[Evaluation objective]

1, answer simple application questions skillfully, and tell the quantitative relationship according to the meaning of the questions. Clear arithmetic.

2. We can use step-by-step formulas and comprehensive formulas to solve general application problems, understand the practical significance of each step formula, and analyze the problem-solving ideas of application problems with comprehensive methods and analytical methods.

[knowledge review]

1, simple application problem

Simple application problems, which only contain a quantitative relationship, can be solved in one step. But it is the basis to solve all application problems.

(1) Find the sum of two numbers.

Addition is the operation of combining two numbers into one number. There are two situations: one is to know the number of two parts and find the total; The other is to know what one number is and how much the other number is more than it, and then find another number.

(2) Find out the difference between two numbers

Subtraction is an operation to find another addend by knowing the sum of two numbers and one of them. It is the inverse of addition. There are three situations: one is to know the total number of two numbers and what one of them is, and find the other number; Second, we know what two numbers are and find out how much one is more (or less) than the other; Third, knowing how much one number and another number are smaller than it, we can calculate another number (smaller number) by subtraction.

(3) Find the product of two numbers

Multiplication is a simple operation, which can find the sum of several identical addends. One is to know the number of copies and the number of copies and find the total; The other is how many times a number is.

(4) Find the quotient of two numbers

Division is an operation to find another factor by knowing the product of two factors and one of them. One is to divide a number into several parts on average and work out how much one part is; The other is to find how many other numbers a number contains. The former is called "equal division" and the latter is called "inclusive division".

The quantitative relationship of multiplication and division application problems can be summarized as follows:

Number of copies × number of copies = total number of copies

Total copies/copies = each score

Total copies/number of copies = number of copies

2. General compound application questions

A compound application problem is an application problem that contains two or more basic quantitative relations, that is, it is solved by two or more operations. In fact, compound application problems are composed of several simple application problems, so solving compound application problems is based on simple application problems.

The key to solve this kind of application problem is to decompose the compound application problem into several simple application problems on the basis of analyzing the quantitative relationship. The steps to solve the problem are as follows:

(1) Find out the meaning of the problem and the problem with known conditions and requirements;

(2) Analyze the quantitative relationship in the questions to find out the intermediate questions, and determine what to calculate first, then what to calculate, and finally what to calculate;

(3) List the calculation formula;

(4) Test and write the answer.

[test analysis]

[Example 1] In the "hand in hand" activity in our school, only 6 1 students from Class 5 (1) donated money to Qiongjiang Primary School11.52 yuan last year. What is the average donation per person?

Analysis: Divide11.52 yuan into 6 1 portions on average, and find out how much each portion costs. It is found that 1 1 1.52 divided by 6 1 cannot be divided completely, because the smallest unit of money is "minute", so two decimal places should be reserved.

111.52÷ 61≈1.83 (yuan)

A: The average donation per person is about 1.83 yuan.

Red Star Bicycle Factory originally planned to produce 2,000 bicycles in 30 days, and 60 bicycles every day in the first 20 days. To finish the task on time, how many bicycles will be produced on average every day in the last 10 day?

Analysis: According to "60 vehicles were produced every day in the first 20 days", we can find out how many vehicles have been produced, and then according to "2,000 vehicles are planned to be produced", and finally we can find out how many vehicles will be produced every day in the next 10 day.

Column synthesis formula calculation:

(2000-60×20)÷ 10

=(2000- 1200)÷ 10

=800÷ 10

=80 (vehicles)

A: After 10, 80 vehicles will be produced on average every day.

[Example 3] A factory stores 160 tons of coal, originally

Burning 1.5 tons per day. After burning for 20 days, the rest only burns 1.3 tons per day due to coal saving measures. How many days can the remaining coal burn?

Analysis: this is a general compound application problem, and there is no certain law to solve it. Usually it is divided into several simple application problems, and the indirect problems are solved separately. Generally, analytical method, comprehensive method or analytical comprehensive method are used for analysis. Now, there are two methods for analysis, as follows:

(1) analysis method: it is to start with the problem and gradually analyze the known conditions in the problem.

(2) synthesis method: that is, from known conditions to unknown conditions step by step, until the solution.

( 160- 1.5×20)÷ 1.3

=( 160-30)÷ 1.3

= 130÷ 1.3

= 100 (days)

A: The remaining coal can be burned for 10 days.

Exercise 1

1. The installation team needs to install 4 140 seats. 12 days has been installed, with an average of 180 seats installed every day. The rest should be installed within 9 days. How many seats should be installed on average every day to complete the task on schedule?

2. The brick factory has 5 1 ton of coal, which has been burned for 15 days, with an average of 1.4 tons per day. If 1.2 tons is burned every day, how many days can the remaining coal be burned?

3. To build the canal, it is planned to build12m every day, and it will be completed in 25 days. In fact, it took only 20 days to finish the task. How many meters are built more than planned every day?

4. Two cars of Party A and Party B leave from Party A at the same time, heading in opposite directions, and meet in 4 hours. After the encounter, car A continued to drive for 3 hours to reach the second place, and car B was driving at a speed of 24 kilometers per hour. How many kilometers are there between these two places?

A factory wants to produce 3000 machines and starts to produce 40 machines every day. After 15 days, the equipment was improved and the work efficiency was doubled. How many days will it take to complete these tasks?

6. A garment factory originally planned to produce 1200 garments in 20 days, but actually produced 960 garments in 12 days. At this rate, how many days can the task be completed ahead of schedule?

7. A reservoir has 50 cubic meters of water. The first water pipe produces 4.5 cubic meters of water per minute, and the second outlet pipe produces 3.5 cubic meters more water per minute than the first one. When the two pipes are closed, how many minutes can the water in the pool be drained?

The toy factory originally planned to produce 900 toys in 45 days, but it was actually completed in 30 days. How many more toys are actually produced every day than originally planned?

9. The garment factory sent 300 meters of cloth, half of which was used to make 30 sets of adult clothes and the other half was used to make 50 sets of children's clothes. How many meters are there in each set of adult clothes than children's cloth?

10, three big boats and two small boats can seat 26 people, and three big boats and five small boats can seat 38 people. How many people can sit in each big boat and each small boat?

1 1, the school bought 6 tables and 8 chairs, and * * * paid 477.6 yuan. Each table is 34.8 yuan more expensive than each chair. How much is a table and a chair?

12, master Zhang produces in 3 days 184 pieces. Compared with the planned daily production tasks, the first day 14, the second day 16, and the third day 2. How many parts do you plan to produce every day?

13. The master processed 80 parts, less than the apprentice 10. How many parts did the apprentice process?

14, Team A and Team B simultaneously excavated a 770-meter-long tunnel. Team A digs10m from one end every day; Team B starts from the other end and chisels 2 meters more than Team A every day. How far are the two teams from the midpoint to meet?

15. A worker plans to process 960 parts in 48 hours. After improving the technology, the plan was completed in half the original time, and 72 more were made. How much more is produced per hour than planned after improving the technology?

Second, the typical application problems

[Evaluation objective]

1, master the basic structural characteristics and analysis methods of average application questions, normalized application questions and travel problem application questions, and be able to answer these application questions skillfully.

2. Learn to use line graphs to analyze the application of travel problems.

[knowledge review]

1, average application problem

A typical application problem is an application problem with unique structural characteristics and unique solving rules.

The basic quantitative relationship of averaging is:

Total quantity/total number of copies = average value

When solving this kind of application problems, we should first try to find the total amount, then find the "total number of copies" corresponding to the "total amount", and then find the average value.

2. Application of normalization problem.

The key to solve the regularization problem is to find out a unit quantity (that is, the workload per unit time, the distance traveled per unit time, the output per unit area, the unit price of goods, etc.). ) according to the known conditions, then calculate the required quantity.

3. Application of trip problem.

When applying the trip problem, we should first understand the words such as "relative", "relative", "opposite", "encounter", "simultaneous" and "same direction", and then understand the structural characteristics of the trip problem.

Direction of motion: same direction or opposite direction?

Departure location: same place or two places?

Departure time: At the same time or separately?

Speed: the speed of one object or the speed of two objects.

Sports results: meet, leave, or leave in the opposite direction after meeting.

Finally, we must master the law of solving each application problem. The law to solve the problem is:

(1) Moving towards each other-it means that two objects have different starting points and move in opposite directions. The closer they get, the closer they get. It can also be divided into two situations: meeting and different.

The basic formula is as follows:

Meeting time = meeting distance/speed and

Meeting distance = speed × meeting time

Speed Sum = Meeting Distance/Meeting Time

(2) Co-directional movement-refers to the movement of two moving objects in the same direction, but the starting point can be the same or different, so it can be divided into two situations: the same direction in the same place and the same direction in different places.

① Same place and same direction: characterized by the same starting point and the same moving direction. Because of the speed, it is getting farther and farther. The formula is:

Distance = speed difference × time

② Same direction in different places: characterized by different starting points and the same movement direction. If the slow one comes first, the fast one can catch up. This is called catching up. The formula is:

Catch-up time = catch-up distance ÷ speed difference

Catch-up distance = speed difference× catch-up time

Speed difference = catching distance/catching time

If the fast is ahead and the slow is behind, the two will go further and further, and they can't catch up. Formula: distance = separation distance+speed difference × time.

(3) Backward movement-refers to two objects moving in opposite directions, but the starting point can be the same or different. The formula is:

Distance = speed and x time

[test analysis]

[Example 1]

Below is the scale of the line segment, and the line segment of 1 cm represents the actual distance of 40 kilometers. On this map, it is measured that the railway line between A and B is 20.4 cm long. A bus and a truck leave from A and B at the same time. The bus travels 80 kilometers per hour and the truck travels 70 kilometers per hour. How many hours do two cars meet?

0 40 80120km

Analysis: This is an encounter problem involving scale knowledge, which is not directly talked about by the railway chiefs of Party A and Party B, but should be solved by using the relevant knowledge of scale. According to the significance of line segment scale, 1 cm means 40 km, and the line segment of 20.4 cm should be (40×20.4) km, and then it can be obtained by the relationship of "time = distance ÷ speed sum".

(1) How many kilometers is the railway?

40× 20.4 = 8 16 km

(2) A few hours later, two cars met?

8 16÷(80+70)

=8 16÷ 150

= 5.44 hours

A: After 5.44 hours, the two cars met.

[Example 2]

A workshop processed 1620 parts before June and 120 parts every day after June. How many parts are processed every day in June?

Analysis: Solving the average application problem can be directly analyzed from the relationship of "total amount ÷ total number of copies = average". According to the question required by the topic, "total number of copies" should be the total number of days in June; "Total Quantity" refers to the total number of parts processed in June, but it is divided into two parts. The number of treatments before 16 days and after 14 days. It should be noted that the processing quantity for the next 14 days is not directly given, but is obtained by two conditions: 14 days and "average daily processing 120 pieces". Many students often neglect to calculate the number of parts processed in 14 days, which leads to wrong answers.

Column synthesis formula calculation:

( 1620+ 120× 14)÷( 16+ 14)

=3300÷30

= 1 10 (piece)

A: In June, 1 10 parts were processed on average every day.

Exercise 2

1. A shoe factory produced 3600 pairs of shoes in January, 4000 pairs in February and 5000 pairs in March. How many pairs of shoes did it produce on average every month in the first quarter?

2. A factory produced 18 units three days ago and 20 units in the next five days. How many machines are produced on average every day?

3. A road team built 240m in the first three days and 86m in the last three days. How many meters does this road repair team repair on average every day?

4. Rebecca scored 94 points in Chinese and mathematics, 98 points in music, 90 points in nature, 85 points in physical education and 9 1 point in fine arts. What was her average score in the last exam?

5. A factory has three workshops. There are 20 people in the first workshop, and each person produces 450 parts on average. There are 10 people in the second workshop, and each person produces 5 10 parts on average; There are 30 people in the third workshop, and each person produces 600 parts on average. How many parts are produced by each of these three workshops?

6. During the "Civilization Activity Month", students did good things for the society, and Class One in Grade Six did 32 less than Class Two. It is known that there are 50 people in class one, each of whom makes 4 pieces on average, and there are 46 people in class two. How many good deeds does each class do on average?

7. Two cars leave from two cities at the same time. It takes four hours for a car to drive from A to B, and six hours for another car to drive from B to A.. How many hours did two cars meet on the road?

8. Three looms can weave 265,438+00 meters in five hours. According to this calculation, how many meters can be woven by adding six looms at the same time?

9. The distance between City A and City B is 565 kilometers, and a local train from City A to City B runs 55 kilometers per hour; Two hours later, an express train departed from B city and headed for A city, with a speed of 75 kilometers per hour. A few hours after the express train left, the two trains met.

10, the school carried out water-saving activities, saving 8.4 tons of water every day in the first four days of the week, and saving water in the last three days 14.7 tons. How many tons of water are saved on average every day this week?

1 1, the sum of a and b is 54, and the average value of c and d is 19. What is the average of these four numbers?

12. Last semester, Li Jun got an average score of 93 in Chinese, mathematics and science, including 100 in mathematics and 89 in natural science. What's his Chinese score?

13, Train A and Train B travel relatively from two places. Car A travels 75 kilometers per hour and car B travels 69 kilometers per hour. Car A takes 2 hours to leave, and the two cars meet again after 3 hours. How long is the railway between these two places?

14, border guards patrol, * * * line 18km. Hiking in the mountains for the first 3 hours, with an average of 3.5 kilometers per hour; Later, I walked on the flat ground for 1.5 hours. How many kilometers did I walk every hour on average?

15, there is a project, 1 1 day 7 people completed. How many people do you need to add if you want to finish it four days in advance?

16, 8 people in the road repair team, 2 160m in 5 days. According to this calculation, if 10 people are added, how many days will it take to repair 4860m?

17, a washing machine factory planned to produce 2400 washing machines last year, and completed the task in 10 month. At this rate, how many units did the actual output increase over the planned output last year?

18. In a 35-meter swimming pool, A and B start from the starting point at the speed of 2 meters per second and 1.5 meters per second respectively. How many seconds later, when A swam to the finish line and returned, she met B?

19, the train from a to b, with a speed of 75 kilometers per hour, is expected to arrive at 1 1 hour. The train broke down halfway and was repaired in 30 seconds. If we want to get to B within the scheduled time, how many meters per hour must we drive for the rest of the journey?

20. It takes 6 hours by bike and 2 hours by car from Party A and Party B.. Cars travel 30 kilometers more than bicycles every hour. How many kilometers does this bike travel per hour?

2 1. This furniture factory produced 2756 pieces of furniture in the first four days of last week, and 920 pieces every day in the last three days. How many pieces of furniture did it produce every day last week?

22. The distance between A and B is 465 kilometers. Two cars, A and B, set out from two cities at the same time, face to face, and three hours later, the two cars met. Car A is 80 kilometers per hour, and car B is how many kilometers per hour?

Third, the general application of scores and percentages

[Evaluation objective]

1, understand and master the quantitative relationship and problem-solving methods of fractional addition and subtraction application problems.

2. Focus on understanding and mastering the quantitative relationship and solving methods of three basic types of application problems: fraction and percentage.

3. Be able to analyze complex sums, fractions, percentages and application questions, and flexibly use what you have learned to answer them.

[knowledge review]

1, score addition and subtraction, method application problem

The meaning of fractional addition and subtraction is the same as that of integer addition and subtraction, so the practical problems solved by fractional addition and subtraction are basically the same as those solved by integer addition and subtraction.

2. Fraction and percentage multiplication and division application problems

(1) Application problem of finding fractions and percentages (i.e. finding fractions or percentages of one number to another).

The application of grading rate and percentage is closely related to the actual production, and its solution method has certain rules, so how to determine the unit "1" is the key to solve this kind of problem. Because fractions and percentages are obtained by dividing two similar quantities, whoever is the divisor in division is the standard quantity (the quantity in "1").

For example: A is B, and B is the quantity of "1"; B is more than a 15%, and a is a real measurement, which is "1"; This year is a few percentage points lower than last year. Last year was the amount comparison, and last year was the unit "1". Because this unit "1" is generated by fractions and percentages, you should look for the unit "1" in the sentence of finding fractions and percentages in fractions and percentages or questions.

(2) Find the fraction or percentage of a number. This kind of application problem is characterized by knowing the quantity and score of the unit "1" and finding the actual quantity corresponding to the score. The key to solving the problem is to accurately judge the quantity of the unit "1", find out the score corresponding to the problem, and then correctly formulate it according to the meaning of multiplying a number by a score. The law to solve the problem is:

Number of unit "1" × fraction (percentage) = number of parts corresponding to fraction (percentage)

(3) Know what the score of a number is and find this number. The characteristic of this kind of application problem is to know an actual quantity and its corresponding score, and to find the quantity in units of "1". Solve by division. Solve this kind of application problems by arithmetic or equation. When solving problems by arithmetic, we must find the corresponding relationship between quantity and fraction (percentage), and use this relationship: quantity ÷ corresponding fraction (percentage) = quantity in unit "1"; When solving problems with equations, it is generally necessary to set the quantity of unit "1" as the unknown χ. We can solve problems by multiplication, using the relationship: the quantity of unit "1" × the fraction (percentage) = the partial quantity corresponding to the fraction (percentage). You can also answer according to the equivalence relation in the question.

3. Methods and skills to solve the application problems of fractional and percentage multiplication and division.

The above three types of application problems reflect the same set of quantitative relations, namely:

① Number of unit "1" × fraction (percentage) = number of parts corresponding to fraction (percentage)

② Quantity ÷ corresponding score (percentage) = quantity in "1";

(3) The quantity corresponding to the score ÷ "1"equals the score.

When solving these three kinds of application problems, you must have the following basic skills:

(1) Accurately determine the unit "1".

For example, "the number of boys in the class" is a sentence with scores. From this sentence, we can find that "class size" is the unit "1".

Another example is: "A road has been repaired", which means that the repaired road accounts for the length of this road, so the length of this road is "1".

(2) Grasp the relationship between the three quantities.

If the quantity of the unit "1" is known, find a fraction of the unit "1" and calculate by multiplication;

The quantity of unit "1" is unknown, and the fraction of unit "1" and the partial quantity corresponding to this fraction are known, so it is calculated by division;

Find the fraction of a quantity in the unit "1" and then divide this quantity by the unit "1".

Of these three types of application problems, the latter two are the most easily confused, so we should focus on whether the quantity of the unit "1" is "known" or "unknown", so as to determine whether it is a multiplication or division problem.

For the first question type, you can generally "sit in the right place" from the question, that is, find a score of B and divide A by B, where the unit "1" is divided by.

(3) Determine the corresponding relationship.

The first two categories are mainly emphasized here.

The corresponding relationship of multiplication questions is as follows:

Number of units "1" × score = number of parts corresponding to the score.

That is, multiply by whose score, you get whose weight. If you multiply the scores of boys, you get the number of boys; Multiply the scores of girls to get the number of girls; Multiply the difference between the number of boys and girls, and you get the difference between the number of boys and girls.

From this, we can think that whoever gets the weight is the score.

The relationship between division application problems is as follows:

Partial Quantity Score = Quantity in "1"

consistent

In other words, if we know whose quantity belongs to whom, we must divide it by whose score. For example, if the known quantity is the number of boys, it must be divided by the boys' scores; The known quantity is the number of girls, so it must be divided by the score of girls; If the known quantity is the difference between boys and girls, it must be divided by the difference between boys and girls.

Knowing the relationship between these three quantities, we can determine the unit "1" and find the corresponding relationship, then the application problem of fractions and percentages can be solved. Even if the required conditions are not given directly, but indirectly, it is easy to list the formulas correctly.

For questions that do not directly give the required score, you should have certain associative ability. Think from one place to another, so that you can quickly find the score you need.

For example, when you see the condition of "men are superior to women", you should immediately associate it with women being superior to women, that is, (1-).

If you see the condition of "repair on the first day, repair on the second day", you should immediately associate it with: two days and one * * * repair+; The difference between these two days is-; What is not fixed is 1-.

If you see the condition that there are more boys than girls, you should immediately think of the situation that boys account for girls 1+.