This kind of problem to find out the number of changes, common problems 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 1. There are several workers in Workshop A and Workshop B respectively. If 100 people are transferred from Workshop B to Workshop A, the number of people in Workshop A is 6 times that of the remaining people in Workshop B; If 100 people are transferred from workshop A to workshop B, then the number of people in the two workshops is equal, and the original number of people in workshop A and workshop B is found.

Example 2. School spring outing, if each car takes 45 people and 28 people don't get on the bus; If there are 50 people in each car, one car is available, and one car can also take 12 people. How many students and cars are there?

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?

Question Type 2, Proportional Problem

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.

For example, the ratio of three positive integers is 1: 2: 4, and their sum is 84, so what is the largest of these three numbers?

Question 3: 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.

For example, a two-digit number, the number of one digit is twice that of ten digits. If the number of ten digits is reversed with the number of one digit, the obtained two digits are 36 larger than the original two digits, so find the original two digits.

Question 4, engineering problem:

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 1. 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?

Example 2. As we all know, the pool has a water inlet pipe and a water outlet pipe. The water inlet pipe can fill the empty pool 15 hours, and the water outlet pipe can fill the pool for 24 hours.

(1) How much water can be injected per hour if the water inlet pipe is opened separately?

(2) If the outlet pipe is opened separately, how much water can be released per hour?

(3) If two pipes are opened at the same time, what is the effect per hour? How to form?

(4) For an empty pool, if the water inlet pipe is opened for 2 hours first, and then the two pipes are opened at the same time, how long will it take to fill the pool?

Question 5. Travel problem

(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 1. 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.

Example 2. A ship is sailing between two docks. The current speed is 3 kilometers per hour. It takes 2 hours to sail downstream and 3 hours to sail upstream. How to find the distance between two docks?

Question six, 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

For example, Bayi Gymnasium designed a sign made of the same cube (as shown in the figure). The side length of each cube is 1m, and its exposed surface (excluding the bottom surface) is made of five plywood nails and then painted. Each five-plywood can be painted on both sides, with 500 grams of paint per square meter.

(1) The building materials store marked 40% off the cost price, 5 plywood, and 20% off for sale. As a result, each Reng Zhang earned 4.8 yuan (five splints must be purchased as a whole):

(2) The paint shop launched the activity of "20 get 20 free, buy more and send more", and the price of paint purchased was 34 yuan per kilogram. How much did it cost to buy five pieces of plywood and paint?

Question 7: Savings.

(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

Sum of principal and interest = principal+interest

Interest tax = interest × tax rate (20%)

Example: A classmate deposited 250 yuan money in the bank for six months. 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)

Question 8: 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 released by Xinhua News Agency on March 28th, 2000 1, as of 0: 00 on October 28th, 20001,the population with primary school education per 10,000 population in China was 3,5701,while 6,5701.

Example 2. An enterprise conducts an English test for candidates. The test questions consist of 50 multiple-choice questions. According to the grading standard, if the answer to each question is correct, it will get 3 points, if it is not selected, it will get 0 points, and if it is wrong, it will be deducted 1 point. It is known that someone didn't do 5 questions and got 103. How many questions did the man make wrong?

Question 9. Setting auxiliary unknowns: problems

Example 1. A concert hall decided to hold a special concert for students during the summer vacation in early May. Admission tickets are divided into group tickets and retail tickets, of which group tickets account for the total number of votes. If you buy tickets in advance, you will be given different degrees of discount. May group tickets 12 yuan each, * * If group tickets are on sale, retail tickets 16 yuan each. The price of group tickets is 16 yuan, and all the remaining tickets are planned to be sold in June. So how much should retail tickets be priced to make the ticket revenue of these two months even?