Classification of application problems of linear equations with one variable
Solving practical problems with equations is one of the important contents of junior high school mathematics. Many practical problems boil down to solving an equation or equations, so listing equations or equations to solve practical problems is an important aspect of integrating mathematics with practice and solving practical problems; The following teachers explain common math problems from the following aspects, hoping to help students.
1. sum, difference, multiplication and division:
(1) multiplicity relation: it is reflected by the key words "how many times, how many times, how many times, what percentage, growth rate …".
(2) How much relationship: it is reflected by the key words "more, less, harmony, difference, lack, surplus ……".
Example 1. According to the statistics of the fifth census published by Xinhua News Agency on March 28th, 2006, the population with primary school education per 1 10,000 population in China is 357,065,438+.
Analysis: The equivalence relation is:
2. Equal product deformation problem:
"Equal area deformation" is based on the premise that the shape changes but the volume remains unchanged. The commonly used equivalence relation is:
(1) The shape area has changed, but the perimeter has not changed;
② Raw material volume = finished product volume.
Example 2. A cylindrical glass (filled with water) with a diameter of 90mm was poured into a cuboid iron box with a bottom area of 81mm. How much mm did the water in the glass drop? (Results are rounded)
Analysis: The equivalent relationship is: the volume of cylindrical glass = the volume of cuboid iron box.
3. Labor deployment: This kind of problem needs to find out the change of the number of people. Frequently asked questions are:
(1) can be transferred in and out;
(2) Only the transfer-in did not turn out, the transfer-in part changed, and the rest remained unchanged;
(3) Only the transfer-out has not been transferred in, some of the transfers have changed, and the rest remain unchanged.
Example 3. There are 85 workers in the processing workshop of the machinery factory, each of whom processes 16 large gears or 10 small gears on average every day. It is known that two large gears and three small gears form a group. How many workers should be arranged to process the big and small gears separately to make the big and small gears processed every day just match?
4. Proportional distribution:
The general idea of this kind of problem is: let one of them be x and write the corresponding algebraic expression by using the known ratio.
Common equivalence relation: sum of parts = total.
Example 4. The ratio of three positive integers is 1: 2: 4, and their sum is 84, so what is the largest of these three numbers?
5. Numbers.
(1) Need to know the representation method of numbers: the hundredth digit of a three-digit number is A, the tenth digit is B, and the first digit is C (where A, B and C are integers, 1≤a≤9, 0≤b≤9, and 0≤c≤9), then this three-digit number is represented as: 65438.
(2) Some representations in the number problem: the relationship between two consecutive integers, the larger one is larger than the smaller one1; Even numbers are represented by 2n, and continuous even numbers are represented by 2n+2 or 2n-2; Odd numbers are represented by 2n+ 1 or 2n- 1.
Example 5. For two-digit numbers, one digit is twice as much as ten digits. If the tenth digit is reversed, the two digits obtained are 36 larger than the original two digits. Find the original two digits.
Equivalence relation:
6. Engineering problems:
Three quantities in engineering problems and their relationship are: total work = working efficiency × working time.
Often when the total workload is not given in the title, the total workload is set to 1.
Example 6. For a project, it takes 15 days for Party A to do it alone and 12 days for Party B to do it alone. Now that Party A and Party B have cooperated for three days, Party A has other tasks, and the remaining projects will be completed by Party B alone. How many days does it take Party B to complete all the projects?
The analysis assumes that the total amount of the project is 1, and the equivalent relationship is:
7. Travel problems:
(1) Three basic quantities in the travel problem and their relationships: distance = speed × time.
(2) The basic types are
(1) meeting problems; (2) follow up the problem; Common ones are: running for opponents; Navigation problems; Circular runway problem.
(3) The key to solve this kind of problem is to grasp the time relationship or distance relationship between two objects, so that the problem can be solved as a whole. And often sketch to analyze and understand the trip problem.
Example 7. The distance between Station A and bilibili is 480 kilometers. The local train departs from Station A at a speed of 90km per hour, and the express train departs from bilibili at a speed of140km per hour.
(1) The local train starts first 1 hour, and the express train starts again. The two cars are driving in opposite directions. How many hours after the express train leaves, will the two cars meet?
(2) After two cars started at the same time and walked in opposite directions for several hours, the two cars were 600 kilometers apart?
(3) Two cars start at the same time, and the local train runs in the same direction behind the express train. How many hours later, the distance between the express train and the local train is 600 kilometers?
(4) Two cars leave in the same direction at the same time, and the express train is behind the local train. How many hours will the express catch up with the local train?
(5) After the local train 1 hour, the two cars are driving in the same direction, and the express train is behind the local train. How many hours after the express train leaves, will it catch up with the local train?
The key to this problem is to understand the meaning of opposite direction, opposite direction and same direction, and to understand the driving process. So it can be combined with graphic analysis.
(1) parsing: When encountering a problem, draw it as:
The equivalence relation is:
(2) Analysis: The running direction is opposite, and the drawing is expressed as:
The equivalence relation is:
(3) Analysis: The equivalent relationship is: the distance traveled by the express train-the distance traveled by the local train +480km = 600km.
Solution: Suppose the distance between two cars is 600 kilometers after X hours.
(4) Analysis: Trace the problem and draw it as follows:
The equivalent relationship is: express distance = local distance+480km.
Solution: Set the express train to catch up with the local train after X hours. Judging from the meaning of the question,
(5) Analysis: the rear-end collision problem, the equivalent relationship is: express distance = local distance+480km.
Solution:
8. profit and loss problem
(1) The quantities that often appear in sales problems are: purchase price, sale price, bid price, profit, etc.
(2) Relationship:
Commodity profit = commodity price-commodity purchase price = commodity price × discount rate-commodity purchase price
Commodity profit rate = commodity profit/commodity purchase price
Commodity price = commodity price × discount rate
Example 8. A store will increase the purchase price of a certain clothing by 40%, then price it and sell it at a 20% discount. As a result, each piece of clothing still earned 15 yuan. What is the purchase price of each piece of clothing?
Analysis: It is the key to explore the implied conditions in the topic, and the cost can be directly set to X yuan.
Purchase discount rate, bid price, preferential price and profit
Twenty percent off (1+40%)x twenty percent off (1+40%)x 15.
Equivalence relationship: (profit = discount price-purchase price) discount price-purchase price = 15.
Solution: let the purchase price be x yuan,
9. Savings problem
(1) The money deposited by the customer in the bank is called the principal, and the reward paid by the bank to the customer is called interest. Principal and interest are collectively referred to as the sum of principal and interest, the time of deposit in the bank is called the number of periods, and the ratio of interest to principal is called the interest rate. Interest tax is paid at 20% of interest.
(2) Interest = principal × interest rate × number of periods and principal and interest = principal+interest tax = interest × tax rate (20%)
Example 9. A classmate deposited 250 yuan in the bank for half a year. After half a year, * * * got the principal and interest and 252.7 yuan. What is the annual interest rate of the bank for half a year? (excluding interest tax)
Analysis: equivalence relation: sum of principal and interest = principal ×( 1+ interest rate)
Solution: let the real interest rate for half a year be x,
A:
There are also "scheme decision-making problems, chicken and rabbit in the same cage, ticket purchase problems, integral problems, navigation problems" and so on
Finally, I wish you progress in your study ~ ~ ~ ~ ~ ~ ~ ~ ~