First, the travel problem.

For example, two vans depart from two cities, A and B, which are 900 kilometers apart, and head for each other. It is known that it takes 10 hour for a bus to travel from city A to city B, and 15 hour for a truck to travel from city B to city A.. How many hours after the two cars left?

2. A car and a motorcycle set out from two cities, A and B, which are 240 kilometers apart, and walked in opposite directions. It is known that it takes 3 hours for a car to travel from city A to city B, and 6 hours for a motorcycle to travel from city B to city A.. How many hours after the two cars left?

3. The van departs from Party A and Party B, which are 480 kilometers apart, and heads for each other, and meets in four hours. It is known that it takes six hours for a bus to complete the journey, and how many hours does it take for a truck to complete the journey?

4. The vans depart from the cities of Party A and Party B, which are 800 kilometers apart, and go in opposite directions. The bus travels 80 kilometers per hour and the truck travels 60 kilometers per hour. How many kilometers are the two cars apart after 4 hours?

For example, students from Team A and Team B start from two places 18km apart and go in opposite directions. A classmate keeps cycling back and forth between the two teams at the speed of 15 km per hour. Team A travels 5 kilometers per hour, while Team B travels 4 kilometers per hour. How many kilometers did the cyclists walk when the two teams met?

1. Xin Wang and Lvliang walked in opposite directions from two places 2000 meters apart. Xin Wang walks 1 10 meters per minute, and Lvliang walks 90 meters per minute. If a dog walks in the direction of Xin Wang at the same time, it walks 500 meters per minute. After meeting Lvliang, he immediately ran back to Xin Wang, and then ran to Lvliang when he met Wang Xin. This was repeated until Xin Wang and Lvliang met. How many meters did the dog walk?

2. The distance between A and B is 400 kilometers, and two cars, A and B, start from both places at the same time. Car A travels 38 kilometers per hour, while car B travels 42 kilometers per hour. A swallow set off with car A at a speed of 50 kilometers per hour and flew to car B. When it met car B, it turned back and flew to car A. How many kilometers did the swallow fly before the two cars met?

At the same time, the two motorcades of Party A and Party B travel in opposite directions from two places 330 kilometers apart. Team A travels 60 kilometers per hour, Team B travels 50 kilometers per hour, and a motorcyclist travels 80 kilometers per hour. They communicated back and forth between the two motorcades and asked how many kilometers the motorcycles had traveled when the two motorcyclists met.

Example 3. Car A travels 6 kilometers per hour and car B travels 5 kilometers per hour. The two cars go back and forth at the same time between two places 10 km apart. A few hours later, the two cars were 65 kilometers apart.

1. A walks 7 kilometers per hour and B walks 5 kilometers per hour. They walked back and forth at the same time in two places separated by 18 km. How many hours later, they are 54 kilometers apart?

2. Party A travels 9 kilometers per hour and Party B travels 7 kilometers per hour. Party A goes south from Nanzhuang and Party B goes north from Beizhuang. After three hours, the distance between them is 60 kilometers. How many kilometers is the distance between the two villages?

The distance between east and west towns is 20 kilometers. Party A and Party B set out from two towns at the same time, and the distance traveled by Party A per hour was twice that of Party B. After three hours, the distance between them was 56 kilometers. What is their speed?

Example 4. Party A and Party B travel in the same direction from AB, with a distance of 36 kilometers. Party B is behind the former. Party A travels15km per hour and Party B travels 6km per hour. How many hours before Party A can catch up with Party B? (distance difference ÷ speed difference = catch-up time)

1. Party A and Party B are moving eastward at the same time from two places 24 kilometers apart, with Party B in front and Party A behind. Party A rides bike every hour13km, and Party B walks 5km every hour. How many hours before Party A can catch up with Party B?

2. A PLA unit set out from the camp and headed for its destination at a speed of 6 kilometers per hour. Eight hours later, the troops were in a hurry and sent correspondents to contact at a speed of 54 kilometers per hour. How long will it take the correspondent to catch up with the team?

3. Liang Xiao's home in Obana is 380m away. Two people set out from home at the same time and walked on the same straight road. Xiaohua walks 65 meters per minute, and Liang Xiao walks 55 meters per minute. How many meters may they be apart in three minutes?

Example 5. The circular runway is 400 meters long. Xiao Qiang runs 300 meters per minute, and Xiao Xing runs 250 meters per minute. Two people set out for the same place and the same direction at the same time. How long did it take Xiao Qiang to catch up with Xiaoxing for the first time? (distance difference ÷ speed difference = catch-up time)

1. Party A and Party B run along the track of the sports ground. Party A runs 290 meters per minute and Party B runs 270 meters per minute. This runway is 400 meters long. If two people run in the same direction from the starting line at the same time, how long does it take for A to catch up with B for the first time?

Bright primary school has a 200-meter-long circular runway, where Liangliang and Jingjing start from the starting line at the same time. Liangliang runs 6 meters per second, Jingjing runs 4 meters per second. Q: How many meters did Liangliang run when she first caught up with Jingjing?

3. Party A and Party B compete around a circular square with a perimeter of1000m. It is known that Party A walks 125m per minute, and Party B's speed is twice that of Party A. Now Party A is 250m behind Party B, how many minutes does it take for Party B to catch up with Party A?

Example 6. A car goes from city A to city B. After 3 hours, it will be repaired for 30 minutes due to car failure, and then it will continue to drive at a speed of 5 kilometers per hour. After 6 hours, it will arrive at B city on time. How many kilometers are there between the two cities?

1. A car travels 560 meters per minute from place A to place B, and it is expected to arrive in 30 minutes, but halfway through, the car needs to refuel and fill it up in 3 minutes. If it still needs to arrive at the scheduled time, how many kilometers per minute must it travel for the rest of the journey?

2. From city A to city C, you have to pass through city B. The speed of car A from city A to city B is 60 kilometers and 420 kilometers per hour, and the speed is 5 kilometers from city B to city C. When you arrive at city C, the journey is 10 hour. What's the distance between two cities in AC?

3. The distance between the two ports is 3. AB is 5 15km, and two speedboats, A and B, set off from two ports at the same time, with opposite directions. Two hours later, Ship A returned to Port A on business, and then sailed for three hours before the two ships met. It is known that B boat has a speed of 50 kilometers per hour, and how many kilometers per hour does A boat have?

Second, general application questions

Example 1. The department store shipped 300 pairs of sports shoes in two wooden cases and six cartons respectively. If two cartons contain the same number of sports shoes in a wooden box, how many pairs of sports shoes are there in each wooden box and each carton?

1. A toy factory packed 630 toys in 5 plastic boxes and 6 cartons respectively. 1 There are as many toys in the plastic box as there are in the three cartons. How many toys are there in each plastic box and carton?

2. xinhua primary school bought two tables and five chairs and handed them over to 195 yuan. As we all know, the price of each table is four times that of each chair. How much is each table?

Uncle Wang bought 3kg litchi and 4kg longan, and paid 156 yuan. As we all know, the price of 5 kg litchi is equal to the price of 2 kg longan. How much is litchi and longan per kilogram?

Example 2. A basket of pears weighs 38 kilograms. Half eaten, there are still 20 kilograms in the basket. Q: How much do pears and baskets weigh?

1. Weight of a barrel of oil180kg. Half of the oil is used, and there is still 100 kg in the barrel. Q: How many kilograms do oil and barrels weigh?

2. A basket of apples, 35 kg, give half to the kindergarten children first, and then half to the first-grade children. The remaining apples weigh 1 1 kg. How much does this basket of apples weigh?

There is some oil in the oil drum. If the oil is doubled, the oil drum will weigh 38 kilograms. If the oil is returned four times and the barrel weighs 46 kilograms, how many kilograms of oil is there in the crude oil barrel?

Example 3. There are six baskets of pears, and the number of pears in each basket is equal If you take out 40 pears in each basket, the sum of the remaining pears in the six baskets is exactly the same as the original two baskets. How many pears are there in each basket?

1. There are 5 boxes of tea. If 200g is taken out of each box, the remaining tea leaves in 5 boxes are exactly equal to the original 4 boxes of tea leaves. How many grams are there in each box?

There are as many oranges in five wooden boxes. If you take out 60 oranges from each wooden box, the sum of the oranges left in five wooden boxes is equal to the sum of the oranges in the original two wooden boxes. How many oranges are there in each wooden box?

There are 8 boxes of biscuits in a food store. If you take 20kg per box, the remaining biscuits in 8 boxes are exactly equal to the original 3 boxes. How many kilograms of biscuits are there in each box?

Example 4. The TV factory has received a batch of production tasks, and plans to produce 90 TV sets every day, which can be completed on schedule. In fact, five more TV sets were produced every day, and the task was completed 1 day in advance. How many TV sets are there in this batch?

1. A timber factory produced a number of tables. It was originally planned to produce 60 sheets a day, but in fact, 4 sheets were produced more than originally planned. As a result, the task was completed one day ahead of schedule. How many desks were originally planned to be produced?

Xiaoming read a story book and plans to read 12 pages every day. In fact, he reads eight more pages every day, and he finishes reading them two days in advance. How many pages does this story book have?

3. To build an expressway, it was originally planned to build 60 meters a day, but actually it was built 15 meters more than planned. It was finished four days in advance. How many meters were built?

Example 5. There are two bags of flour, the first bag is 24 kg, and the second bag is 18 kg. How many kilograms are taken from the first bag and put into the second bag, so that the weight of flour in the two bags is equal?

1. There are two boxes of pushpins, 72 in box A and 48 in box B. How many pushpins should be taken out of box A and put into box B to make the pushpins in the two boxes equal?

2. There are two boxes of glass balls, 72 in box A and 48 in box B. How many times have you taken four from box A and put them in box B?

There are two bags of sugar, one bag is 68, and the other bag is 20. Take 6 capsules out of more bags and put them into fewer bags at a time. How many times do you have to eat to make these two bags of sugar have the same content?

Example 6 Master and apprentice add 840 parts at the same time. If the master adds 9 days and then the apprentice processes for 2 days, the task can be completed. If the apprentice processes for six days first and the master processes for another six days, the task can be completed. How many parts are processed by the master and the apprentice respectively?

1. Master Zhang processes 750 parts at the same time. If Master Zhang handles it for 7 days and Li Shifu handles it for 4 days, the task can be completed. If Li Shifu handles it for five days first, and then Master Zhang handles it for another five days, the task can be completed. How many parts did the two masters process respectively?

2. Workshop A and Workshop B process 540 garments at the same time. If workshop A is processed for 2 days and workshop B is processed for 5 days, the task can be completed. If workshop A processes for 3 days first and then workshop B processes for 3 days, the task can be completed. How many garments are processed in each workshop?

3. The gas company plans to lay the gas pipeline from 10 workers in 16 days. After 4 days of construction, workers 10 will be added. If the daily work efficiency of each worker remains the same, how many days can the original laying task be completed in advance?

Third, the elimination method to solve the problem

Example 1. The school bought 3 water bottles and 20 teacups for the first time, and spent 134 yuan. I bought the same three water bottles and 16 teacups for the second time and spent 1 18 yuan. What is the unit price of the water bottle and the teacup?

1. My younger brother bought four exercise books and three pencils, one * * bought them at 90 cents from 2 yuan, my sister bought the same four exercise books and 1 pencils, and one * * bought them at 30 cents from 2 yuan. Find the unit price of each exercise book and pencil.

2. I bought 3 Jin of tea and 5 Jin of sugar for the first time, and used them in 420 yuan for the first time. I bought the same 5 Jin of tea and 5 Jin of sugar for the second time, and I used more 220 yuan than the first time. How much is tea and sugar per kilogram?

In order to award prizes, Class One bought two pens and five ballpoint pens with 39 yuan, while Class Two bought the same three pens and four ballpoint pens with 48 yuan. How much is a pen more expensive than a ballpoint pen?

Example 2. Buy three basketballs and five footballs * * * 480 yuan, buy the same six basketballs and three footballs * * * 5 19 yuan. What is the unit price of basketball and football?

1. bright primary school bought two tables and five chairs, which were handed over to 1 10 yuan; 240 yuan from Yucai Primary School bought the same 6 tables and 6 chairs. How much is each table and chair?

2. A store sells 660 kilograms of apples and oranges a day. It is known that it sold 3 boxes of apples and 2 boxes of oranges in the morning, *** 180 kilograms; I sold 4 boxes of apples and 8 boxes of oranges in the afternoon. How many kilograms does each box of apples and oranges weigh?

3.3 Cattle and 6 sheep graze 93kg a day, and 6 cattle and 5 sheep graze 130kg a day. How many kilograms does each cow eat more than each sheep every day on average?

Example 3.4 * * 560 science and technology books and 6 story books, 450 same 3 science and technology books and 5 story books. How many books are there in each package?

1.5 kg of fruit candy and 2 kg of biscuits ***76 yuan, the same 3 kg of fruit candy is more expensive than 2 kg of biscuits in 20 yuan. How much are fruit candy and biscuits per kilogram?

Students go boating in the park. Four big boats and three small boats take 29 people, and four big boats take more than three small boats 1 1 person. 1 How many people can the big boat and 1 small boat hold?

3.3 Two towels *** 154 yuan, and the same three towels 17 1 Yuan is less than three. 1 How much are towels and 1 sheets?

Example 5.7 Cattle and 5 horses eat 155 kg of grass every day, and 5 cows and 7 horses eat 145 kg of grass every day. How many catties does each cow and horse eat every day?

1.4 The total price of walking tractors and 5 threshers is 17000 yuan, and the total price of 5 walking tractors and 4 threshers is 19000 yuan. How much is the walking tractor How much is the thresher?

2. For the first time, the canteen brought in 5 bags of rice and 7 bags of flour, * * *1350kg; I brought 7 bags of rice and 5 bags of flour for the second time, * * * 1650kg. How many kilograms are there in each bag of rice and flour?

3. Xiaohong went to the store to buy prizes, 12 pens and 15 pencil cases. She wanted 300 yuan. Xiaohong bought the same pen and pencil box by mistake, and as a result, she paid 2 1 yuan. How much is each pen and pencil box?

Example 6. In the school gymnasium, I bought two basketballs, soccer balls and volleyball at one time and used 340 yuan. I bought four basketballs, three footballs and two volleyballs for 495 yuan for the second time. For the third time, I bought five basketballs, four footballs and two volleyballs for 605 yuan. How much are basketball, football and volleyball?

1. Mr. Wang bought three picture clips, three boxes of pigments and three brushes for the first time. 195 yuan, the second time he bought three picture clips, four boxes of pigments and three brushes. * * It cost 2 13 yuan, and the third time he bought four picture clips, five boxes of pigments and three brushes. What is the unit price of picture clip, paint and brush?

A group of friends went boating. It is known that 2 small boats, 3 medium boats, 1 large boat * * take 23 people, 3 small boats, 4 medium boats and 2 large boats * * take 35 people, and 1 small boat, 2 medium boats and 3 large boats * * take 26 people. Q: Every small boat, 4 medium boats and 3 large boats * * *.

3. Xiaoming bought two pens, 1 ballpoint pen and two mechanical pencil * * used 26 yuan, Xiaohua bought 1 pen, three ballpoint pens and four mechanical pencil also used 26 yuan. It is known that the price of 1 ballpoint pen is exactly the price of 4 mechanical pencil. How much are pens, ballpoint pens and mechanical pencil?

Example 7. Aunt Zhou bought a coat, a pair of trousers and a pair of leather shoes in the department store, and spent 904 yuan. It is known that coat is 3 12 yuan more expensive than trousers, and coat plus trousers is 392 yuan more expensive than leather shoes. Do you know how much Aunt Zhou spent on her coat, trousers and leather shoes?

1. Mao Qi bought a pencil case, a pen and a box of watercolor pens in 44 yuan, Xinhua Stationer. It is known that pencil cases are more expensive than watercolor pens in 6 yuan, and pencil cases and watercolor pens are more expensive than pens in 6 yuan. Do you know the unit price of pencil case, pen and watercolor pen?

Xiao Wang spent 58 yuan to buy some pears, apples and bananas to visit his teacher. It is known that three kinds of fruits weigh 2290g, pears are lighter than bananas180g, and pears and bananas weigh 770g than apples. Do you know how many grams of pears, apples and bananas in Little Wang Mai weigh respectively?

Mom bought two bottles of Head & Shoulders shampoo and three pieces of Lux soap in the supermarket, 48 yuan. At the time of payment, my mother changed two pieces of soap into the same bottle of shampoo, and the cashier asked her to look at it again. 10 yuan. Please calculate the unit price of Head & Shoulders shampoo and Lux soap.

Fourth, the average problem.

Example 1. The math interest group held a test. Eight students in Class Four (1) got 82, 75, 95, 98, 100, 80, 87, 79 respectively. What is the average score of eight students?

1. There are 8 elderly people in the nursing home. Their ages are 78, 76, 77, 8 1, 78, 76, 80 and 78 respectively. Find the average age of these eight old people.

2. The weekly temperatures measured by the meteorological team at 8: 00 every morning are as follows: 13, 13, 14, 15, 14, 16. Find the average temperature of a week.

3. Seven students from Class 6 (3) took part in the math contest. Two students scored 99 points, three students scored 96 points, and the other two students scored 97 points and 89 points respectively. What is the average score of these seven students?

Example 2. The bicycle factory produced 4,500 bicycles in the first 65,438+00 days of April and 8,400 bicycles in the last 20 days. How many bicycles are produced on average every day this month?

1. Xiaoming took the math exam. The total score of the first two exams is 170, and the total score of the last three exams is 270. What is the average score of Xiaoming's five exams?

2. The students in Class Two (1) are divided into three groups to plant trees. The first group of 8 people planted 80 trees, the second group of 6 people planted 66 trees and the third group of 6 people planted 54 trees. How many trees are planted per person on average?

3. In the handicraft class, the students in Class 3 (4) are divided into three groups to make flowers. There are 8 people in the first group, and the average cost per person is 10. There are 6 people in the second group, with an average of 1 1 per person. There are 6 people in the third group, with an average of 9 flowers per person. How much does Class Three (4) cost per person?

Example 3. Xiao Wang is doing climbing exercises. When going up the mountain, he walks 65,438+0,200m and 30m per minute. When going down the mountain, return along the original road, 60m per minute, and find the average speed of Xiao Wang going up and down the mountain.

Xiao Wang does mountain climbing. When he goes up the hill, he walks 30 meters every minute. When going down the mountain, go back along the original road and walk 60 meters per minute to find the average speed of Xiao Wang going up and down the mountain.

1. The road from the top of the mountain to the bottom of the mountain is 36 kilometers long. The car takes 4 hours to reach the top of the mountain, and it takes only 2 hours to reach the foot of the mountain by returning along the original road. Find the average speed of this car going back and forth.

2. Xiao Qiang walks from home to school every minute120m,10min. At noon, he went home along the same road, walking 80 meters per minute, and found out the average speed of going to and from school in Xiao Qiang.

3. Xiao Liang climbs the mountain at a speed of 2 kilometers per hour. After reaching the top of the mountain, he immediately went down the mountain at a speed of 6 kilometers per hour. What is the average speed of going up and down the mountain in Xiao Liang?

Example 4. Xiaojun participated in three math competitions with an average score of 84. It is known that the average score of the first two competitions is 82 points. How many points did he get in the third exam?

1. Li Hua took part in the sports standard test, and the average score of five events was 85. If the throwing score is not included, the average score is 83 points. How many points did Li Hua score in throwing?

2. Before the final exam math scores were announced, Xiaoli scored an average of 92 in four subjects. After his math scores were announced, his average score dropped 1. How many points did Xiaoli get in the math exam?

There are seven numbers in a row, and their average is 3 1. The average of the first three numbers is 29, and the average of the last five numbers is 34. Find the third number.

Example 5. If the average age of three people is 22, and none of them is younger than 18, how old is the oldest person?

1. If the average age of four people is 23, and no one is younger than 18, how old is the oldest person?

2. If the average age of four people is 29, and no one is over 32, what is the age of the youngest person?

3. If the average age of four people is 25 years old, and none of them is younger than 16 years old, and these four people are not equal in age, then who is the oldest person?

Example 6. Liu Yun scored an average of 85 points in six subjects in the final exam. The math score is higher than the average score of seven subjects 12 points. How many points did Liu Yun get in the math exam?

1. Xiaoming scored an average of 89 in five subjects in the final exam last semester, and the average score in English was 5 points higher than that in six subjects. How many points did he get in the English exam?

In a foreign language test, the scores of students A, B and C were 87, 75 and 93 respectively. Ding's foreign language score is 6 points higher than the average score of A, B, C and D. How many points did Qiu Ding get in the foreign language exam? What is the average score of four people?

Class 3.4 (1) has a boat race. The teacher divided these students into three groups: A, B and C. 12 was rented, and the cost of chartering was shared equally among the three groups. Group A paid the rent for seven ships in advance, while Group B paid the rest, while Group C did not pay for the time being. After rowing, the teacher settled, and group C chartered the boat 12. How much should group C give to groups A and B?

Verb reduplication (abbreviation of verb)

Yes 1. In the fifth grade, 122 people took the Chinese and math exams, and everyone got excellent grades in at least one subject. Among them, there are 65 excellent students in Chinese and 87 excellent students in mathematics. How many people are excellent in Chinese and mathematics?

1. There are 48 students in Class One. The head teacher asked at the class meeting, "Who finished the Chinese homework? Please raise your hand! " Thirty-seven people raised their hands and asked, "Who finished the math homework? Please raise your hand! " 42 people raised their hands. Finally, I asked, "Who hasn't finished their Chinese and math homework?" No one raised their hands. Ask the number of students in this class to finish Chinese and math homework.

There are 54 students in Class One, Grade Four. There are 13 people who subscribe to "Excellent Composition for Primary School Students" and "Mathematical World". There are 45 people who subscribe to the excellent composition of primary school students. Everyone subscribes to at least one reading material. How many people have subscribed to Mathematical World?

Everyone in the school art troupe can play at least one musical instrument. It is known that there are 24 people who can play the accordion and 17 people who can play the electronic organ, among which 8 people can play both instruments. How many people are there in this art troupe?

Example 2. There are 40 students in Class Five (1), including 25 students in the math group and 23 students in the science and technology group. 19 has both groups. So, how many people didn't participate in either group?

1. There are 36 students in one class. In one test, 25 students answered the first question correctly, 23 students answered the second question correctly, and 15 students answered two questions correctly. How many people answered both questions wrong?

2. There are 55 students in one class, 32 of whom subscribe to Primary Mathematics Newspaper, 29 subscribe to China Youth Daily and 25 subscribe to two newspapers. How many people have not subscribed to any newspaper?

3. A school selected 50 students to participate in the district composition contest and math contest. As a result, three students won both competitions, and 27 students didn't win both competitions. It is understood that 14 students won the composition competition. How many people won the math contest?

There are 36 people in a travel agency, including 24 who can speak Japanese, 18 who can speak Russian, and 4 who can neither. How many people can both?

1. There are 56 students in one class, 28 students take part in the Chinese contest and 27 students take part in the math contest. If 25 students didn't take part in both subjects, how many students took part in the Chinese and math competitions?

2. A club 103 people, including 69 people who can play China chess, 52 people who can play chess, and neither of them 12 people. How many people can play both chess?

3. There are 40 students in Class One, Grade Three, and 20 students in the dance team. Both the chorus and the dance team are 14, and neither team is 10. Please calculate how many people are there in this class.

Example 4. How many natural numbers from 1 to 200 are not multiples of 5 or 8?

1. From 1 to 100, how many natural numbers are neither multiples of 5 nor multiples of 6?

2. How many natural numbers from 1 to 130 are not multiples of 4 or 9?

Class 3.5 (1) is doing broadcasting exercises. The whole class is divided into four rows, and each row has an equal number of people. The position of the small flower row is: the fifth in front and the eighth in the back. How many students are there in this class?