Several common problems in solving application problems by enumerating linear equations of one variable are used to make use of their characteristics.

1, the problem of bank savings.

2. Travel problems;

3. Engineering problems

4, distribution and supporting problems

5. Ask age questions

6. Quantity problem

7. Equal product problem

8. Profit problem

9. Deployment issues.

10, solution preparation.

1 1. Discuss the problem in groups.

The above 1 1 common questions and their characteristics are designed to help students deepen their understanding and memory and make their knowledge organized. They must never use it as a "crutch" for rote memorization. They should cultivate the ability of analyzing and solving problems and master the general method of solving application problems by using equations. There are no other types of problems except the common problems mentioned above. The key is to clarify the basic quantitative relationship between various problems.

First, the issue of savings.

Its quantitative relationship is: interest = principal interest rate deposit period; Principal and interest = principal+interest, interest tax = interest tax rate. Note that interest rates include daily interest rate, monthly interest rate and annual interest rate, with annual interest rate = monthly interest rate × 12 = daily interest rate ×365.

Example 1. Xiaoming's father saved a two-year time deposit the year before last, with an annual interest rate of 2.43%. After the expiration of this year, after deducting the interest tax, the interest earned just bought Xiao Ming a calculator worth 48.60 yuan. How much did Xiaoming's father save the year before last?

Solution: Suppose Xiaoming's father saved X yuan the year before last.

2.43%x( 1-20%)=48.60

X=2500

Example 2. Xiaoli has a time deposit of 2500 yuan in the bank. Calculated by the annual interest rate of 65,438+0.98%, the total principal and interest due for this deposit is 2,648.5 yuan. How many years has this deposit been kept? If 20% interest tax is deducted, how much will Xiaoli's principal and interest be?

Solution: Suppose this deposit has been kept for X years.

2500( 1+ 1.98% x)= 2648.5

X=3

If tax is deducted, the sum of principal and interest is:

26 18.8

Example 3. Some people deposit a yuan in the bank for one year in the form of educational savings, with an annual interest rate of b; Take it out one year later and deposit the sum of principal and interest in the bank in the form of one-year regular education savings. The annual interest rate is still B, so the sum of principal and interest after maturity is

Targeted training:

1. Li Yong's family saved 3,000 yuan in two forms, and after one year, all the interest was 43.92 yuan. Since the interest rates of the two kinds of savings are 2.25% and 0.99% respectively, ask him how much his family has saved. (Considering that the interest tax is 20%)

Let the deposit with an annual interest rate of 2.25% be X yuan.

2. Xiao Wang saved 2000 yuan and 1000 yuan respectively in two forms. After one year's withdrawal, after deducting the interest income tax, he can get the interest of 43.92 yuan. It is known that the sum of the annual interest rates of the two kinds of savings is 3.24%. What is the annual interest rate of these two kinds of savings? (Interest income tax payable by citizens = interest amount ×20%)

Suppose the annual interest rate of 2000 yuan deposit is

X=2.25

3. The original tax method for personal published articles and books is: (1) The tax amount is not higher than that of 800 yuan; (2) If the membership fee is not higher than 4,000 yuan in 800 yuan, the 800 yuan will pay 14% of the membership fee; (3) If the contribution fee is more than 4,000 yuan, a tax of 1 1% of the total contribution fee shall be paid. Now I know that Mr. Ding received the manuscript fee and paid personal income tax to 420 yuan. How much is this manuscript fee for Teacher Ding?

(1) Analyze the scope of tax payment. If it exceeds 4,000 yuan, it should be taxed in 440 yuan. So the manuscript fee should be between 800-4000 yuan.

Let this manuscript fee be x yuan,

Second, the trip.

To master the basic relationship in the trip: distance = speed time.

(1) meet problem (opposite). The arithmetic relation of this kind of problems is: the sum of the distances that each person walks is equal to the total distance, or the time that two people walk at the same time is equal to the arithmetic relation.

(2) Catch-up problem (driving in the same direction). The equivalent relationship of this kind of problems is that the distance difference between two people is equal to the distance of reminiscence or the relationship is equal to the time of reminiscence.

(3) Meeting and catching up on the circular runway: the equivalent relationship between two people walking in the same place is that the sum of the distances traveled by two people is equal to one circle; The equivalent relationship of walking in the same place and the same direction is that the distance difference between two people is equal to the distance of a circle.

(4) Navigation problem: The relationship between relative motion of combined speed is

Downstream velocity = still water velocity+current velocity; Velocity = still water velocity-water velocity.

Travel problems should be understood by drawing a schematic diagram, and pay attention to the time and place when two people travel.

Example 4. Xiao Zhang and his father are scheduled to go to the railway station to visit their grandfather by bus at home. Halfway through, Xiao Zhang asked the driver about the driving time. The driver estimated that the train had just left when he continued to take the bus to the railway station. According to the driver's suggestion, Xiao Zhang and his father immediately got off the bus to take a taxi, doubled the speed and arrived at the train station 15 minutes before the train started. It is understood that Xiao Zhang and his father arrived at the railway station.

Solution 1: Let the distance between Xiaozhangjia and the railway station be X kilometers, and the meaning of the question is as follows:

(Set the amount of knowledge directly according to the equal time before and after)

Option 2: Suppose Xiao Zhang takes a bus for x hours.

The distance from Xiaozhangjia to the train is:

(Unknown number is indirectly set according to the equal distance between front and back)

Targeted training:

1. Xiaohua's family plans to take a taxi from home to the railway station. If the taxi travels at a speed of 50 kilometers per hour, it will be 24 minutes late; If the taxi travels at a high speed of 75 kilometers per hour, it can arrive at the train station 24 minutes in advance, so as to find out the distance from Xiaohua's home to the train station.

The distance from Xiaohua's home to the railway station is X kilometers.

2. A player wants to go from place A to place B, and the distance is 18km. There is only one car, so all the members are divided into two groups: group A and group B. First, group A takes a car, while group B walks, and at the same time drives to place C on the way. When the car went back to pick up group B, group A arrived at location B at the same time. If the speed is 60 kilometers per hour and the walking speed is 4 kilometers per hour, Group A will arrive at Point B at the same time.

Method 1: Let AC be x kilometers apart.

Method 2

Example 5. In order to celebrate the opening of the school sports meeting, the students in Class Two, Grade One accepted the task of making a small national flag. Originally, half of the students planned to participate in the production and make 40 noodles every day. After completing one third, the whole class will participate together. As a result, the task was completed one and a half days ahead of schedule. Assuming that everyone's production efficiency is the same, how many noodles has * * made?

Solution: Let * * be the X plane of the flag, which is deduced from the meaning of the question: (equivalence relation: planned time = actual time+/-time difference)

Targeted training:

1. A school organized a spring outing for teachers and students. If multiple 45-seat buses are rented separately, they are just full; If you rent only 60 buses, you can rent less 1 car, and there are 30 empty seats. Ask the number of people who will take part in the spring outing in this school.

Method 1: Unknown indirect setting.

A 45-seat car needs x cars.

Method 2: Set the unknown directly.

Let's assume that the person who takes part in the spring outing is X.

2. A worker produced 20 parts per day according to the original plan, but 65,438+000 parts could not be completed before the scheduled date. If the work efficiency is increased by 25%, 50 parts will be overfulfilled by the due date. How many parts did the worker originally plan to produce?

Method 1: Assuming that the original planned output is x days,

Method 2: Assuming that Y parts were originally planned to be produced,

Example 6. Motorboats travel 36 kilometers downstream and 24 kilometers upstream, which takes 3 hours. Find the speed and current speed of the motorboat in still water.

Solution: Let the speed of motorboat in still water be x km/h, from the meaning of the question:

(Equivalent relationship: speed along the water × time along the water = distance along the water × time against the water = distance against the water)

The water flow speed is:

Suppose the speed of a ship in still water is x km/h.

Targeted training:

A car is driving back and forth on a sloping road. Uphill speed 10 km/h, downhill speed 20 km/h, find the average speed of the car.

Thought: divide all the way by all the time.

Let the length of the slope be one kilometer.

Example 7. Party A and Party B practice running on the circular track. It is known that the length of a circular track is 400 meters, the speed of Party B is 6 meters per second, and the speed of Party A is twice that of Party B. If A starts in the same direction at the same time 8 meters in front of B, how many seconds will it take them to meet for the first time?

Analysis (figure): circular runway, driving in the same direction, can be regarded as a catch-up problem. The first encounter is the first pursuit, and the distance of pursuit is 400 meters. The speed of this question is fast, and it is obvious that A is chasing B. Because A is 8 meters in front of B and heading in the same direction at the same time, the pursuit distance of this question is actually (400-8) meters.

Solution: suppose that after x seconds, two people meet for the first time.

Targeted training:

1. On the expressway, a car with a length of 4m and a speed of 1 10km/h will overtake a truck with a length of 12m and a speed of 10km/h. How many seconds does it take for the car to catch up with and overtake the truck?

Method 1: Assuming it takes x hours, Method 2: Use relative speed.

2. It is known that the length of a railway bridge is 1000m, and there are trains passing through. According to the calculation, it takes 1 minute for the train to get on the bridge and completely cross it. The whole train takes 40 seconds on the bridge. Find the speed and length of the train.

Let the train be x meters long and the speed is:

second

3. Ring Road A 18km, and A rides a bike along the road at a speed of 550m per minute; B Run along the road, running 250 meters per minute. Two people start from the same place and the same direction at the same time. How many hours did they meet again?

Let two people meet in x hours (pay attention to the unity of the unit)

250m/min =15km/hr 550m/min = 33km/hr.

15x+ 18=33x

X= 1

4. Party A and Party B practice running on the circular track with a circumference of 400 meters. If they start from opposite directions at the same time, they will meet every 2.5 minutes. If they start in the same direction at the same time and meet every 10 minute, assuming that their speed is constant, A is fast and B is slow, find the speed of both sides.

Let the speed of A be x m/min, then the speed of B is: 400 ÷ 2.5–x (according to the combined speed).

Three. Engineering problems

Its basic quantitative relationship: total work = working efficiency and working time; Joint operation efficiency = sum of individual operation efficiency. When the total workload is not given, the permanent total workload is "1", and a list or drawing can be used to help understand the meaning of the problem.

Example 8. Master and apprentice repair a gas pipeline, and it takes 15 hours for the master to complete it alone, and 15 hours for the apprentice to complete it alone.

(1) If two people cooperate, how many hours will it take?

(2) If the apprentice works for five hours first, and then the master works with him, how many hours will it take to finish?

(3) If two people work together for five hours first, and then the apprentice works alone, how many hours will it take to finish?

Solution: ① Suppose it takes X hours to complete, so: ② It takes Y hours to complete, so:

(3) the apprentice must work alone for z hours to complete, so:

Targeted training:

1. Open the pipe and fill the tank with water, which can be filled in 5 minutes. Pull out the bottom plug after filling, so that the water in the cylinder can be used up within 10 minutes. Once, I opened the pipe and filled the empty cylinder with water. After a few minutes, I found that the bottom plug was not plugged. It took too long to fill it. How long did it take to fill the tank?

Set x minutes to fill the water tank.

minute

Two candles with the same length, one can burn for 6 hours, the other can burn for 4 hours, and both candles are lit at the same time. A few hours later, one candle is twice as long as the other.

Let x hours later, the length of one candle is twice as long as the other, and let the length of the candle be "1".

Four. Assignment and matching problem

This kind of problem has no basic quantitative relationship, but the key is to see how different quantities are distributed and matched.

Example 9. A group of students are boating in the park. If there are five people on each boat, then the other two can't get on board. If there are six people on each boat, there are still three seats. Ask for the number of students and the number of charters.

Solution: Law 1: With classmate X, you can get:

Rule number two: hire a y boat,

This question is easy to confuse the symbols of 2 and 3. The simplest way to check is to solve the equation. If the symbol is wrong, it is unrealistic to solve a negative number.

Targeted training:

1. At present, the workload of project A is twice that of project B, with the first group 19 and the second group 14 (assuming the average work efficiency is the same). How to allocate two groups of people so that two projects can be started and completed at the same time?

Analysis: If A has twice as many people as B, it can start and end at the same time.

Let's transfer x people from the second group to the first group.

2. There are 27 people working in A, and there are 19 people working in B. Now, 20 people are transferred to support, so the number of A is twice that of B. How many people should A and B be transferred respectively?

Let's assign x people to a,

Example 10. A workshop can produce 120 Class A parts or 100 Class B parts every day. Only three or two parts of A and B can be made into one set, and the most complete set of products should be produced within 30 days. Q: How to arrange the number of days to produce parts A and B?

Analysis: The ratio of Part A to Part B is 3: 2.

Solution: If the production plan of Part A is x days, then:

(1) when x= 16

A: 640 sets, B: 700 sets, so you can produce 640 sets.

② When x= 17,

A: 680 sets, B: 650 sets, so you can produce 650 sets.

Example 1 1. Use white cardboard as the packaging box, each cardboard can be made into 16 box or 43 box bottoms, and one box can be made into a set of two box bottoms. At present, there are 100 pieces of cardboard. How many boxes can be made into certificate sets?

Solution: suppose the box is an x-ray, which is deduced from the meaning of the question.

Targeted training:

1. One workshop processes shafts and bearings, and each person can process 12 shaft or 16 bearing on average every day. 1 shaft and 2 bearing are a set. There are 90 people in the workshop. How to allocate manpower to make the bearings and shafts produced every day just match?

Set the production axis x (pay attention to the quantity relationship when matching and establish the equivalence relationship).

2. A square table consists of a desktop and four legs. If you can make 50 tables or 300 tables with 1 m3 of wood, and there are 5 m3 of wood, how many m3 of wood can you make a table top and how many m3 of wood can you make table legs, just to make a complete set? Then figure out how many sets you can make.

Set the x cube as the desktop.

The problem of finding the age of verb (verb's abbreviation)

This problem usually involves the ages of two people at different times, which can be based on the following points: (1) The age difference between people is always the same;

(2) They grew up at the same age; Equations can be listed according to one of these two equivalence relations.

Time Jiayi

Before (1) (1)

Now, now

In the future, in the future

Example 12. A said to B, "When my age is your present age, you are only 4 years old", and B said to A, "When my age is your present age, you will be 6 1 year old". Q: How old are A and B now?

Solution: Suppose A is now X years old.

(Analysis: A and B are four years apart, so A plus four years old will be twice as big as B)

So B's age is: years old, and the equation is:

Example 13. This year, the two brothers add up to 55 years old; One year, my brother's age was the age of my brother this year. At that time, my brother was just twice as old as my brother. Q: How old are my younger brother and younger brother this year?

According to the age difference between two people, the equation is always the same. ) time brother once this year, brother once this year.

Solution: Suppose my brother is X years old this year.

(Reciprocal relationship: brother age+brother age = total age; Brother's age this year-brother's age this year = brother's age-brother's age)

Targeted training:

This year, Xiao Li's age is 65438+ 0/5 of his grandfather's. Xiao Li found that after 12, his age became his grandfather's 1/3. Try to find out Xiao Li's age this year.

Let Xiao Li be x years old this year.

Sixth, the quantity problem.

A multi-digit: abc=a× 100+ b× 10+ c (e.g. 547=5× 100+4× 10+7).

Abc=a× 100+ bc (for example, 547=5× 100+47)

To correctly distinguish the two concepts of "number" and "number", this kind of problems usually adopt indirect methods, and the common problem-solving thinking analysis is to grasp the relationship between numbers or between new numbers and original numbers and find the equivalent relationship. The premise of the column equation must also correctly express the algebraic expression of multi-digits. A multi-digit is the sum of the products of each digit and its counting unit.

Example 14. Once Xiaohong wrote down the ten digits and the single digits of the answer to a question, and the result was 27 fewer than the correct answer, and the ten digits of the correct answer were twice as many as the single digits, so she sought the correct answer.

Solution: Let the ten digits in the correct answer be X, which is derived from the meaning of the question:

Example 15. For a three-digit number, the sum of the hundredth digit and its last two digits is 58. If the hundredth digit has reached the end of this number, the new three digits are 306 larger than the original number, and the original three digits are found.

Solution: Let the original hundred digits be X and the last two digits be Y. From the meaning of the question:

One of the problems is to be accurate, whether to set the original number or the new number; The second is to be familiar with the representation of multiple digits.

Targeted training:

1. For a three-digit number, the number on the hundredth digit is 2 larger than the number on the tenth digit, and the number on the tenth digit is 2 times. Switch the number on the digit with the number on the hundredth digit to get a new three-digit number, which is 99 less than the original three-digit number. Find the original three digits.

2. The unit number of a three-digit number is 7. If the unit number is moved to the first place, the new number is 86 times more than the original number. Find this three-digit number

Seven, equal product problem

Here, equal product means equal area or volume. The basic quantitative relationship is: volume before deformation = volume after deformation. You must master the area and volume formulas of common geometric figures.

Example 16. How long does it take to cut a round steel with a diameter of 4cm and forge a cylindrical blank with a diameter of 80mm and a height of 30mm in the factory?

Solution: Let the round steel with a diameter of 4cm be x mm, and it is concluded from the meaning of the question: (V cylinder =πr2×h).

There are two misunderstandings in solving this problem: first, pay attention to the unity of the unit; Second, don't take the diameter as the radius.

Example 17. Fill a cylindrical bottle with a bottom diameter of 5cm and a height of 18cm with water, and then pour the water in the bottle into a cylindrical glass with a bottom diameter of 6cm and a height of 10cm. Can it be completely filled? If not, how high is the water level in the bottle? If it is not full, find the distance from the water surface in the cup to the mouth of the cup.

Solution: ① The volume of cylindrical bottle is: the volume of cylindrical glass is: the comparison of the two volumes:

(2) The water level in the bottle is still x cm high, so:

Targeted training:

1. A cylindrical glass with a diameter of 90 mm is filled with water. Put the water in the glass into a rectangular iron box with a bottom area of (13/kloc-0 /×1) mm2 and a height of 81mm. When the iron box is filled with water, the water in the glass will be filled. (accurate to 0. 1 mm)

2. There is a rectangular iron sheet with a length of 40cm and a width of 30cm, which is used to make the side of the cylindrical iron drum, and another iron sheet with a large enough size is used as the bottom of the drum. How to maximize the volume of iron drum?

Eight, the profit problem

Basic quantitative relationship: profit = selling price-buying price; Profit rate of commodities =%.

Selling price = pricing × discount;

Example 18. If the goods are sold at a discount due to seasonal changes, they will be sold at 75% of the pricing, and 25 yuan will be compensated; If you sell it at 90% price, you will earn 20 yuan. Ask the price of the goods.

Solution: Let the commodity price be X yuan, and deduce it from the meaning of the question: (using the formula of constant principal)

Example 19. A commodity is sold at a price that increases the cost by 25%, and then the price needs to be reduced due to the backlog of inventory. If you still want to make a profit of 10% on each item, how much discount will you give to the original price?

Solution: If the cost is one yuan, it should be sold at X copies of the original price when the price is reduced. From the meaning of the question, we know that there are many unknowns in this question, but we know that the price is = (1+25) cost, so we can set the cost to one yuan first, and then the price is = (1+25) one yuan.

(At this time, both sides of the equation can be divided by a at the same time, and a can be omitted. )

Targeted training:

1. If a store sells a sweater at 20% of the list price, it can still make a profit of 20%. If the purchase price of a sweater of this brand is 100 yuan, what is the list price of each sweater?

2. The purchase price of a commodity is 400 yuan, and the bid price is 550 yuan. The goods are sold at 20% of the bid price. What is the profit rate?

3. The price of a commodity is 900 yuan per piece. In order to participate in the market competition, the store will give 40 yuan a 10% discount on the selling price and still make a profit of 10%. What is the purchase price of this commodity?

A vendor bought several baskets of apples at the purchase price of 3 yuan per kilogram, and then sold them at the price of 4 yuan per kilogram. When he sold half of the apples, he recovered the cost. How many baskets of apples did he buy?

5. The cost of clothes A and B * * In 500 yuan, in order to make a profit, the store decided to price clothes A 50% and clothes B 40%. In actual sales, at the request of customers, both clothes are sold at a 10% discount, so the store makes a profit 157 yuan. What is the cost of clothes A and B?

IX. Deployment Issues

Looking for equivalence relationship from the quantity relationship after deployment is often regarded as the relationship of "sum, difference, times and points", and attention should be paid to the direction and quantity of deployment objects.

Example 20. There are 57 workers in factory A and 75 workers in factory B. Now some workers are transferred from factory A to factory B, so the number of workers in factory B is twice that of factory A. How many people are transferred from factory A to factory B?

Try to practice (cooperative learning):

There are 57 workers in Factory A and 75 workers in Factory B. Now we need to transfer 42 workers from Factory B to support other factories. After the job transfer, the number of workers in Factory A is half that of Factory B. How many workers in Factory A and Factory B have transferred jobs respectively?

It is suggested that if the unknown number is set, the number of people transferred from factory A can be set directly, and the number of people transferred from factory B can be set indirectly.

Consolidation exercise:

There are 73 cars in Team A and 65 cars in Team B. How many cars should be transferred from Team B to Team A to make the number of cars in Team A double that of Team B?

Improved exercises:

There are 39 people in the processing workshop of a garment factory, and each worker can process 5 coats or 8 pairs of trousers every day. How to allocate the number of people processing coats and trousers to match coats and trousers?

X. Solution preparation problems

Its basic quantitative relationship is: solution mass = solute mass+solvent mass; Solute = solution × concentration (), solution = solute+solvent. Solute mass = the mass of solute contained in the solution.

This kind of problem often finds the equivalent relationship according to the solute quality or solvent quality before and after preparation, and the tabular method can be used to help understand the meaning of the problem.

Example 2 1. There are 8 kilograms of 98% sulfuric acid solution, and how many kilograms of 20% sulfuric acid solution can be added to make 60% sulfuric acid solution.

Variant exercise

1. How much water does it take to change 9000g of solution containing 60% alcohol into solution containing 40% alcohol?

2. A middle school laboratory needs 20% iodine. 25% iodine contains 350g iodine. How many grams of pure alcohol should I add?