application problem

First, the issue of savings.

Basic quantitative relationship: interest = principal × interest rate× time pure interest = interest × (1-20%)

Principal and interest (principal and interest) = principal+interest

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.

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.

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

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

For this kind of problem, the known quantity and unknown quantity are used to express the basic quantities such as interest, principal and interest rate, and then the basic quantity relationship is directly used.

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%)

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%)

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?

Second, the trip.

Basic quantitative relationship: speed × time = distance

The problem of sailing in the river: V Shun = V Ship +V Water V Inverse = V Ship -V Water

(v) The speed of the ship traveling along the water (v) The speed of the ship traveling against the water (v) The speed of the ship in still water (v).

Basic analytical problem solving method: draw a line segment diagram

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.

Analysis (Figure):

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.

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.

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 in the question:

(Equivalence: 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.

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?

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 and the current speed be y 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)

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.

Example 7. Party A and Party B practice running on the circular track. It is known that one circle of the circular orbit is 400 meters long, the speed of Party B is 6 meters per second, and the speed of Party A is twice that of Party B. If A sets off 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?

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.

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? 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.

Three. Engineering problems

Basic quantitative relationship: working efficiency × working time = workload.

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 Assuming it takes x hours to complete, it needs:

(2) To finish in y hours, you must:

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

For this kind of engineering problems, there is no specific amount of total workload, so we can regard the total workload as the unit of 1 and analyze everyone's work efficiency and working time. The equation is listed according to workload A+ workload B = 1.

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?

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.

Four. Assignment and matching problem

This kind of problem has no basic quantitative relationship. 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: There are X students and Y boats are rented, so you get:

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?

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?

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 of Part A takes X days and Part B takes Y days, we can get:

(The second equation can be simplified:)

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 X-piece is used as the box body and Y-piece is used as the box bottom.

The solution is x=, y =；; If a piece of cardboard can't be both the box and the bottom, then: x=, y =；;

This question is the same as the last one, but the result must consider two possibilities. )

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?

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.

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

This kind of problem usually involves the age of two people at different times, which can be based on: ① the age difference between two 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

Early times

at present

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 and B is now Y years old.

(List the equations according to their age difference. )

Time brothers brothers

this year

once

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? Solution: Suppose my brother is X years old and my brother is Y years old.

(Reciprocal relationship: elder brother's age+younger brother's age = total age.

Elder brother's age this year-younger brother's age this year = elder brother's age-younger 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 be younger 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)

Example 14. Once Xiaohong wrote down the ten digits and 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 he sought the correct answer.

Solution: Let the ten digits in the correct answer be X, and the digits in the correct answer be Y, which is derived from the meaning in 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: the volume before deformation = the volume after deformation.

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

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:

Volume comparison between the two:

(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; Selling price = pricing × discount; Profit margin =

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, which is derived from the meaning in the question:

(Using principal invariant formula)

Example 19. The pricing cost of a commodity rises by 25%, and then it needs to be reduced due to the backlog of inventory. If each product still wants to make a profit of 10%, how much discount will it be sold at the original price when the price is reduced?

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, then the price is = (1+25)% one yuan, and the actual price is = (65433).

(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?

solve an equation

3X+ 189=52 1

4Y+ 1 19=22

3X* 189=5

8Z/6=458

3X+77=59

4Y-6985=8 1

87X* 13=5

7Z/93=4 1

15X+863-65X=54

58Y*55=27489

z*(z-3)=4

The root of the equation x2 = is.

2. The general form of the equation (x+1) 2-2 (x-1) 2 = 6x-5 is.

3. One root of the quadratic equation x2+mx+3=0 about x is 1, so the value of m is.

4. Given that the quadratic trinomial x2+2mx+4-m2 is completely flat, then m=.

5. It is known that +(b- 1) 2 = 0. When k is, the equation kx2+ax+b=0 has two unequal real roots.

6. The equation mx2-2x+ 1 = 0 about x has only one real root, then m=.

7. Please write that one root is 1 and the other root satisfies-1

8. Equation X X2-(2m2+m-6) X-m= 0 If two mutually opposite numbers, then m=.

9. It is known that the two roots of the unary quadratic equation (A- 1) x2+x+A2- 1 = 0 are x 1, x2, x 1+x2=, then x 1, x2=.

The original timber stock of 10 timber yard is one cubic meter. As we all know, wood grows at a rate of 20% every year. When the amount of timber felled in winter is X cubic meters each year, the stock of timber is cubic meters after one year and B cubic meters after two years. Try to write the relationship between a, b and m: