ratio, and the weight of the two solutions and changes in their concentration are contrast.

Solution: Substances (such as sugar, salt, alcohol, etc.) are called solutes in other substances.

Solvent: Solubate (e.g., water, gasoline, etc.) called a solvent.

Solution: Liquid (e.g., brine, sugar, etc.), which is mixed with solute and solvent, is called a solution.

Basic formula: solution weight = solute weight + solvent weight;

Solute weight = solution weight × concentration;

Concentration = × 100% = × 100%

Theoretical part of the small practice: other formulas of the three solutes, solutions, solvents.

32. Economic issue

The percentage of profits = (selling price – cost) ÷ cost × 100%;

Sold price = cost × (1 + profit percent);

n

= M

= M

Common methods:

basic method:

2

2

2

2

2

2

2

= X

The basic idea:

The basic idea:

Experience Summary: There is such a reverse proportional relationship in the process of ratio, and the weight of the two solutions and changes in their concentration are contrast.

Cost = selling price ÷ (1 + profit percent);

The pricing of goods is determined in accordance with the expected profits;

Pricing = cost × (1+ expectation profit percentage);

This gold: the amount of savings;

Interest rate: interest and proportional ratio;

Interest = principal × interest rate × current number;

Tax price = no tax price × (1+ value-added tax rate);

33. Simple process

Algebraic: The letters or numbers that are connected with an operation symbol (addition and subtraction).

Equation: The equation containing unknown equations.

Column places: connect two or several equal algebraic use the equal sign.

Column equation Key Problem: The same number is expressed in two or more different algebraes.

Al equity properties: add or subtract a number on both sides, the equation is constant; the equation is simultaneously multiplied or divided by one (divided by 0), the equation is constant.

Movement: Move from the equation equal to the other after the number or formula change the symbol;

Movement rules: First move, then change, then multiply; first go to the braces, then go to brackets, and finally go to small brackets.

Package rules: In the case of only the decrease of the calculation, if the parentheses are “+”, then add, parentheses, the operation symbols inside the parentheses are constant; if the parentheses are “-“, add, Brackets, the calculation symbols in parentheses should be changed; there is no “+” or “-” in front of the brackets, all “+” processing.

Movement Key Problem: Use the nature of the equation, movement rules, plus, and parentheses rules.

Multiplication Rate: A (B + C) = AB + AC

Solution Code: 1 to the point; 2 to brackets; 3 movement; 4 merge same item; 5 solution;

Equation Group: A set of equations composed of several binary one equations.

Steps of the Code of Solution: 1 Email; 2 Press the first equation step.

Method: 1 Add or subtilize the embolism; 2 generation into the yuan.

34. Unexpected equation

One definite equation: contains two unknown equations called binary equations, because its solution is not unique, so it is also called binary and one defined equation;

General methods: observation, testing, enumeration method

Multiplexed equation: The equation containing three unknown equations is a three-yuan equation, and its solution is not unique;

Multi-diverse equation solution: Determine an unknown value according to known conditions, or eliminate an unknown number, this will turn the three-yuan equation into a binary and undeitated equation, according to the binary, no equation is determined;

Recommended knowledge points: column equation, number of tensile, size comparison;

The step of solving the equation: 1, column equation; 2, embolization; 3, write expressions; 4, determine the scope; 5, determine features; 6, determine the answer;

Strike summary: a

35. Circulating decimal

First, the rules of the decimal part of the circulatory decimal

1 Pure circulating decimal part of the fractional number of components: the number of components of a circulation as a molecule, the number of denominates is the number of 9, 9 is the same as the number of bits of the cycle, and the re-cut removal of the final can be given.

2 Mixed-circular decimal fractional components: Molecule is the number of numbers composed of the number of numbers of the previous decimal part of the second cycle, and the number of heads of the denominator is 9, 9 The number is the same as the number of bits of one cycle, and the number of attempts is 0, 0 is the same as the number of bits of the non-circular portion.

Second, the score is transformed into a circulating decimal method:

1 One of the simplest scores, if the denominator contains both the quality factors 2 and 5, but also the number of cyclic factors other than 2 and 5, then this score is a decimal amount of the mixed circulation.

2 One of the simplest scores, if only 2 and 5 are only 2 and 5, then this score is a pure cycle decimal.