1. The following operation is correct ()
A.3﹣2=6B.m3？ m5=m 15C。 (x﹣2)2=x2﹣4D.y3+y3=2y3
2. Among the four numbers-,π, 3.21221221..., the number of irrational numbers is ().
A. 1B.2C.3D.4
There are two sticks, their lengths are 20cm and 30cm respectively. If you want to order a tripod, you should choose the following four kinds of sticks ().
A.10cm b.30cm c.50cm d.70cm
4. The following statement is true ()
The square root of is-﹣3B.9 is 3.
The arithmetic square root of C.9 is 3D. The arithmetic square root of C.9 is 3.
5. The purchase price of a commodity is 10 yuan, and the selling price is 15 yuan. In order to promote sales, we have decided to sell at a discount now, but if the profit of each piece is not lower than that of 2 yuan, we will sell at most ().
A.6 discount B.7 discount C.8 discount D.9 discount.
6. As shown in the figure, ab∨CD, ∠ CED = 90, EF⊥CD and F are vertical feet, then the complementary angle of ∠EDF in the figure is ().
A.4 B.3 C.2 D. 1
Fill in the blanks (3 points for each small question, 30 points in total)
The cube root of 7.-8 is.
8.x2？ (x2)2=。
9. If am=4 and an=5, then AM-2n =.
10. Please express the number 0.0000 12 as scientific notation.
1 1. If a+b=5 and A-B = 3, then A2-B2 =.
12. If the solution of the equation 2x-y+3k= 0 about x and y is, then k=.
13.N The sum of the inner angles of a polygon is at least greater than the sum of its outer angles 120, and the minimum value of n is.
14. If A and B are adjacent integers.
15. Xiao Liang places two pieces of rectangular paper as shown in the figure, so that a vertex of the small rectangular paper just falls on the edge of the large rectangular paper. If ∠ 1 = 35 is measured, ∠ 2 = 0.
16. If the inequality group has a solution, the range of a is.
Third, solve the problem (this big problem has 10 small articles, 52 points)
17. Calculation:
( 1)x3÷(x2)3÷x5
(x+ 1)(x﹣3)+x
(3)(﹣)0+()﹣2+(0.2)20 15×520 15﹣|﹣ 1|
18. Factorization:
( 1)x2﹣9
b3﹣4b2+4b.
19. Solve the equation:
①;
②.
20. Solve the inequality group: and express the solution set of the inequality group on the number axis.
2 1.( 1) Solving inequality: 5 (x ﹣ 2)+8
If the smallest integer solution of inequality in (1) is the solution of equation 2x-ax = 3, find the value of a. 。
22. As shown in the figure, the vertices of △ABC are all on the grid points on the square paper with each side length of 1 unit. Move △ABC to the right by 3 squares, and then move up by 2 squares.
(1) Please draw the translated' b' c in the picture;
△ The area of △ABC is:
(3) If the length of AB is about 5.4, find the height on the side of AB (the result is an integer).
23. As shown in the figure, if AE is the height on the edge of △ABC, then the angular bisector AD of ∠EAC intersects with BC at D, and ∠ ACB = 40, and ∠ADE is obtained.
24. If the solution set of the inequality group is-1.
(1) Find the value of the algebraic expression (A+ 1) (B- 1);
If a, b and c are three sides of a triangle, try to find the value of | c-a-b |+| c-3 |.
25. As shown in the figure, straight line AB and straight line CD, straight line BE and straight line CF are all cut by straight line BC. Please choose two of the following three formulas as topics and the other as a conclusion to form a true proposition and prove it.
①AB⊥BC、CD⊥BC,②BE∥CF,③∠ 1=∠2.
Title (known):.
Conclusion (verification):.
Prove:
26. A shopping mall buys two kinds of goods, A and B, and the price is 6.5438+0.8 million yuan. The buying price and selling price are as follows:
ab blood type
Purchase price (RMB/piece) 1200 1000
Price (RMB/unit) 1380 1200
(1) How many A's and B's will the mall buy if it makes a total profit of 30,000 yuan after the sale;
If the purchase quantity of Class B goods is not less than 6 times that of Class A goods, and each kind of goods must be purchased.
(1) How many kinds of purchasing plans are there?
(2) In order to ensure the profit, which procurement scheme do you choose?
Reference answers and analysis of test questions
1. Multiple choice questions (3 points for each question, totaling 18 points, each question has one and only one correct answer. )
1. The following operation is correct ()
A.3﹣2=6B.m3？ m5=m 15C。 (x﹣2)2=x2﹣4D.y3+y3=2y3
Test center: complete square formula; Merge similar projects; Multiplication with the same base; Negative integer exponential power.
Analysis: According to the exponential power of negative integer, same base powers's multiplication and complete dichotomy formula, the solution can be obtained by combining similar terms.
Answer: solution: a, so it is wrong;
b、m3？ M5=m8, so it is wrong;
C, (x-2) 2 = x2-4x+4, so it is wrong;
D, correct;
Therefore, choose: d.
Note: This topic examines negative integer exponential powers, same base powers's multiplication, complete dichotomy formula and merging similar items. The key to solving this problem is to memorize the relevant rules.
2. Among the four numbers-,π, 3.21221221..., the number of irrational numbers is ().
A. 1B.2C.3D.4
Test center: irrational number.
Analysis: Irrational numbers are infinite acyclic decimals. To understand the concept of irrational number, we must also understand the concept of rational number, which is a general term for integers and fractions. That is, finite decimals and infinite acyclic decimals are rational numbers and infinite acyclic decimals are irrational numbers. Therefore, we can judge the options.
Solution: solution: it is a fraction and a rational number;
π, 3.21221221… are irrational numbers;
So choose C.
Comments: This topic mainly examines the definition of irrational numbers, among which the irrational numbers studied in junior high school are π, 2π and so on. Endless prescriptions; And numbers such as 0.101001001.
There are two sticks, their lengths are 20cm and 30cm respectively. If you want to order a tripod, you should choose the following four kinds of sticks ().
A.10cm b.30cm c.50cm d.70cm
Test center: triangular trilateral relationship.
Analysis: First, get the value range of the third stick according to the trilateral relationship of the triangle, and then further find the qualified answer.
Solution: Solution: According to the trilateral relationship of a triangle, we can get
The length of the third stick should be greater than10cm and less than 50cm.
So choose B.
Comments: This question examines the solution of the triangular trilateral relationship; The key is to find the range of the third side.
4. The following statement is true ()
The square root of is-﹣3B.9 is 3.
The arithmetic square root of C.9 is 3D. The arithmetic square root of C.9 is 3.
Test center: arithmetic square root; square root
Analysis: A, B, C and D can be judged according to the definitions of square root and arithmetic square root respectively.
Answer: Solution: A and -9 have no square roots, so option A is wrong;
The square root of b and 9 is 3, so option b is wrong;
The arithmetic square root of c and 9 is 3, so the c option is wrong.
The arithmetic square root of d and 9 is 3, so the d option is correct.
Therefore, choose: d.
Comments: This topic mainly investigates the application of the concepts of square root and arithmetic square root. If x2=a(a≥0), then x is the square root of a. If a >；; 0, which has two square roots in opposite directions. We call the positive square root the arithmetic square root of a. If a=0, there is a square root, that is, the square root of 0 is 0, the arithmetic square root of 0 is also 0, and the negative number has no square root.
5. The purchase price of a commodity is 10 yuan, and the selling price is 15 yuan. In order to promote sales, we have decided to sell at a discount now, but if the profit of each piece is not lower than that of 2 yuan, we will sell at most ().
A.6 discount B.7 discount C.8 discount D.9 discount.
Test site: the application of one-dimensional linear inequality
Analysis: If the profit of each piece is not lower than that of 2 yuan, the corresponding relationship is: profit-purchase price ≥2, which can be substituted into the relevant value.
Answer: Solution: Let's sell at a discount of X, and the profit per piece is not lower than that of 2 yuan. According to the meaning of the question, we can get:
15×﹣ 10≥2,
Solution: x≥8,
A: Up to 20% discount.
So choose: C.
Comments: This topic mainly investigates the application of one-dimensional linear inequality. The key to this problem is the relationship between getting profits. Note that "not less than" is expressed as "≥" with mathematical symbols.
6. As shown in the figure, ab∨CD, ∠ CED = 90, EF⊥CD and F are vertical feet, then the complementary angle of ∠EDF in the figure is ().
A.4 B.3 C.2 D. 1
Test site: the nature of parallel lines; Complementary angle and complementary angle.
Analysis: According to ∠ CED = 90, EF ∠ CD, ∠ EDF+∠ DEF = 90, ∠ EDF+∠ DCE = 90, then we can know ∠ DCE = ∠ from the properties of parallel lines.
Solution: solution: ∫∠ced = 90, EF⊥CD.
∴∠EDF+∠DEF=90，EDF+∠DCE=90。
∫AB∨CD，
∴∠DCE=∠AEC，
∴∠AEC+∠EDF=90。
So choose B.
Comments: This question examines the nature of parallel lines, and the knowledge points used are: two straight lines are parallel and the internal angles are equal.
Fill in the blanks (3 points for each small question, 30 points in total)
The cube root of 7-8 is -2.
Test center: cube root.
Analysis: It can be solved by the definition of cube root.
Answer: Solution: ∵(﹣2)3=﹣8,
The cube root of ∴ 8 is 2.
So the answer is: 2.
Comments: This question mainly examines the concepts of square root and cubic root. If the cube of a number X is equal to A, that is, the cubic power of X is equal to a(x3=a), then this number X is called the cube root of A, also called the cube root. It is pronounced as "cube root number A", where A is called root number and 3 is called root index.
8.x2？ (x2)2=x6。
Test center: the power of power and the power of products; Multiplication with the same base.
Analysis: According to the multiplication property and the property of the power of the same base number, the solution can be obtained.
Answer: Solution: x2? (x2)2=x2？ x4=x6。
So the answer is: x6.
Comments: This question examines same base powers's multiplication and power, and it is the key to find out the change of index.
9. If am=4 and an=5, then AM-2n =.
Examination center: the division of power with the same base; Power and products.
Analysis: according to the division of the same radix power, subtract the radix constant exponent; Multiply the power by the exponent of the same base to get the solution.
Solution: solution: am-2n =,
So the answer is:
Comments: This question examines the division of the same base, and the powers are easily confused. You must remember the rules before you can do it.
10. Please use scientific notation to express the number 0.0000 12 as 1.2× 10 ~ 5.
Test center: scientific notation-indicating smaller numbers.
Analysis: Positive numbers with absolute value less than 1 can also be expressed by scientific notation, and the general form is a× 10-n, which is different from the scientific notation of large numbers in that it adopts negative exponential power, and the exponent is determined by the number of zeros before the first non-zero number from the left of the original number.
Solution: 0.000012 =1.2×10 ~ 5.
So the answer is: 1.2× 10 ~ 5.
Comments: This question examines small numbers expressed by scientific notation, and the general form is a×10-n, where 1 ≤| A | < 10, and n is determined by the number of zeros before the first non-zero number from the left of the original number.
1 1. If a+b=5 and A-B = 3, then A2-B2 = 15.
Test center: factorization-using formula method.
Analysis: first decompose with the square difference formula, and then substitute it into the known solution.
Answer: Solution: ∵a2﹣b2=(a+b)(a﹣b),
When a+b=5 and a-b = 3, the original formula =5×3= 15.
So the answer is: 15.
Comments: This topic mainly examines the use of formula method to decompose factors and algebraic evaluation. Correct decomposition of factors is the key to solve the problem.
12. If the solution of the equation 2x-y+3k = 0 about x and y is, then k =- 1.
Test site: the solution of binary linear equation.
Special topic: calculation problems.
Analysis: Substitute the known values of X and Y into the equation to get the value of K..
Solution: 4- 1+3K = 0 by substituting into the equation.
Solution: k =- 1,
So the answer is:-1
Comments: This question examines the solution of a binary linear equation. The solution of the equation is an unknown number that can make both equations in the equation hold.
13.N The sum of the inner angles of a polygon is at least greater than the sum of its outer angles 120, and the minimum value of n is 5.
Test center: polygon inner corner and outer corner.
Analysis: Is it the sum of the internal angles of an N-polygon (n ~ 2)? 180, the sum of the outer angles of an n-polygon is 360, and the sum of the inner angles is at least greater than the sum of the outer angles 120, so an inequality can be obtained: (n﹣2)? 180﹣360>； 120, you can find the range of n and the minimum value of n.
Answer: Solution: (n ~ 2)? 180﹣360>； 120, the solution is: n >；; 4.
So the minimum value of n is 5.
Comments: The inequality relation is known and can be solved by inequality.
14. If A and B are adjacent integers.
Test center: estimate the size of irrational numbers.
Analysis: the estimated interval can determine the values of a and b, and the solution can be obtained.
Answer: Solution: ∫ and
∴a=2,b=3，
∴b﹣a=，
So the answer is:
Comments: This question examines the estimation method of irrational number: find out two complete square numbers adjacent to this number, so as to determine the size range of this irrational number.
15. Xiao Liang put two rectangular pieces of paper together, as shown in the figure, so that a vertex of the small rectangular paper just falls on the edge of the large rectangular paper. If ∠ 1 = 35 is measured, then ∠ 2 = 55.
Test site: the nature of parallel lines.
Analysis: If point E is EF∪AB, then AB∪CD∪EF can be obtained from AB∪CD, so the number of times ∠4 can be obtained, and then the number of times ∠3 can be obtained, from which a conclusion can be drawn.
Solution: solution: as shown in the figure, the passing point e is EF∨AB,
∫AB∨CD，
∴AB∥CD∥EF.
∵∠ 1=35 ,
∴∠4=∠ 1=35 ,
∴∠3=90 ﹣35 =55 .
∫AB∨EF，
∴∠2=∠3=55 .
So the answer is: 55.
Comments: This question examines the nature of parallel lines, and the knowledge points used are: two straight lines are parallel and the internal angles are equal.
16. If the inequality group has a solution, the value range of a is a >；; 1.
Test site: inequality solution set.
Analysis: According to the meaning of the question, the range of a can be obtained by taking the solution set of the inequality group.
Solution: solution: the inequality group has a solution.
∴a>； 1,
Therefore, the answer is: a> 1.
Comments: This question examines the solution set of inequality, and mastering the method of obtaining the solution set of inequality group is the key to solve this question.
Third, solve the problem (this big problem has 10 small articles, 52 points)
17. Calculation:
( 1)x3÷(x2)3÷x5
(x+ 1)(x﹣3)+x
(3)(﹣)0+()﹣2+(0.2)20 15×520 15﹣|﹣ 1|
Test site: mixed operation of algebraic expressions.
Analysis: (1) Calculate the power first, and then calculate the division of same base powers;
First use algebraic multiplication calculation, and then further merge;
(3) Calculate the exponential power, negative exponential power and absolute value of the product of 0, and then add and subtract.
Solution: Solution: (1) Original formula =x3÷x6÷x5.
=x﹣4；
Original formula = x2-2x-3+2x-x2
=﹣3;
(3) The original formula = 1+4+ 1- 1
=5.
Comments: This question examines the mixed operation of algebraic expressions, and mastering the operation order and calculation method is the key to solving the problem.
18. Factorization:
( 1)x2﹣9
b3﹣4b2+4b.
Test center: the comprehensive application of common factor method and formula method.
Special topic: calculation problems.
Analysis: (1) The original formula can be decomposed by the square difference formula;
B is extracted from the original formula and decomposed by the complete square formula.
Solution: Solution: (1) Original formula = (x+3) (x-3);
Original formula = b (B2-4b+4) = b (b-2) 2.
Comments: This topic examines the comprehensive application of common factor method and formula method, and mastering factorization is the key to solve this problem.
19. Solve the equation:
①;
②.
Test site: solving binary linear equations.
Analysis: In this problem, we can use the exclusion method. First of all, we can eliminate an unknown quantity, turn it into a linear equation with one variable, find its solution, then substitute the solution into the original equation and find another one, and then we can get the solution of the equations.
Solution: Solution: (1)
①×2, so: 6x-4y = 12③,
②×3, so: 6x+9y=5 1④,
Then ④-③: 13y=39,
Solution: y=3,
Substituting y=3 into ① gives 3x-2x3 = 6.
Solution: x=4.
Therefore, the solution of the original equation is:
Multiply both sides of Equation ② by 12 at the same time to get: 3 (x-3)-4 (y-3) = 1,
Simplify and get: 3x ~ 4y = ~ 2③,
①+③, we get 4x= 12,
Solution: x=3.
Substituting x=3 into ① gives 3+4y= 14.
Solution: y=
Therefore, the solution of the original equation is:
Comments: This question examines the solution of binary linear equations, which is solved by elimination method. The topic is simple, but it needs to be careful.
20. Solve the inequality group: and express the solution set of the inequality group on the number axis.
Test site: solving a set of linear inequalities; Represent the solution set of inequality on the number axis.
Special topic: calculation problems.
Analysis: Solve two inequalities separately to get X.
Answer: Solution:,
X
X of solution ② is more than or equal to 3,
So the solution set of inequality group is 3 ≤ X.
Represented by the number axis as:
Comments: This question examines the unary linear inequality group: when solving the unary linear inequality group, the solution set of each inequality is generally found first, and then the common part of these solution sets is found, and the solution set of the inequality group can be intuitively expressed by using the number axis. Take the small as the big; Small, large and medium search; Can't find the big one or the small one.
2 1.( 1) Solving inequality: 5 (x ﹣ 2)+8
If the smallest integer solution of inequality in (1) is the solution of equation 2x-ax = 3, find the value of a. 。
Test center: solving one-dimensional linear inequality; The solution of one-dimensional linear equation; Integer solution of one-dimensional linear inequality.
Analysis: (1) According to the basic properties of inequality, the brackets are removed first, and then the solution set of the original inequality is obtained by shifting terms and merging similar terms;
Determine the minimum integer solution of x according to the value range of x in (1); Then substitute the value of X into the known equation, and list the unary linear equation 2×(﹣2)﹣a×(﹣2)=3 about the coefficient A. By solving this equation, the value of A can be obtained.
Solution: solution: (1) 5 (x-2)+8
5x﹣ 10+8<； 6x﹣6+7
5x﹣2<； 6x+ 1
﹣x<； three
x & gt﹣3.
From (1), the smallest integer solution is x =-2,
∴2×(﹣2)﹣a×(﹣2)=3
∴a=.
Comments: This topic examines the solution of one-dimensional linear inequality, the integer solution of one-dimensional linear equation and one-dimensional linear inequality. The solution of inequality should be based on the basic properties of inequality:
(1) The direction of addition and subtraction of the inequality of the same number or algebraic expression on both sides remains unchanged;
Both sides of inequality are multiplied or divided by the same positive number at the same time, and the direction of inequality remains unchanged;
(3) When both sides of the inequality are multiplied or divided by the same negative number at the same time, the direction of the inequality changes.
22. As shown in the figure, the vertices of △ABC are all on the grid points on the square paper with each side length of 1 unit. Move △ABC to the right by 3 squares, and then move up by 2 squares.
(1) Please draw the translated' b' c in the picture;
The area of △ABC is 3;
(3) If the length of AB is about 5.4, find the height on the side of AB (the result is an integer).
Test center: drawing-translation conversion.
Analysis: (1) Just draw the translated △ a ′ b ′ c ′ according to the nature of graphic translation;
According to the area formula of triangle, we can draw a conclusion;
(3) Let the height on the side of AB be h, and we can draw a conclusion according to the area formula of triangle.
Solution: solution: (1) as shown in the figure;
S△ABC=×3×2=3。
So the answer is: 3;
(3) Let the height on the side of AB be h, then AB? h=3，
That is ×5.4h=3, and the solution is h≈ 1.
Comments: This topic examines the mapping-translation transformation, and understanding the translation invariance nature of graphics is the key to solve this problem.
23. As shown in the figure, if AE is the height on the edge of △ABC, then the angular bisector AD of ∠EAC intersects with BC at D, and ∠ ACB = 40, and ∠ADE is obtained.
Test site: triangle interior angle sum theorem; The bisector, midline and height of a triangle.
Analysis: ∠CAE can be obtained according to the complementarity of the two acute angles of a right triangle, ∠DAE=∠CAE can be obtained according to the definition of the bisector of the angle, and then ∠ADE can be obtained.
Solution: solution: ∵AE is the height of the edge of △ABC, ∠ ACB = 40,
∴∠CAE=90 ﹣∠ACB=90 ﹣40 =50，
∴∠DAE=∠CAE=×50 =25，
∴∠ADE=65。
Comments: This question examines the theorem of the sum of angles in a triangle, and the definition of the bisector of the angle is the basic question. Memorizing theorems and concepts and accurately understanding maps are the key to solving problems.
24. If the solution set of the inequality group is-1.
(1) Find the value of the algebraic expression (A+ 1) (B- 1);
If a, b and c are three sides of a triangle, try to find the value of | c-a-b |+| c-3 |.
Test site: solving a set of linear inequalities; The trilateral relationship of a triangle.
Analysis: First, take A and B as known conditions, find the solution set of the inequality group, and then compare with the known solution set to find the values of A and B. 。
(1) directly substitute the value of ab to get the algebraic value;
Judging the symbol of c﹣a﹣b according to the trilateral relationship of the triangle, and then removing the absolute symbol. Just merging similar projects.
Answer: Solution:,
From ①, x
From ②, X & GT2B-3,
The solution set of the inequality group is ∵ 1
∴=3,2b﹣3=﹣ 1，
∴a=5,b=2.
( 1)(a+ 1)(b﹣ 1)=(5+ 1)=6；
∫a, B and C are three sides of a triangle.
∴5﹣2
∴c﹣a﹣b0，
Original formula = A+B-C+C-3
=a+b﹣3
=5+2﹣3
=4.
Comments: This question examines the solution of a set of linear inequalities, and is familiar with "taking the same big; Take the small as the big; Small, large and medium search; The principle of "size cannot be found" is the key to solve this problem.
25. As shown in the figure, straight line AB and straight line CD, straight line BE and straight line CF are all cut by straight line BC. Please choose two of the following three formulas as topics and the other as a conclusion to form a true proposition and prove it.
①AB⊥BC、CD⊥BC,②BE∥CF,③∠ 1=∠2.
Title (known): ① ②.
Conclusion (verification): ③.
Proof: ellipsis.
Test sites: propositions and theorems; Determination and properties of parallel lines.
Special topic: calculation problems.
Analysis: ① ② We can get ③: Because AB⊥BC and CD⊥BC can get abc, while ∠ABC=∠DCB and be∨cf can get ∠EBC=∠FCB.
Answer: As shown in the figure, AB⊥BC, CD⊥BC, BE ∨ CF.
Verification: < 1 = < 2.
Evidence: ∵AB⊥BC, CD⊥BC,
∴AB∥CD，
∴∠ABC=∠DCB，
And ∵BE∨CF,
∴∠EBC=∠FCB，
∴∠ABC﹣∠EBC=∠DCB﹣∠FCB，
∴∠ 1=∠2.
So, the answer is ① ②; ③; Omit.
Comments: This topic examines propositions and theorems: the sentence that judges things is called a proposition; The correct proposition is called true proposition, and the wrong proposition is called false proposition; The true proposition after reasoning is called theorem, and the properties of parallel lines are also investigated.
26. A shopping mall buys two kinds of goods, A and B, and the price is 6.5438+0.8 million yuan. The buying price and selling price are as follows:
ab blood type
Purchase price (RMB/piece) 1200 1000
Price (RMB/unit) 1380 1200
(1) How many A's and B's will the mall buy if it makes a total profit of 30,000 yuan after the sale;
If the purchase quantity of Class B goods is not less than 6 times that of Class A goods, and each kind of goods must be purchased.
(1) How many kinds of purchasing plans are there?
(2) In order to ensure the profit, which procurement scheme do you choose?
Test site: the application of one-dimensional linear inequality; Application of binary linear equations.
Analysis: (1) From the meaning of the question, we can know the equivalence relationship of this question, that is, "the total cost of two commodities is1.8000 yuan" and "the total profit is 30000 yuan". According to these two equivalence relations, we can list the equations and then solve them.
List the inequality groups according to the meaning of the question and answer them.
Solution: Solution: (1) Suppose you buy X pieces of A goods and Y pieces of B goods.
According to the meaning of the question
Simplify,
Solve,
A: The mall bought 65,438+000 A-class goods and 60 B-class goods.
Suppose you buy X A goods and Y B goods.
According to the meaning of the question:
Solution:,,,
Therefore, there are five purchase schemes.
ab blood type
Scheme 1: 25 pieces, 150 pieces.
Option 2: 20 pieces of 156.
Scheme III 15 pieces 162 pieces
Scheme IV 10 pieces 168 pieces
Scheme 5: 5 pieces 174 pieces.
② Because the profit of B is large, if you want to ensure the profit, choose 5 items of A and 174 items of B. 。
Comments: This question examines the application of binary linear equations and univariate linear inequalities. The key to solve this problem is to connect the events in real life with mathematical ideas, understand the meaning of the problem, find out the equivalence relationship and solve the equations.