In daily life, such problems often occur: a certain number of items are distributed to a certain number of people, and there are more people, and the items are not enough; There are fewer people, so there are more things. Profit and loss problem determines the total number of items and the number of people participating in the distribution when the profit and loss are known.

The key to profit and loss problem's solution is to find out the relationship between profit and loss.

The quantitative relationship of profit and loss problem is:

(1) (profit+loss) ÷ twice distribution difference = number of copies.

(big profit-small profit) ÷ twice distribution difference = number of copies

(big loss-small loss) ÷ twice distribution difference = number of copies

(2) Number of shares per time × number of shares+surplus = total number of shares

Number of shares per time × number of shares-loss = total number

Example 1: A tree planting group planted trees. If each person plants 5 trees, there will be 14 trees left; If there are seven trees for each race, there will be four fewer trees. How many people are there in this tree planting group? How many trees are there?

As can be seen from the meaning of the question, the number of people planting trees is the same as that of trees. Comparing the two distribution schemes, the difference is 14+4 = 18, that is, the result of scheme 1 is more than that of scheme 2 18. This is because the difference between the two distribution schemes is 7-5 = 2. Therefore, the tree planting group has 18÷2=9 people, and one * * has 5× 9+ 14 = 59 trees.

Exercise 1

1, the kindergarten gives some building blocks to the children. If each child is divided into two, there are still 20 left. If everyone is divided into three parts, the difference is 40. How many children are there in kindergarten? How many building blocks does a * * have?

2. A school arranges dormitories. If there are 6 people in each room, 16 people have no beds. If there are 8 people in each room, there will be an extra bed of 10. How many dormitories are there? How many students are there?

3. One class goes boating. They calculated that if one more boat was added, each boat could only hold six people. If one boat is reduced, it happens that each boat will take nine people. Q: How many students are there in this class?

Exodus 2: The school awarded a batch of pencils to three outstanding students. If everyone wins 9, they will lose 45; If everyone wins 7, there will be 7 less. How many students are there in Sanhao? How many pencils are there?

Analysis and answer: This is a lose-lose problem. From the meaning of the question, we can know that the number of students and pencils in Sande is unchanged. Comparing the two distribution schemes, the difference is 45-7 = 38. This is because the difference between the two distribution schemes is 9-7 = 2 pencils. So there are 38÷2= 19 students and 9× 19-45 = 126 pencils.

Exercise 2

1, put Chinese rose flowers in some vases. If 8 flowers are inserted in each bottle, 15 flowers are missing; If there are 6 flowers in each bottle, you will be short of 1 flower. Find out the number of vases and the number of Chinese roses.

2. Miss Wang distributes drawing paper to the students in the art interest group. If each person sends 5 cards, 32 cards will be less; If each person is dealt three cards, there will be two less cards. How many students are there in the art interest group? How many pictures and pieces of paper does Miss Wang have?

The teacher distributed some exercise books to the students in the class. If each person sends 10 copies, two students will not be assigned; If each person sends eight copies, it will be over. How many students are there? How many exercise books?

Ex. 3: Some young pioneers went to the mountain to plant a batch of trees. If everyone plants 16 trees, there are still 24 trees that are not planted; If each race 19 trees, there are still 6 trees without seeds. How many young pioneers are there? How many trees are there?

Analysis and answer: This is a mutually beneficial question. According to the meaning of the question, the number of young pioneers and the number of trees are unchanged. Comparing the two distribution schemes, the difference is 24-6 = 18 trees, which is because the difference between the two distribution schemes is 19- 16 = 3 trees. So there are 18÷3=6 young pioneers, 16× 6+24 = 120 trees.

Exercise 3

1, Xiao Hu listened to the bullets being fired in the enemy window: one person said that each person would give 45 rounds and 260 rounds more; Another person said that if everyone carries 50 rounds, they can get an extra 200 rounds. How many enemies are there? How many bullets?

Teacher Yang gave a stack of exercise books to the students in the first group. If each person is divided into 7 books, there are 7 more books; If each person is divided into eight parts, it is just finished. Please count, how many students are there in the first group? How many exercise books are there in this stack?

3. Teacher Cui gave some colored pens to the students in the art interest group. If everyone is divided into five parts, there will be more12; If everyone is divided into eight, there are still three. Excuse me, how many pens are divided for each person, just enough to divide the markers?

Example 4: The school allocates dormitories to a group of freshmen. If there are 12 people in each room, there is no room for 34 people; If each room is occupied by 14 people, four rooms will be vacant. How many dormitories are there? How many boarding students are there?

Analysis and solution: if each room has 14 people, then four rooms are vacant; if each room has 14 people, then there will be less 14×4 = 56 people. Comparing the two distribution schemes, the difference is 34+56 = 90 people, while the difference in each room is 14- 12 = 2 people. The number of rooms is 90÷2=45 and the number of students is 12× 45+34 = 574.

Exercise 4

1, a school has several student dormitories. If there are 6 people in each dormitory, there will be 34 more. If there are 7 people living in each dormitory, there will be 4 more dormitories. How many rooms are there in the dormitory? How many boarding students are there?

2. Yucai pupils go for a spring outing by car. If there are 65 people in each car, 15 people can't take the bus; If there are five more people in each car, there will be one more car. How many cars are there? How many students are there?

3. The school allocates student dormitories. If there are 6 people living in each room, there will be 2 fewer dormitories; If there are 9 people in each room, there are 2 rooms available. How many dormitories are there? How many boarding students are there?

Example 5: Young Pioneers go to plant trees. If everyone digs five tree pits, there are still three pits that no one digs. If two of them each dig four holes and the rest each dig six tree pits, they can dig all the tree pits. How many tree pits did the Young Pioneers dig?

Analysis and solution: If everyone digs six tree pits, there will be (6-4) × 2 = 4 tree pits less, and the difference between the two times is 4+3 = 7 tree pits. This is because the difference between the two distribution schemes is 6-5 = 1 tree pit. So there are 7 young pioneers1= 7 people, and one * * * digs 5× 7+3 = 38 tree pits.

Exercise 5

1, the kindergarten teacher gives the children apples. If each child is divided into two, there are still 30; If 12 children are divided into three parts each, and the rest are divided into four parts, it will be just over. How many apples are there?

In a general cleaning, the teacher assigned several people to clean the glass. If two of them wipe 4 pieces each and the others wipe 5 pieces each, there are 22 pieces left; If you wipe 7 yuan each, it's over. Find out the number of people cleaning glass and the number of glass fragments.

Xiaohong bought a basket of oranges and distributed them to the whole family. If two of them are divided into four and the rest are divided into two, there will be four more; If one of them is divided into six parts and the others are divided into four parts, 12 is missing. How many oranges did Xiaohong buy? How many people are there in Xiaohong's family?