First, method guidance.
1. Standardization problem
According to the known conditions, when solving a problem, we must first find out how much a copy is (normalized), such as workload per unit time, output per unit area, unit price of goods, distance per unit time, etc. And then find out the application problem of this problem is called normalization problem. Normalization can be divided into positive normalization and anti-normalization
(1) is normalized to 1
Total Quantity ÷ Quantity = Single Quantity
Single Quantity × New Quantity = New Total Quantity
Comprehensive formula: total quantity ÷ quantity × new quantity = new total quantity.
(2) denormalization
Total Quantity ÷ Quantity = Single Quantity
New Total ÷ Single Quantity = New Quantity
Comprehensive formula: new total ÷ (total ÷ quantity) = new quantity.
2. The problem of induction
Inductive problem refers to the application problem that the total amount should be calculated first (called "total amount") and then the required amount should be calculated. The problem of induction implies that the "total" is unchanged, that is, the product is unchanged, so this kind of problem can also be solved by inverse proportion knowledge.
The key to solving the problem of induction is to find "total" first, which is always equal.
The problem of induction is also composed of two groups of similar quantitative relations.
Second, typical cases
Example 1: It costs 375 yuan to buy five identical basketballs at school. According to this calculation, how much does it cost to buy a basketball like 13?
Analysis: Through reading the questions, we know that this is an application problem that is being corrected one by one. We can calculate the unit price of basketball first, and then calculate the total price of 13 basketball.
Solution:
Step by step:
375 ÷ 5 = 75 (yuan)
75× 13 = 975 (yuan)
Column synthesis formula:
375÷5× 13
=75× 13
= 975 (yuan)
A: It costs 975 yuan to buy a basketball like 13.
Example 2: Uncle Li installed a batch of computers and installed 12 computers every day for 30 days. If 15 is installed every day, how many days can it be completed?
Analysis: according to the meaning of the question, the total number of computers in this batch is certain, and it will take several days to complete. It is necessary to know how many computers are installed in this batch of units and how many computers are installed every day. Now that we know that 15 computers are installed every day, we must first find out how many computers there are in this batch.
Solution:
How many computers are there in this batch?
12× 30 = 360 (unit)
How many days will it take to complete?
360 ÷ 15 = 24 (days)
Comprehensive formula:
12×30÷ 15
=360÷ 15
= 24 days
A: It can be completed in 24 days.
Example 3: Four cows ate 240 kilograms of grass in five days. According to this calculation, how many Jin of grass does 18 cattle eat in 9 days?
Analysis: This is a quadratic normalized application problem. First, find out the unit quantity, that is, the average quality of grass eaten by each cow every day, and then find out the quality of grass eaten by 18 cows for 9 days.
Solution:
240÷4÷5× 18×9
= 12× 18×9
= 1944 (kg)
Answer: 18 cows need to eat 1944 kg of grass for 9 days.
Example 4: The farm tool factory produces a batch of small clothes and tools. It was originally planned to produce 0/20 pieces per day, and the task could be completed in 28 days. Actually, 20 pieces are produced every day, so how many days can we finish the task ahead of schedule?
Analysis: It is required to complete the task a few days in advance, how many days have actually been produced first, and then you must know how many pieces these small farmers have and how many pieces they actually produce every day. According to the original plan, 120 pieces can be produced every day, and the task can be completed in 28 days. The total number of these small farm tools can be calculated, and then the actual daily output can be calculated, so the problem is solved.
Solution:
How many small farm tools are there in this batch?
120× 28 = 3360 (pieces)
How many pieces are actually produced every day?
120+20 = 140 (pieces)
How many days did it actually produce?
3360 ÷ 140 = 24 (days)
Finish the task a few days in advance.
28-24 = 4 days
Comprehensive formula:
28- 120×28÷( 120+20)
=28- 120×28÷ 140
=28-24
= 4 (days)
You can finish the task four days in advance.
Third, actual combat drills.
Question 1: two cars 1 200kg gasoline1month, how many kilograms of gasoline do five cars use for 8 months * * *? There are 36,000 kilograms of gasoline now. How many cars can a quarter use?
Question 2: It takes 8 people 10 day to build the 840m highway. At this rate, how many days will it take 20 people to build a 4200-meter highway?
Question 3: A conference room is paved with tiles and square bricks, with a side length of 4 mm, 400 yuan. How many bricks do you need if you replace it with a square brick with a side length of 8 decimeters?
Question 4: A job was originally planned to be completed in 15 days, and 20 people worked 8 hours a day. Later, five people were added and the working hours were reduced by two hours a day. How many days can you actually finish the work?
Question 5: The canteen uses 50 kilograms of rice every day, and the stored grain can last 18 days. If you use 5 kilograms less every day, how many days can you use the stored food?
Question 6: A project is scheduled to be completed within 30 days. Actually, it took 1 person 18 days to complete the project12. How many more people do you need if you want to finish on time?