I. Meeting and catching up in a straight line

Example 1. A and B both start from east and west at the same time. A travels 56 kilometers per hour and B travels 48 kilometers per hour. The two cars met at a distance of 32 kilometers from the midpoint of the two places. Q: What is the distance between east and west? (A key middle school junior high school examination questions in 2007)

1, an express train and a local train leave from Station A and bilibili at the same time in opposite directions. After the 6-hour rendezvous, the express train will continue to travel for 3 hours and then arrive in bilibili ... It is understood that the local train travels 45 kilometers per hour. How many kilometers are the two stops apart?

Party A and Party B walk from A and B to each other at the speed of 3km and 5km per hour respectively. After meeting, they walked on. If A departs from the assembly point and arrives at Line B in 4 hours, how many kilometers is there between A and B?

3. Two brothers leave home for school at the same time, and the distance is 1400 meters. My brother rides a bike 200 meters per minute, and my brother walks 80 meters per minute. On the trip, my brother met his brother who had just returned to school. How many meters did my brother walk from departure to meeting? How many meters is the venue from the school?

The distance between a and b is 38 kilometers. A and B start from two places at the same time and go in opposite directions. A travels 8 kilometers per hour, and B travels 1 1 km per hour. Go back to B as soon as A arrives at B, and B will go back to B as soon as B arrives at A, and meet you on the road a few hours later? How far is it from A when we meet?

5. Two cars, A and B, leave from A and B respectively at the same time. After five hours of meeting, they left separately. Three hours later, a car arrived at b, and at this time, b car was still 120 km away from a place. What are the speeds of A and B respectively?

Example 2. Two swimmers are swimming back and forth in a 30-meter swimming pool. The speed of A is 1 m per second, and the speed of B is 0.6 m per second. They started from both ends of the swimming pool at the same time and swam back and forth for 5 minutes. How many times have two people met during this time, if not counting the turning point? (A key middle school junior high school examination questions in 2006)

2. The train passes by, crosses the bridge, and the wrong car

Example 3. It takes 25 seconds for a bus to pass through a 250-meter tunnel and 23 seconds for a 2 10-meter tunnel. It is known that there is a truck in the same direction in front of the bus, with a length of 320 meters and a speed of 17 m/s. Find out when the train and truck meet and leave. (A key middle school in 2005 fifth grade last semester final exam questions)

Example 4. A PLA captain is 450m, and the speed is1.5m per second. How long does it take a soldier to walk from the end of the platoon to the head of the platoon at a speed of 3 meters per second, and then return to the end of the platoon immediately? (A key middle school junior high school examination questions in 2008)

1, Xiaoming sat on the moving train and saw the oncoming truck from the window, 6 seconds. Known truck length168m. Later, it took 12 seconds to see the train passing through a bridge 180 meters long from the window. How many kilometers does this truck travel per hour?

2. An army marched in a line of 65,438+0,200m. The correspondent on the other end of the phone had to contact the battalion commander at the front. He ran for six minutes and caught up with the battalion commander. In order to get back to the end of the line, he waited for 24 minutes where he caught up with the battalion commander. If he runs from the front of the line to the end, it only takes a few minutes.

3. A PLA unit dispatched 80 vehicles to take part in the field work, and on the way, it had to pass through a tunnel with a length of120m. If the length of each vehicle is10m, and the distance between two adjacent vehicles is 20m, how many minutes will it take for the motorcade to pass through the tunnel at a speed of 500m per minute?

The express train and the local train are going in opposite directions. The length of the express train is 50 meters, the length of the local train is 80 meters, and the speed of the express train is twice that of the local train. If it takes five seconds for a person sitting in the local train to see the express train pass through the window, how long does it take for a person sitting in the local train to see the local train pass through the window?

Three. Move between multiple objects

Example 5. There are three people: A, B and C. A walks 100 meters per minute, B walks 80 meters per minute and C walks 75 meters per minute. Now A starts from the East Village, and B and C start from the West Village at the same time. On the way, A meets B for 6 minutes, and A meets C again. So, what is the distance between the two villages?

1, Party A and Party B go from A to B, and Party C goes from B to A. They start at the same time. A cycling speed is 8 kilometers per hour, and C cycling speed is 10 kilometers. Five hours later, A and C meet, 1 hour later, B and C meet. Find the speed of B.

2. The walking speeds of Party A, Party B and Party C are 30m, 40m and 50m per minute respectively. A, B are in place A and C are in place B, walking in opposite directions at the same time. Party C meets Party A 10 minutes after meeting Party B. How many meters is there between A and B?

3. Party A, Party B and Party C should go in groups of three. A walks 60 meters per minute, B walks 50 meters per minute and C walks 40 meters per minute. A starts from A, B and C start from B. After A and B meet, 15 minutes later meets C, thus finding the distance between A and B. ..

4. Ring problem and clock problem

Example 6. Two people, A and B, ride bicycles from the same place on the ring road at the same time, with their backs to each other. It is now known that it takes 70 minutes for A to walk once. If A and B meet 45 minutes after departure, how many minutes does it take for B to take a trip?

1. The circular track of the stadium is 400 meters long. Xiao Gang and Xiao Hua are on the same starting line of the track and start in opposite directions at the same time. Xiaogang runs 152 meters per minute, and Xiaohua runs 148 meters per minute. A few minutes later, they met for the third time.

2. Party A and Party B run towards each other around the circular runway at a uniform speed, starting at both ends of the circular diameter. If they start at the same time and meet for the first time when A runs 60 meters, and meet for the second time when B is 80 meters short, what is the length of the runway?

Example 7. The clock now shows 10 hour. Then, how many minutes later, the minute hand and the hour hand coincide for the first time; How many minutes later, the minute hand and the hour hand coincide for the second time?

1. When does the hour hand and minute hand of a clock form a right angle between four o'clock and five o'clock?

Grandpa goes out for a walk at 7: 00 pm. When he went out, the hour hand and the minute hand were in a straight line. When he came back, the hour hand and the minute hand coincided. How long has grandpa been out for a walk?

The hour hand and minute hand of a clock are between 4 and 6, and the "5" on the clock face is right in the center of the hour hand and minute hand. What time is it now?

Xiao Liang aimed at his watch at nine o'clock in the evening. Arrived at school at 8 am and was late 10 minutes. Then how many minutes does Xiao Ming's watch lose every hour?

Verb (abbreviation of verb) is a problem of running water.

1. When the ship is sailing in the river, the downstream speed is12km/h and the upstream speed is 6km/h.. How many kilometers per hour is the ship speed and how many kilometers per hour is the water speed?

The speed of the ship in still water is 265,438+0 kilometers per hour. The ship sailed upstream from city A for 8 hours and arrived at city B, with a distance of 1.44 km. How many hours does it take for this ship to return from B to A?

A ship is sailing at the speed of 20 kilometers per hour in calm water. It takes 12.5 hours to get the distance between two docks in a river with a flow speed of 4 kilometers per hour.

5, a ship, the first time sailing 56 kilometers downstream, 20 kilometers upstream, * * * need 12 hours; The second time, at the same time, I sailed 40 kilometers downstream and 28 kilometers upstream. Find the speed of the ship in still water.

Exclusion principle (3)

1. How many * * * in an integer not exceeding 20 are multiples of 2 or multiples of 3?

There are 23 members in a class. There are 20 boys in this class. How many more girls are there in this class than boys who are not league members?

There are 25 people who get 90 points in math in one class. There are 2 1 person with a language score of 90 or above; There are at least one of the two subjects, and there are 38 people over 90. How many people scored above 90 in both subjects?

As shown in the figure, a square with a side length of 2 and a square with a side length of 3 are placed on the desktop. How big is the area they cover?

5. A piece of paper with an area of 7 and a square piece of paper with a side length of 2. Put these two pieces of paper on the table, covering an area of 8. What is the overlapping area of two pieces of paper?

6. There are 65,438+000 passengers, of whom 65,438+00 know neither English nor Russian, 75 know English and 83 know Russian. How many people know both English and Russian?

7. A school organizes chess competitions, which are divided into three groups: Go, China Chess and Chess. There are 42 participants in Weiqi, 5 1 person in China and 30 in chess. There are 65,438+03 people who take part in Weiqi and China chess competitions at the same time, 7 people who take part in Weiqi and chess competitions at the same time, and 65,438+065,438+0 people who take part in China chess competitions at the same time, among which 3 people take part in all three chess competitions. How many people are there in the chess game?

8. As shown in the figure, put three squares with sides of 6, 5 and 2 on the desktop. Ask them what area they cover.

9. A class of students have red, yellow and blue balls in their hands. There are 34 red balls, 26 yellow balls and 8 blue balls/kloc-0. Among them, there are six people with red, yellow and blue balls in their hands. There are 9 people who have only red and yellow balls, 4 people who have only yellow and blue balls and 3 people who have only red and blue balls. So how many people are there in this class?

Of the natural numbers from 10 to 100,

(1) How many numbers are not divisible by 6 and 10?

(2) How many numbers are divisible by at least one of 2, 3 and 5?

1 1, from 1 to 200, how many natural numbers are not divisible by any of 2, 3 and 5?

12, one day in midsummer, 10 students went to a cold drink shop and handed the waiter a cold drink demand statistics: coke, sprite and juice for five people each; Three people want coke and sprite; Two people want coke and juice; There are two people who want sprite and juice; Only 1 person wants all three. Prove that there must be 1 of them who didn't want any of these three drinks.

13. 100 the results of the survey of students' extracurricular activities are as follows: 32 students participated in the math group; 20 people take part in the English group and 45 people take part in the biology group. Among them, 15 people participated in both the math group and the biology group; There are 7 people in English group and math group; 10 people participated in the English group and the biology group. There are still 30 people who have not participated in any of the above subject groups.

(1) Find out the number of participants in all three topic groups.

(2) Fill in the corresponding number of students in eight areas of venn diagram. Among them, A, B and C represent the number of students participating in math, English and biology groups respectively. The number of 100 students surveyed is i.

Special lecture on the competition of the principle of inclusion and exclusion

14. How many natural numbers from 1 to 1000 are not divisible by 5 or 7?

15. How many natural numbers from 1 to 100 are not divisible by 2, 3 and 8?

16. A class of students took the exams in mathematics, science and chemistry, with 30 excellent students in mathematics, science and chemistry, 28 excellent students in mathematics, physics and chemistry, 20 excellent students in mathematics, physics and chemistry, 16 excellent students in all three subjects 17. Q: How many people are excellent in at least one of mathematics, science and chemistry?

17. There are 48 students in the class, among whom 27 can swim, 33 can ride bicycles and 40 can roller skate. Q: How many students can do at least all three sports?

17. On a hot summer day, several children went to a cold drink shop, and each of them ordered at least one cold drink. Six of them asked for popsicles, six for soda and four for Sprite. As long as there are three popsicles and soda, as long as there are no popsicles and sprite, as long as there are 1 soda and sprite; All three have 1 person. Q: * * * How many children went to the cold drink shop?

18. All the students in a class took part in sprint, swimming and basketball tests. Four students failed to achieve excellent results in these three items, and the rest of the students each achieved excellent results in at least one item. The number of students who have won excellent projects is as follows:

Ask how many students there are in this class.

Fill in the blanks: (basic training)

1, 3.85 cubic meters = () cubic decimeter 4 liters 40 ml = () liters.

2. Weld a cubic frame (excluding joints) with 48cm long iron wire. The surface area is () square centimeters and the volume is () cubic centimeters.

3. Fill in the appropriate unit name in brackets:

The volume of an eraser is about 8 (), and a classroom accounts for about 48 ().

The volume of the automobile fuel tank is 30 (), and the length of each step is about 60 ().

There are () odd numbers and () even numbers among natural numbers within 4.20 (including 20).

5. In14,6, 15 and 24, () is divisible by (), and () and () are prime numbers.

6. The largest two-digit number that can be divisible by 2, 3 and 5 at the same time is (), and the prime factor to decompose it is ().

7, 5□ When () is filled to the maximum, this number can be divisible by 3, and the divisor of this number is ().

8. If A is divisible by B, the greatest common divisor of A and B is (), and the least common multiple of A and B is ().

9. It is known that A = 2× 2× 3× 5 b = 2× 5× 7, the common prime factor of A and B is (), and their greatest common divisor is ().

10, a 2-meter-long rectangular steel, cut into two sections along the cross section, and the surface area is increased by 0.6 square decimeter. The volume of this rectangular steel is () cubic decimeter.

Second, judgment: 5 points

1, nonzero natural numbers are either prime numbers or composite numbers. ( )

2. The multiple of a number must be greater than its divisor. ( )

3. The product of two prime numbers must be a composite number. ( )

Application of engineering problems

[Example 1] For a project, team A 12 days completed the task, team B 15 days completed the task, team A completed it alone, and the rest were completed by both parties. How many days did it take to complete the task?

[Example 2] For a project, it takes 20 days for Team A to do it alone and 30 days for Team B to do it alone. Now the two teams cooperate for a few days, and the remaining B team 10 days will be completed. How many days did Team A and Team B cooperate?

[Example 3] A job takes 6 days by myself and 8 days by team B. Now that team C has completed all the projects, the rest are completed by teams A and B. How many days does it take to complete the task?

Example 4: A pool has three water pipes, A, B and C. Opening the first pipe can fill the empty pool within 6 hours, opening the second pipe can fill it within 4 hours, and opening the third pipe can fill it within 12 hours. How many hours does it take to fill an empty pool with three pipes at a time?

Correspondence practice

Fill in the blanks

1. A project can be completed in 4 days by both parties, 8 days by Party A and () days by Party B. ..

A project will take 65,438+00 days for Team A, 20 days for Team B, and () days for both parties. ..

3. A project can be completed by both parties in 6 days, with one person from Party A 15 days. Party A and Party B will cooperate for () days, and the rest will be completed by Party B's team in 5 days.

4. From Station A to bilibili, it takes 5 hours by bus and 6 hours by truck. The speed of a bus is ()% faster than that of a truck.

5, processing a batch of parts, a single hour, b single hour, two people do () hours to complete.

6. Party A completes a project in 6 days, and Party B completes it in 0/2 days. ..

(1) Party A and Party B complete all the projects in one day ();

(2) Party A and Party B shall complete it within () days;

(3) Party A and Party B jointly complete the whole project within 3 days ();

(4) The ratio of work efficiency between A and B is ().

Second, answer the following questions

1, a pile of goods, car A needs to deliver in hours, and car B needs to deliver in hours. How many hours will it take if two cars are shipped together?

2. It takes 6 days for A to finish a job alone, and B is twice as efficient as A ... How many days can two people do it at the same time?

3. For work A, A will do it alone 15 days, B will do it alone 18 days, A will do it for 5 days first, and B will do the rest alone. How many days?

4. It takes 10 hour for Party A to produce a batch of parts, but Party B can only produce this batch of parts at the same time. How many hours does it take Party B to do this batch of documents alone?

5. For a job, Team A 12 days, Team B 15 days, Team A alone, and the rest are jointly completed by Party A and Party B. How many days does it take to complete the task?

6. Build a 30-kilometer expressway. Team A only takes 10 days to complete, and team B only takes 15 days to complete. How many days can the two teams finish it together?

7. There is a project. It takes 8 days for Team A to complete it alone, and it takes 12 days for Team B to complete it alone. How many days does it take for Party A and Party B to cooperate in this project?

8. When the swimming pool is filled with water, the single-port pipe is filled for 10 hour, and the single-port pipe is filled for 8 hours. If two pipes are opened at the same time, how many hours can the pool be filled?

9. For typing a 5400-word manuscript, Party A will type it all in 3 hours, Party B will type it all in 2 hours, and both parties will type it together 1 hour. How many words does Party A type more than Party B?

10, it takes 30 days for Party A to do a job, and the time for Party B to do it alone is the time for Party A. If two people do it together, how many days will it take to complete the whole project?

Name of comprehensive review of engineering problems in Xiaoshengchu:

1. A project needs 15 days for team A, 20 days for team B, and 12 days for team C.

(1) What fraction of the project is completed by each of the three teams every day? (2) How many days does it take for three teams to cooperate to complete the project?

(3) How many days will it take for three teams to cooperate to complete 1/4 of this project? (4) How much will Party A and Party B repay each other after 3 days of the remaining works?

(5) How many days will 1/2 of this project remain after the cooperation of three teams? (6) After three teams cooperate for two days, the rest will be done by Team A alone. How many days will it take?

(7) After two days, the rest shall be completed by both parties. How many days will it take to complete?

(8) How many days will it take for Team A to work for three days, and the rest will be completed by three teams?

(9) After Party A and Party C work together for two days, the rest will be done by Team B alone. How many more days will it take?

Party A, Party B and Party C will complete a project within 8 days. Now Party A and Party B do it 1 day, and Party C alone does the rest 15 days. Seek the working efficiency of C.

3. There are two water pipes in a reservoir. The water in the whole reservoir can be filled within 10 minute by opening the water inlet pipe alone, and the water in the whole reservoir can be drained within 15 minute by opening the water outlet pipe alone. How many minutes does it take for two pipes to open at the same time to fill the whole pool?

It takes 8 hours for the local train to go from Station A to bilibili, and 6 hours for the express train to go from bilibili to Station A ... The two cars are driving in opposite directions. It takes two hours for the local train to leave from Station A, several hours for the express train to leave from bilibili, and several hours for the express train to meet the local train.

It takes 8 hours for the express train to go from city A to city B, and 12 hours for the local train to go from city B to city A. Two trains leave from two directions at the same time. When they met, the express train traveled more than the local train 180 km. How many kilometers are the two stations apart?

6. For a manuscript, Party A types 2/ 15 of the manuscript every hour, and it takes 4 hours for Party B to type the manuscript alone. If two people type this manuscript together, how many hours can it be completed?

7. It takes 40 days for Team A to complete a project alone, and the efficiency of Team A is twice that of Team B. If the two teams cooperate, how many days will it take to complete the project?

8. To build an expressway, it will take 8 days to repair the perfect armor, Party B 10 days, and Party A and Party B will work together for 4 days, with 72 meters left to be repaired. What is the total length of this highway?

Engineering problems of midway departure (midway exchange)

9. For a project, Party A will complete it alone for 75 days and Party B will complete it alone for 50 days. During the joint work, Party A walked for several days and the whole project was completed in 40 days. A How many days did you walk halfway?

10. For a batch of goods to be delivered separately, Party A needs 10 hour, and Party B needs 15 hour. After a period of time, Party A will deliver the goods first, and then Party B will deliver them. It took hours to get it all away. How long did it take to transport it?

1 1. Party A will finish a project in 20 days and Party B will finish it in 30 days ... Now they are working together. Party A has a rest 1 day, and Party B has a rest for 6 days. How many days did they work at the same time when the project was completed?

13. A slender candle is lit in 4 hours, and a short candle is lit in 6 hours. After two candles are lit at the same time for 2 hours, the remaining length is exactly the same. It turns out that the stubby candle is a fraction of the long thin candle.

14. The pool is equipped with water inlet pipe and drainage pipe. Open the inlet pipe for 4 minutes to fill the pool, and open the drain pipe for 6 minutes to empty the pool. There is 1/3 dirty water in the pool now. Li Shifu had to drain the dirty water first, but he forgot to turn off the drain when he drained the clean water. How long does it take to fill up the clear water?

15. Team A and Team B cooperated for 24 days. Team a will work for six days first and team b will work for four days. Only 1/5 of the project can be completed. How much does it cost for two teams to do it independently?

(2).a does it alone, and it will take 6 days to finish it. A does it for 3 days and B does it for 4 days. A does it for x days first, and B continues to do it. How many days will it take?

(3) The two trains leave from AB relatively simultaneously. It takes 20 hours for the express train to complete the journey, 30 hours for the local train, 15 hours for the departure, 4 hours for the express train and several hours for the local train.

(4) The distance between. AB is 120km, and the car started from A 10 minutes later and traveled the whole distance15. How many kilometers has this car traveled, and the ratio of the remaining distance to the driving distance is 3: 1?

(6).a work alone 12 hours. Now AB works together for 2 hours, and the remaining work will be completed in 5 and a half hours. How many hours will it take if all the work is done by B?

Correspondence practice

1, it takes 6 days for Party A to do a job alone, and the work efficiency of Party B is twice that of Party A ... How many days can two people do it at the same time?

2, A work, A does it alone 15 days, B does it alone 18 days, A does it for 5 days first, and B does the rest alone. How many days?

3. build a 30-kilometer expressway. Team A only takes 10 days to complete, and team B only takes 15 days to complete. How many days can the two teams finish it together?

4. When the swimming pool is filled with water, the single-port pipe is filled for 10 hour and the single-port pipe is filled for 8 hours. If two pipes are opened at the same time, how many hours can the pool be filled?

5. Type a 5400-word manuscript. Party A will separately enter all 65,438+0/5 for 3 hours, and Party B will separately enter all 65,438+0/4 for 2 hours. They both have to play for an hour. How many words does Party A type more than Party B?

b grade

1. If Team A goes it alone, it can be finished in 6 days, Team A goes it alone for 3 days and Team B goes it alone for 2 days. How many days does Team B need to finish?

2. A project will take 30 days for team A and 40 days for team B.. Team a will do it for a few days first, and then team b will do it. * * * 35 days to complete the task. How many days did Team A and Team B each do?

3. A project will take 5 days for Team A and Team B, 6 days for Team B and Team C, and 6 days for Team A and Team C. How many days will it take for Team A, Team B and Team C to complete now?

4. To build a highway, Team A can finish it in 20 days, and Team B can finish it in 30 days. Now two teams are working together. Team A has a rest of 2.5 days and Team B has a rest of several days, so it will take 14 days to finish, and Team B has a rest of several days.

Class c

1, a project, it takes 20 days for a team to do it alone, and it takes 12 days for a team to do it alone. It is known that this project was first done by team A for a few days, and then continued by team B, and it took 14 days from the beginning to the completion. So how many days did team a do it first? How many more days did team b do it?

2. There is a pool. When the first pipe is opened, it can be filled with water 1 hour, and when the second pipe is opened, it can be filled with water for 40 minutes. After opening two pipes at the same time 10 minutes, 4 tons of water will be injected. How many tons of water can this pool hold?

3, A work, A alone 15 hours to complete, B alone 10 hours to complete. Now two people work together for a few hours, and the rest will be done by B alone for another five hours. How many hours did they work together?

A bus and a truck leave from Station A and bilibili at the same time. After the 6-hour meeting, the two cars continue to move at the original speed, and it takes another 4 hours for the bus to reach bilibili. Q: How many hours does it take for the truck to reach Station A after the meeting?

Hello, this is my carefully collected question, I hope it will help you.

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