[Examination Highlights and Frequently Asked Questions]

This paper examines the ability of solving application problems with series equations (groups), with emphasis on solving application problems with series quadratic equations or series fractional equations. The exercises are mainly engineering problems and travel problems, and some economic problems have appeared in recent years, so we should pay attention to them.

Fill in the blanks

1. The price of a commodity is 165 yuan. If the price is reduced by 10% (that is, the discount is 10%), the profit can still be 10% (relative to the purchase price). So the purchase price of the goods is

2. Party A and Party B invest in establishing a joint venture and agree to distribute profits according to the investment proportion. In view of the fact that the investment ratio of Party A and Party B is 3: 4, and the profit in the first year is 38,500 yuan, the profits that Party A and Party B can get are RMB yuan and RMB yuan respectively.

3. In 1996, a company earned 1 35000 USD, which increased by a% every year in 1997 and 1998, so the company earned110000 USD in 1998.

4. A city has an existing population of 420,000. It is planned that after one year, the urban population will increase by 0.8% and the rural population will increase by 1. 1%, so that the population of the whole city will increase by 1%. Find the existing urban population and rural population of this city. If the existing urban population is x million and the existing rural population is y million, the equation listed below is as follows.

5. In agricultural production, it is necessary to select seeds with salt content of 16%. There are 200 kilograms of salt water with a salt content of 24%. How many kilograms of water should I add?

Solution: Suppose you need X kilograms of water. According to the meaning of the question, the equation is. To solve this equation, you have to answer.

6. A TV factory paid 4 million yuan in profits and taxes to the state in 1994, and it increased to 4.84 million yuan in 1996, so the average annual growth rate of profits and taxes paid by the factory in two years.

7. The purchase price of a commodity is X yuan per piece, and the retail price is 900 yuan per piece. In order to adapt to the market competition, the store reduced the price by 10% on the basis of the retail price, allowed to sell in 40 yuan, and still made a profit of 10% (relative to the purchase price), so X = yuan.

8. A wholesale and retail stationery store stipulates that anyone who buys more than 30/kloc-0 pencils at one time (including 30/kloc-0 pencils) can pay at the wholesale price; If you buy less than 300 pieces (including 300 pieces), you can only pay at the retail price. Xiao Wang, an existing student, came to buy pencils. If you buy 1 block for every junior in the school, you can only pay at the retail price, and you need to use (m2- 1) yuan (m is a positive integer, m2-1>100); If you buy 60 more, you can pay at the wholesale price, which also costs (m2- 1) yuan.

(1) If there are X students in the third grade of our school, the range of (a)x should be

(b) The retail price of each pencil shall be RMB, and the wholesale price shall be RMB.

(represented by an algebraic expression containing x, m)

(2) If the wholesale price of 15 cigarettes is less than the retail price of 15 cigarettes, pay 1 yuan, try to find out how many students are in the third grade of this school and determine the value of m.

2. Solving application problems with equations

1. A store brought 120 air conditioners for sale. Due to the promotion activities, four more air conditioners were sold every day than originally planned. As a result, the sales task was completed five days ahead of schedule. How many air conditioners were originally planned to sell every day?

2. 1995, the number of qualified students in six subjects of junior high school graduation examination in our province was 80,000. 1997 rose to 90,000, so the percentage of average growth in two years was obtained (take = 1.45438+0).

3. When Team A and Team B finish a certain job, it takes 65,438+05 days for Team A to finish it alone than Team B. If Team A works for 65,438+00 days and then Team B works for 65,438+05 days, the job can be completed. How many days does it take for Party A and Party B to finish this work alone?

The headmaster of a school will lead the "three good students" of the school to travel to Beijing in the summer vacation. A travel agency said that "if the principal buys a full ticket, the rest of the students can enjoy a half-price discount", and B travel agency said that "all students including the principal enjoy a 60% discount on the full ticket price (that is, if the principal buys a full ticket in 240 yuan).

(1) Let the number of students be X, the fee of travel agency A be y A, and the fee of travel agency B be y B, and calculate the fees of the two travel agencies respectively (establish expressions).

(2) When the number of students is what, are the charges of the two travel agencies the same?

(3) Discuss which travel agency is more favorable in terms of the number of students X?

5. There is 400g of salt water containing 15%, and Mr. Zhang requested that the salt water content be changed to 12%. A classmate added110g of water due to a calculation error. Ask this classmate to explain how much water is added by calculating the equation, and point out how many grams of water is added?

6.A walks from A to B at 6: 00 in the morning, and B rides a bike from A to B at 3: 00 in the morning/kloc-0. When did B catch up with A?

7. In order to welcome Hongkong's return to China, teachers and students planted 1994 to 1997 trees. It is understood that the school planted 342 trees in 1994 and 500 trees in 1995. If 1996,

8. To build a rectangular chicken farm with an area of 150m2, in order to save materials, one side of the chicken farm is against the original fence, and the other three sides are surrounded by bamboo fences, as shown in the figure. If the fence is 35m long, what is the length and width of the chicken farm (1)? (2) What role does the length a of the wall play in solving the problem?

9. Yongsheng Electronics Co., Ltd. applied for two loans from ICBC, with an annual interest of RMB 680,000.00. The annual interest rate of loan A is 12%, and that of loan B is 13%. What is the amount of these two kinds of loans?

10. Xiaoming deposits 100 earned from work-study program in children's bank for one year, and takes out 50 yuan to buy school supplies after the expiration, and deposits all the remaining 50 yuan and interest due for one year. If the annual interest rate of the deposit remains unchanged, the principal and interest can be obtained after maturity. Find the annual interest rate of this deposit.

1 1. A company borrowed 400,000 yuan from a bank to produce new products. It is known that the annual interest rate of the loan is 15% (excluding compound interest, that is, interest is not repeated every year before repayment). The cost of each new product is 2.3 yuan and the price is 4 yuan, and the taxable amount is 10% of the sales. If 200,000 such products are produced every year, how many years will it take to repay the loan with profit (profit = sales-cost-tax payable)?

12. A workshop processed 130 parts in a specified time. After processing 40 parts, due to the improvement of operation technology, 10 parts are processed more than originally planned every day. As a result, it took five days to complete the task. How many parts do you plan to process every day?

First, column algebra

1.a gram of water is mixed with b gram of salt to form brine, and the percentage of salt in the brine is

2. If the price of a commodity is one yuan after x% reduction, then the original price of the commodity is Yuan Yuan.

3. There is a job that Party A needs a day to complete alone, and Party B needs b days to complete alone. If Party A and Party B cooperate to complete this work, how many days will it take to complete this work?

4. In order to encourage electricity conservation in a certain place, the charging standards for users' electricity consumption are as follows: if the monthly electricity consumption of each household does not exceed 100 kWh, it will be charged at one yuan per kWh; If it exceeds 100 degrees, the excess will be charged twice per degree. A household uses electricity 180 kwh a month, and has to pay RMB this month.

Second, only equations (groups) are incomprehensible.

1.Students from Class A and Class B take part in afforestation. It is known that Class A plants 5 more trees every day than Class B. The number of days for Class A to plant 80 trees is equal to the number of days for Class B to plant 70 trees. If Class A plants X trees every day, the equation is as follows

2. Someone deposits 2,000 yuan in the bank for one year, takes out 1000 yuan for shopping after the expiration, and deposits the remaining 1000 yuan and the interest due in the bank for one year. If the deposit interest remains unchanged, the principal and interest after maturity will be *** 1320 yuan. If the annual interest rate of this deposit method is x, the equation is obtained.

3. There is a conference room with a length of 20m and a width of15m. There is a carpet in the middle of the conference room, and the area of the carpet is half that of the conference room. If the width of the blank area around the carpet is x meters, the equation listed is

A factory plans to produce 65,438+0,000 machine tools in X days. Later, in actual production, it produced 25 more machines every day than originally planned, and the result was completed two days ahead of schedule, so there was an equation.

5.A and B are 60 kilometers apart, and two people ride bicycles from A and B respectively. If A leaves 30 minutes earlier than B, and A travels 2 kilometers less per hour than B, then the distance they travel when they meet is exactly the same. If the circulation speed of a is x kilometers per hour, the equation is obtained.

Third, column inequality.

A bicycle factory produces and sells a new type of bicycle this year, and now we provide you with the following information:

Last year, the factory prepared 1000 wheels of this bicycle. This year, the wheel workshop can produce 1500 wheels per month on average, and each bicycle needs two wheels.

(2) The assembly workshop of our factory (the production workshop of the last process of bicycles) can assemble at least 1 1,000 bicycles per month, but not more than 1 1,200 bicycles;

(3) This year, the factory has received orders for bicycles from customers all over the world *** 14500;

(4) The ex-factory sales unit price of this bicycle is 500 yuan/vehicle.

The sales of this kind of bicycle in this factory this year is one million yuan. Please judge the range of A according to the above information.

Four, column equation to solve application problems:

1. The original price of a commodity in 50 yuan was reduced in June 5438+00 due to poor sales 10%, and the price was increased from June 5438+0 10, and the price in February/6 was 64.8 yuan.

Q: (1) What is the price of this commodity in June? (2) 1 1 and 65438+what is the average price increase rate in February?

2. Team A and Team B each transport 150 tons of goods. It is known that Team A has five more cars than Team B, while Team B has an average of more cars 1 ton of goods than Team A. Both teams have finished loading at one time. How many cars are there in Team A and Team B?

Party A and Party B can complete a task after working together for 6 days. It takes 9 days for Party A to finish it alone than for Party B. How many days does it take for Party B to finish it alone?

The distance between a and b is 30 kilometers. A walks more per hour than B 1 km, and the time from A to B is less than B 1 hour. How many kilometers do A and B walk per hour?

Teachers and students of a school visited a place 28 kilometers away from the school and began to walk at a speed of 4 kilometers per hour. The rest of the journey is by car at 36 kilometers per hour, the whole journey 1 hour. How long does it take to walk?

The following are difficult application problems:

1. Two trains run on two parallel tracks respectively, in which the express train is 100 meters long and the local train is 150 meters long. When two trains are driving in opposite directions, it takes 5 seconds for the express train to pass through a window of the local train (from the time when the front of the express train arrives at the window to the time when the rear of the express train leaves).

(1) Find the sum of the speeds of the two trains, and the time it takes for the local train to pass through a window of the express train when the two trains are driving in opposite directions (from the time when the local train head arrives at the window to the time when the local train tail leaves);

(2) If two cars are driving in the same direction, and the speed of the local train is not less than 8m/s, how many seconds does it take for the express train to catch up with the local train from the back to leave the local train?

2. A project is completed by two teams of Party A and Party B within 6 days. The manufacturer needs to pay Party A and Party B * * 8700 yuan, and Party B and Party C will work together 10 day. The manufacturer needs to pay Party B and Party C 9500 yuan, and Party A and Party C will work together for 5 days to complete 2/3 of the whole project. The manufacturer needs to pay Party A and Party C..

(1) How many days does it take for Team A, Team B and Team C to complete all the projects independently?

(2) It takes no more than 15 days for a project to complete all projects. Which team can complete the project alone with the least money? Please explain the reason.

Five, function application questions:

1. A car drives from Guangzhou to Hunan 300 kilometers away, with an average speed of 80 kilometers per hour. Then the functional relationship between the distance s (kilometers) from Hunan and the driving time t (hours) is

2. A factory plans to use Q tons of coal every month, and consumes an average of one ton of coal every day. If x tons of coal are saved every day, then q tons of coal can be used for y days. Write the functional relationship between y and x as follows

3. The candle is 20cm long and burns 5cm every hour after being lit. The functional relationship between the residual height h(cm) and the combustion time t (hours) is represented by (*).

4. If there are 12 ballpoint pens in each box, and the price is 18 yuan, then the functional relationship between the ballpoint pen price y (yuan) and the number x of ballpoint pens is (*).

According to a fruit wholesale market, when the apple is not less than100kg, the wholesale price is 2.5 yuan per kg. Xiao Wang brought 3000 yuan in cash to this market to buy apples at the wholesale price. If the apple he bought is X kilograms and the remaining cash after Xiao Wang paid is Y yuan, try to write the functional relationship between Y and X, and point out the range of the independent variable X.

6. City A and City B have 12 and 6 machines in stock respectively, and decided to send 8 machines to City C 10 and City D.. It is known that the freight rates from A city to C city and D city are 400 yuan and 800 yuan respectively, and the freight rates from B city to C city and D city are 300 yuan and 500 yuan respectively.

(1) Let machine X be transported from city B to city C, and find the functional relationship between freight W and X;

(2) If the total freight does not exceed 9000 yuan, how many transportation schemes are there?

(3) Find out the transportation scheme with the lowest total freight, and what is the lowest freight?

7. A businessman starts selling goods with a unit price of 8 yuan at 10 yuan, and can sell 100 pieces every day. Now he wants to increase profits by raising the price. It is known that the daily sales of this commodity will decrease by 1 0 for every price increase of1yuan.

(1) Write the functional relationship between the selling price of X yuan and the gross profit of Y yuan per yuan;

(2) What is the daily sales price to maximize the daily profit?

8. A factory has 360 kilograms of raw materials A and 290 kilograms of raw materials B, and plans to produce 50 products A and B with these two raw materials. It is known that it takes 9 kilograms of raw materials A and 3 kilograms of raw materials B to produce a product, which will benefit 700 yuan. To produce a type B product, 4 kg of type A raw materials and 10 kg of type B raw materials are needed, and the profit can be 120 yuan.

(1) Is there any plan to arrange the number of production pieces of products A and B as required? Please design it;

(2) Let the total profit of producing two kinds of products be Y yuan and the quantity of one kind of product be X. Try to write the functional relationship between Y and X, and use the properties of the function to explain which production scheme has the largest total profit (1). What is the maximum profit?

1. A pool contains 84 tons of water, and there are two drainage pipes A and B. The first pipe drains 2.5 tons per hour and the second pipe drains 3.5 tons per hour. If the first pipe is opened and the second pipe is opened after 2 hours and 24 minutes, how many hours can the water in the pool be discharged after the first pipe is opened?

2. The correspondent sent a letter from A to B, and immediately returned to A. * * * It took 3 hours and 52 minutes, with a speed of 30 km/h when he left and 28 km/h when he came back, to find the distance between A and B.

3.A walks 5 kilometers every hour, and after 2 hours, B rides a bike to chase A. ..

(1) If the speed of B is 20km/h, how long will it take for B to catch up with A?

(2) If B is required to go 14km to catch up with A, what is the speed of B?

4. Party A and Party B practice race walking on the 400-meter circular runway. Party B walks 80 meters per minute, and Party A's speed is 1 and 1/4 of Party B's. Now Party A is ahead of Party B 100 meters. How many minutes did they meet for the first time?

1. has a three-digit number, and its unit is three times smaller than the number on the 100th place, and the number on the unit is 1. If a new number is obtained by exchanging the number in the tenth place and the number in the hundredth place of this three-digit number, the original three-digit number is 270 times smaller than the new number, and the original three-digit number is found.

2. The school has a four-story teaching building with six classrooms on each floor and three doors (two front doors and one side door of the same size) entering and leaving the building. The door was inspected during the safety inspection; When a main entrance and a side door are opened at the same time, 400 students can pass through in 2 minutes. If the main entrance has an average of 40 more students per minute than the side entrance.

(1) How many students can pass through a main entrance and a side door every minute on average?

(2) During the inspection, it was found that the efficiency of going out was reduced by 20% because students were too crowded in an emergency. According to the safety regulations, in case of emergency, students in the whole building should be evacuated safely through these three doors within 5 minutes. Suppose that each classroom in this teaching building has a maximum of 45 students. Q: Do these three doors meet the safety requirements? Why?

3. Party A and Party B walk from City A to City B. Party A walks 4 kilometers per hour, while Party B rides 8 kilometers per hour more than Party A. After half an hour, Party B sets out and both of them arrive at City B at the same time to find the distance between the two cities.

4. Several students in Education Middle School asked for cancellation. If there are 4 people living in each dormitory, there are 20 people left; If there are 8 people living in each dormitory, one dormitory is not empty and the others are full. Q: How many dormitories are there in this middle school? How many students are required to live on campus?

Bao Xiao and his parents are playing on the seesaw in the playground. His father weighs 72 kilograms and sits at one end of the seesaw. Bao Xiao, who weighs only half as much as his mother, sits at the other end of the seesaw with his mother. At this time, his father's feet were still on the ground. Later, Bao Xiao borrowed a pair of dumbbells weighing 6 kilograms and put them at the end where he sat with his mother. As a result, Bao Xiao and his mother escaped from danger. Guess how much Bao Xiao weighs? (accurate to 1 kg)

3. It is known that a factory has 70m and 52m fabrics. It is planned to produce 80 sets of A and B fashions with these two fabrics. It is known that the fabrics needed to make a set of A and B fashions are shown in the following table. Can the factory complete the task with existing raw materials? If yes, how many production plans are there? Please design it.

70m 52m A 0.6 0.9m b 1. 1m 0.4m

A batch of goods is transported by several cars with a load of seven tons. If each car only carries four tons, there is still 10 tons of goods left. If each car is full of seven tons, the last car will not be empty. Excuse me: How many cars are there?

It is known that the garment factory has 70m fabric A and 52m fabric B, and now it is planned to produce 80 sets of M and N fashions with these two fabrics. It is known that making a set of M fashions requires 0.6 meters of fabric and 0.9 meters of fabric. To make an N-fashion, fabric A needs 1. 1 m, and fabric B needs 0.4m.. If the number of sets for producing N models is X, then there are several schemes for producing these two models with this batch of cloth.