Application problem refers to the problem of applying what you have learned to real life practice. In mathematics, application problems are divided into two categories: one is mathematical application. The other is practical application. I sorted out the types of math application problems in primary schools for reference!
First, the general application problem.
General application problems have no fixed structure, and there are no rules to follow in solving them. Look for clues to solve the problem by analyzing the quantitative relationship of the problem completely.
Point: Start with conditions? From the question
When analyzing the conditions, we should always pay attention to the problem of the topic.
When analyzing a problem, we should always pay attention to the known conditions of the topic.
Examples are as follows:
A workshop of a hardware factory wants to produce 1 100 parts, which has been produced for five days, with an average of 130 parts per day. If the average daily output is 150, how many days will it take to complete the rest?
Thinking analysis:
It is known that "it has been produced for 5 days, with an average daily production of 130 pieces", and the quantity that has been produced can be calculated.
Knowing "to produce 1 100 machine parts" and the number already produced, and knowing that "the average remaining output is 150 pieces per day", we can draw the conclusion that it will take several days to complete.
Second, the typical application problems
In the application of two-step or multi-step solution, some problems can be solved by specific steps and methods because of their special structure. This kind of application problem is usually called typical application problem.
(A) the average application problem
The law to solve general problems is:
Total quantity ÷ corresponding total number of copies = average value
Note: in this kind of application problems, to grasp the corresponding relationship, you can divide it into different sub-quantities according to the total amount, then find out their respective copies one by one according to the sub-quantities, and finally get the corresponding relationship.
Example 1 is as follows:
A rice mill, 4 hours in the morning 1360 kg, 3 hours in the afternoon 1096 kg. How many kilograms of rice is milled per hour on this day?
Thinking analysis:
How many kilograms of rice should be milled per hour on this day, and the following three problems need to be solved:
1. How many meters did you press this day? A day includes morning and afternoon.
2. How many hours did you work this day? (4 hours in the morning and 3 hours in the afternoon).
3. What is the total amount of this day? What's the total number of copies today? (In this way, the corresponding relationship is found and the problem is solved. )
(2) the problem of standardization
The title structure of standardization problem is:
The first part of the topic is known conditions, which is a set of related quantities;
The second half of the topic is a question and a set of related quantities, one of which is unknown.
The law of solving problems is to find the single quantity first, and then according to the problem, or how many times the single quantity is, or how much the single quantity is.
Examples are as follows:
Six tractors cultivated 300 acres of land in four hours. According to this calculation, how many acres of land can eight tractors cultivate in seven hours?
Thinking analysis:
First, the unit quantity, namely 1 tractor and 1 hour, is calculated, and then the mu of cultivated land of 8 tractors for 7 hours is calculated.
(3) the problem of the meeting
Refers to two moving objects moving in opposite directions from two places at different speeds.
The basic relationship when encountering problems is:
1, meeting time = distance (when two objects are moving) ÷ speed sum.
The example is as follows: the distance between the two places is 500 meters. Xiaohong and Xiaoming came from two places at the same time. Xiaohong walks 60 meters per minute and Xiaoming walks 65 meters per minute. How many minutes did you meet?
2. Distance (when two objects move) = sum of velocities × meeting time.
Examples are as follows: a bus and a truck set off from both parties at the same time, and met on the way 10 hours later. It is known that the average speed of trucks is 45 kilometers per hour, and the speed of passenger cars is 20% faster than that of trucks. How many kilometers is it between Party A and Party B?
3. Speed A = distance (when two objects move) ÷ meeting time-speed B.
The example is as follows: a truck and a bus set off from two places 648 kilometers apart at the same time and met after 4.5 hours. The speed of the passenger car is 80 kilometers per hour, and how many kilometers per hour is the truck?
The question of meeting can vary a lot.
For example, two objects move in opposite directions from two places, but they don't start at the same time;
Or one of the objects pauses in the middle;
Or after two moving objects meet, they continue to walk for a certain distance. These should be analyzed in combination with the specific situation.
In addition, the encounter problem can be generalized as an engineering problem: namely, work efficiency and × joint working time = total work.
Third, the application of scores and percentages
There are three basic application problems of fractions and percentages. Let's talk about the characteristics and problem-solving rules of each application problem first.
(1) What percentage of one number is another?
The structural feature of this kind of problem is that two quantities are known, and the problem is the percentage between these two quantities.
Finding the percentage of one number to another number is essentially the same as finding the multiple or fraction of one number to another number, except that the calculation result is expressed by percentage, so when finding the percentage of one number to another number, it should be calculated by division.
The general rule of solving problems is: let A and B be two numbers, and when A is a few percent of B, the formula is A ÷ B. When solving this kind of application problems, the key is to understand the meaning of the problem.
Examples are as follows:
Aunt Li, a professional pig farmer, raised 350 pigs last year. This year, she raised 60 more pigs than last year. What is the proportion of pigs she raises this year?
Thinking analysis:
The meaning of the question is: there are more pigs raised this year than last year, which is a few percent of last year. Therefore, we should use the number of pigs raised this year more than last year to calculate the number of pigs last year, and then convert the result into a percentage.
(2) Find the fraction or percentage of a number.
Find the fraction or percentage of a number and calculate it by multiplication.
When answering such questions, we should start with the known conditions that reflect the multiple relationship between two numbers, first determine the unit "1", and then determine the score or percentage of the unit "1".
(3) Find the fraction or percentage of a given number.
This kind of application problem can be solved by equation or arithmetic.
When solving by arithmetic, we should use division to calculate.
When solving this kind of application problems, we should also analyze the known conditions that reflect the multiple relationship between two numbers:
Determine the unit "1" first, and then determine the fraction or percentage of the unit "1".
You can draw some slightly more difficult application questions to help analyze the quantitative relationship.
(4) Engineering problems
The engineering problem is to study work efficiency, working time and total work.
The characteristics of this kind of topic are:
The total amount of work is regarded as "1" without giving the actual amount, and the cooperation time is mostly expressed by work efficiency.
Examples are as follows:
For a project, Team A needs 8 days to build and Team B needs 12 days to build. After four days of joint repair by the two teams, how many days will it take for the remaining tasks to be repaired by team B alone?
Thinking analysis:
If the workload of a project is "1", the work efficiency of Party A is 1/8, and the work efficiency of Party B is112.
It is known that the two teams have worked together for four days, so we can calculate the workload of joint repair and then calculate the remaining workload.
Divide the remaining workload by B's work efficiency, that is, it will take several days to complete.
Fourth, the application of ratio and proportion
Ratio and proportional application problems are important parts of primary school mathematics application problems. In primary schools, the application of ratio includes: the application of ratio, the application of proportional distribution, and the application of positive and negative ratio.
(A) the problem of scale application
This kind of application problem is to study the relationship between distance, actual distance and scale on the map.
When solving this kind of application problems, the most important thing is to understand the meaning of scale, that is:
Map distance ÷ actual distance = proportion
According to this relationship, if any two quantities between the three are known, the third unknown can be found.
Examples are as follows:
On the map with the scale of 1: 300000, the distance from city A to city B is 8 cm. What is the actual distance from city A to city B?
Thinking analysis:
Write the scale in the form of fraction, set the actual distance as x, and substitute it into the relationship of scale to solve it. The name of the unknown unit of measurement should be the same as that of the known unit of measurement.
(2) Proportional distribution of application questions
This kind of application problem is characterized by dividing a quantity into two parts or several parts according to a certain proportion, and finding the number of each part.
This is the only problem that students come into contact with in primary school.
The law to solve this kind of application problem is:
First calculate the sum of the shares of each part, and then determine the scores of each part in the total. Finally, according to the fraction of a number, the number of each part is calculated by multiplication.
Proportional distribution can also be solved by normalization.
Examples are as follows:
A pesticide solution is prepared by adding water to medicinal powder, and the weight ratio of medicinal powder to water is1:100. How many kilograms of powder does 2500 kilograms of water need? How many kilograms of water does 5.5 kilograms of powder need?
Thinking analysis:
Knowing the number of parts of medicine and water, we can know the sum of the total parts of medicine and water, and we can also know how much medicine and water each account for the total parts. Knowing the scores, we can also calculate their respective relative quantities.
(3) Positive and negative proportion application questions
To solve this kind of application problem, the key is to judge whether the two related quantities in the problem are directly proportional or inversely proportional.
If the letters X and Y are used to represent two related quantities, and K is used to represent the proportion (certain), when two opposite related quantities are in direct proportion, the following formula is used:
Kx = y (sure).
If two related quantities are inversely proportional, they can be expressed by the following formula:
×y=K (certain).
Examples are as follows:
June 1st Toy Factory will produce 2080 sets of children's toys. 960 units were produced in the first six days. According to this calculation, how many days will it take to complete all the tasks?
Thinking analysis:
Because the total amount of work ÷ working time = working efficiency, it is known that the working efficiency is certain, so the total amount of work is directly proportional to working time.
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