I. Travel Problems
Example. Passenger and freight vehicles start at the same time from two cities A and B, which are 900 kilometers apart , and travel in opposite directions. It is known that it takes 10 hours for the bus to travel from city A to city B, and 15 hours for the truck to travel from city B to city A. How many hours after departure do the two vehicles meet?
2. A car and a motorcycle start at the same time from two cities, A and B, which are 240 kilometers apart, and travel in opposite directions. It is known that it takes 3 hours for the car to travel from city A to city B and 6 hours for the motorcycle to travel from city B to city A. How many hours after their departure did the two cars meet? How many hours after their departure do the two cars meet?
3. Two vehicles, a passenger and a freight, start at the same time from A and B, which are 480 kilometers apart, and travel in opposite directions, meeting each other after 4 hours. How many hours does it take for the bus to travel the full distance and for the truck to travel the full distance?
4. Passenger and freight vehicles from the distance of 800 kilometers from the A and B two cities at the same time, traveling in the same direction. The passenger car travels 80 kilometers per hour, and the truck travels 60 kilometers per hour, how many kilometers will the two cars be apart after 4 hours
Example. Two teams of students A and B start at the same time from two places 18 km apart and travel in opposite directions. A student rides a bicycle at a speed of 15 km/hour and keeps contacting the two teams back and forth. Team A travels 5 km/hour and team B travels 4 km/hour. How many kilometers will the student on the bicycle **** travel when the two teams meet?
1. Wang Xin and Lu Liang two people at the same time from 2000 meters apart from the two places to each other, Wang Xin 110 meters per minute, Lu Liang 90 meters per minute, if a dog and Wang Xin at the same time with the same line, 500 meters per minute,, encountered Lu Liang, immediately turned back to Wang Xin ran to meet Wang Xin and then run to Lu Liang. This continues to go back and forth until Wang Xin and Lu Liang meet, the dog **** line how many meters?
2. A, B two places 400 kilometers apart, A and B two cars at the same time from the two places relative to each other, car A 38 kilometers per hour, car B 42 kilometers per hour, a swallow to 50 kilometers per hour and car A at the same time, fly to the car B, encountered the car B turn back to the car to fly to A. This has been flying, the swallow flew back to the car. How many kilometers will the swallow fly before the two cars meet?
3, A and B two teams at the same time from 330 kilometers apart from the two places to each other, team A 60 kilometers per hour, team B 50 kilometers per hour, a motorcycle 80 kilometers per hour, in the middle of the two teams back and forth liaison, asked the two teams meet, the motorcycle traveled how many kilometers?
Example 3. Car A travels 6 kilometers per hour, car B travels 5 kilometers per hour, and the two cars travel back-to-back at the same time in two places 10 kilometers apart, how many hours after the two cars are 65 kilometers apart?
1. A travels 7 km/hour, B travels 5 km/hour, and they travel back to back at the same time at two places 18 km apart, after how many hours are they 54 km apart?
2, a 9 kilometers per hour, b 7 kilometers per hour, a south from the south bank to the south, at the same time, b from the north bank to the north, after 3 hours, the two are 60 kilometers apart, north and south of the two banks how many kilometers apart?
3. East and west of the two towns are 20 kilometers apart, A and B were two people from the two towns at the same time, back to back, A every hour line is twice as far as B, after 3 hours, the two are 56 kilometers apart, the speed of the two people are how much?
Example 4. A and B two people at the same time from the distance of 36 kilometers AB two places in the same direction, B in front of A in the back, A hourly line 15 kilometers, B hourly line 6 kilometers, how many hours after the A can catch up with B? (Difference in distance ÷ difference in speed = time to catch up)
1. A and B respectively from the distance of 24 kilometers between the two places at the same time to the east, B in front of the A in the back, A riding a bicycle 13 kilometers per hour, B walking 5 kilometers per hour, how many hours after A can catch up with B?
2. A department of the People's Liberation Army (PLA) departs from a camp and proceeds to its destination at a speed of 6 kilometers per hour. 8 hours later, the unit has an emergency and sends a correspondent on a motorcycle to liaise with it at a speed of 54 kilometers per hour, how long will it take for the correspondent to catch up with the team?
3. Xiaohua and Xiaoliang's homes are 380 meters apart, two people from home at the same time, walking on the same straight road, Xiaohua walk 65 meters per minute, Xiaoliang walk 55 meters per minute. 3 minutes after the two people may be how many meters apart?
Example 5: A circular track is 400 meters long, Xiao Qiang runs 300 meters per minute, Xiao Xing runs 250 meters per minute, two people at the same time in the same place in the same direction, after how long Xiao Qiang first caught up with Xiao Xing? (Difference in distance ÷ difference in speed = time to catch up)
1. A and B are running along the track of a sports ground, A runs 290 meters per minute, B runs 270 meters per minute, and a lap of the track is 400 meters long. If they run in the same direction from the starting line at the same time, how long will it take for A to catch up with B for the first time?
2. Bright Elementary School has a 200-meter-long circular track, and Liangliang and Jingjing start from the starting line at the same time. Liangliang runs 6 meters per second and Jingjing runs 4 meters per second. How many meters did each of them run when Liangliang caught up with Jingjing for the first time?
3. A B two people around the circumference of 1000 meters of the circular square race, known as the A walk 125 meters per minute, B's speed is twice as fast as A, now A in the back of the B 250 meters, B catch up with the A need how many minutes?
Example 6. A car from city A to city B, 3 hours after the car breaks down to repair 30 minutes, and then accelerate 5 kilometers per hour to continue to drive, and then after 6 hours on time to city B. How much is the distance between city A and city B? How many kilometers is the distance between city A and B?
1. A car from A to B, 560 meters per minute, is expected to arrive in 30 minutes, but halfway, the car needs to be refueled, with 3 minutes to fill up the gas, if you still need to arrive at the scheduled time, how many kilometers per minute for the rest of the journey must be traveled?
2. from city A to city C to pass through city B, a car from city A to city B 60 kilometers per hour, *** line 420 kilometers from city B to city C accelerated 5 kilometers per hour to city C a *** line for 10 hours, how many kilometers between the two cities AC?
3. AB two ports are 515 kilometers away from each other, A and B two speedboats respectively from AB two ports at the same time, each other and travel. 2 hours later, the A boat has something to return to the A port, and then continue to drive 3 hours after the two boats meet. How many kilometers per hour does boat B travel 50 kilometers per hour, and how many kilometers per hour does boat A travel?
Second, general application problems
Example 1. 300 pairs of sneakers shipped to the department store were packed in two wooden boxes and six cartons. If the same number of sneakers are packed in the same wooden boxes, how many pairs of sneakers are packed in each wooden box and each cardboard box?
1. A toy factory to 630 toys in 5 plastic boxes and 6 cartons, 1 plastic box and 3 cartons of toys in the same number of each plastic box and carton of toys?
2. Xinhua Elementary School bought two tables and five chairs and *** paid 195 yuan. How much does each table cost, knowing that each table costs 4 times as much as each chair?
3. Uncle Wang bought 3 kilograms of lychee and 4 kilograms of cinnamon and *** paid 156 yuan. The price of 5 kilograms of lychee is known to be equal to the price of 2 kilograms of cinnamon. How many dollars per kilogram of lychee and per kilogram of cinnamon?
Example 2: A basket of pears, even the basket weighs 38 kilograms, eaten half of the basket, even the basket has 20 kilograms, ask: how many kilograms of pears and the basket each weigh?
1. A barrel of oil weighs 180 kilograms with the barrel, and after half of the oil is used, there are 100 kilograms left in the barrel, ask: how many kilograms of oil and the barrel weigh each?
2. A basket of apples, even the basket *** weighs 35 kilograms, first take half of the kindergarten children, and then take the remaining half of the first grade children, the remaining apples even the basket weighs 11 kilograms, the basket of apples weighs how many kilograms?
3. An oil drum has some oil, if the oil to the original 2 times, the oil drum even oil weighs 38 kg; if the oil back to the original 4 times, this time and the barrel *** weighs 46 kg, how many kilograms of oil in the original oil drum?
Example 3. There are 6 baskets of pears, each basket of pears equal number, if from each basket out of 40, 6 baskets of pears left in the sum of the number of pears and the original two baskets of pears equal to the number of pears, the original basket of how many?
1. There are 5 boxes of tea, if you take out 200 grams from each box, then 5 boxes of tea left and the original 4 boxes of tea weight is equal to the original box of tea, how many grams of tea per box?
2. There are the same number of oranges in 5 wooden boxes. If 60 oranges are taken out of each crate, then the sum of the number of oranges left in the 5 crates is equal to the sum of the number of oranges in the original two crates. how many oranges were in each crate?
3. A grocery store has 8 boxes of cookies. If 20 kilograms are taken out of each box, then the cookies left in the 8 boxes are equal to the weight of the original 3 boxes of cookies. How many kilograms of cookies were in each box?
Example 4: A television factory receives a batch of production tasks and plans to produce 90 televisions per day, which can be completed on schedule. 5 more televisions are actually produced per day, resulting in the completion of the task 1 day ahead of schedule. How many TV sets *** are in this batch?
1. A wood factory produces a batch of desks. The original plan was to produce 60 desks per day, but the factory actually produced 4 more desks per day than the original plan, and as a result completed the job one day ahead of schedule. How many desks were originally planned to be produced?
2. Xiao Ming reads a storybook and plans to read 12 pages a day, but actually reads 8 more pages a day and finishes 2 days ahead of schedule. How many pages are in this storybook?
3. repair a highway, plan to repair 60 meters per day, the actual daily more than the plan to repair 15 meters, the result is completed 4 days ahead of schedule, a **** repair how many meters?
Example 5.There are two bags of flour, the first bag has 24 kilograms of flour and the second bag has 18 kilograms of flour. How many kilograms are taken out of the first bag and put into the second bag to equalize the weight of the flour in the two bags?
1. There are two boxes of pegs, 72 in box A and 48 in box B. How many pegs can be taken out of box A and put into box B to equalize the pegs in the two boxes?
2. There are two boxes of glass spheres, 72 in box A and 48 in box B. How many times can you take 4 spheres from box A and put them in box B to make the two boxes equal?
3. There are two bags of sugar, one bag of 68 grains, the other bag of 20 grains, each time from the bag of more 6 grains into the bag of less, how many times to make the two bags of sugar the same amount?
Example 6 master and apprentice **** with the addition of ah 840 parts, if the master first plus ah 9 days, and then by the apprentice processing 2 days, the task can be completed; if the apprentice first processing 6 days, and then by the master processing 6 days, the task can also be completed. How many parts are processed by each of the master and apprentice?
1. Master Zhang and Master Li both **** with the processing of 750 parts, if Master Zhang first processing 7 days, and then by Master Li processing 4 days, the task can be completed; if Master Li first processing 5 days, and then by Master Zhang processing 5 days, can also complete the task. How many parts are processed by each of the two masters?
2. A and B two workshops **** with the processing of 540 pieces of clothing, if the A workshop first processing 2 days, and then by the B workshop processing 5 days, the task can be completed; if the A workshop first processing 3 days, and then by the B workshop processing 3 days, can also complete the task. How many pieces of clothing are processed in each of the two workshops?
3. The gas company plans to lay a gas line by 10 workers for 16 days, and after 4 days of work, 10 more people have been added. If the efficiency of each worker per day remains the same, how many days ahead of schedule can the original laying task be completed?
Third, elimination method of solving problems
Example 1. The school bought the first 3 water bottles and 20 teacups, *** with 134 yuan, and the second time to buy the same 3 water bottles and 16 teacups, *** with 118 yuan. What is the unit price of the water bottles and teacups?
1. My brother bought 4 exercise books and 3 pencils for $2.90, and my sister bought the same 4 exercise books and 1 pencil for $2.30. Find the unit price of each exercise book and each pencil.
2. The first purchase of 3 kg of tea and 5 kg of sugar cost 420 yuan, and the second purchase of the same 5 kg of tea and 5 kg of sugar cost 220 yuan more than the first. How many dollars per kilogram of tea and per kilogram of sugar?
3. In order to give out prizes, the first class bought 2 fountain pens and 5 ballpoint pens for 39 yuan, and the second class bought the same 3 fountain pens and 4 ballpoint pens for 48 yuan. How much more expensive is a fountain pen than a ballpoint pen?
Example 2. It costs $480 to buy 3 basketballs and 5 soccer balls*** and $519 to buy the same 6 basketballs and 3 soccer balls***. What is the unit price of each of the basketballs and soccer balls?
1. Bright Elementary School bought 2 tables and 5 chairs and *** paid $110; Yucai Elementary School bought the same 6 tables and 6 chairs and *** paid $240. How many dollars for each table and each chair?
2. A store a day **** sold apples and oranges 660 kilograms, known in the morning sold 3 boxes of apples and 2 boxes of oranges, **** 180 kilograms; afternoon sold 4 boxes of apples and 8 boxes of oranges. How many kilograms do each crate of apples and each crate of oranges weigh?
3. 3 cows and 6 sheep eat 93 kilograms of grass a day***, and 6 cows and 5 sheep eat 130 kilograms of grass a day***. How many kilograms per cow per day on average eat more than per sheep per day on average?
Example 3. 4 packs of science and technology books and 6 packs of story books*** 560 books, and the same 3 packs of science and technology books and 5 packs of story books*** 450 books, how many copies of each of the two types of books are in each pack?
1.5 kg of fruit candies and 2 kg of cookies **** 76 dollars, the same 3 kg of fruit candies are 20 dollars more expensive than 2 kg of cookies. How many dollars per kilogram of fruit candy and per kilogram of cookies?
2. Students go boating in the park. 4 big boats and 3 small boats*** sit 29 people, and 4 big boats sit 11 more people than 3 small boats. How many people can sit in 1 big boat and 1 small boat?
3. 3 towels and 2 sheets*** cost $154, and the same 3 towels cost $171 less than 3 sheets. Find how much each of 1 towel and 1 bed sheet costs?
Example 5. 7 cows and 5 horses eat 155 kg of grass per day, and 5 cows and 7 horses eat 145 kg of grass per day. How many kilograms of each are eaten by each cow and each horse per day?
1. 4 walk-behind tractors and 5 threshing machines total $17,000, and 5 walk-behind tractors and 4 threshing machines total $19,000. How many dollars is a walk-behind tractor? How many dollars for one thresher?
2. canteen first brought in 5 bags of rice, 7 bags of flour, ****1350 kg; the second brought in 7 bags of rice, 5 bags of flour, ****1650 kg. How many kilograms of rice per bag and flour per bag?
3. Xiaohong to the store to buy prizes, buy 12 pens, stationery box 15 *** need to pay 300 yuan, Xiaohong wrongly bought into the same pen 15, stationery box 12, the result is more than 21 yuan. How much is each pen and each stationery box?
Example 6: The school sports room once to buy basketballs, soccer balls and volleyballs each two *** with the expenditure of 340 yuan, the second time to buy 4 basketballs, 3 soccer balls and 2 volleyballs *** with 495 yuan, the third time to buy 5 basketballs, 4 soccer balls and 2 volleyballs *** with the expenditure of 605 yuan. How much is each of the basketballs, soccer balls and volleyballs?
1. Mr. Wang bought 3 clips, 3 boxes of paints and 3 brushes*** for the first time for 195 yuan, 3 clips, 4 boxes of paints and 3 brushes*** for the second time for 213 yuan, and 4 clips, 5 boxes of paints and 3 brushes*** for the third time for 266 yuan. What is the unit price of the clips, paints and brushes?
2. A group of friends to go boating, known 2 small, 3 medium and 1 large boat *** sit 23 people, 3 small, 4 medium and 2 large boat *** sit 35 people, 1 small, 2 medium and 3 large boat *** sit 26 people, ask: how many people sit in each small, medium and large boat?
3. Xiaoming bought 2 pens, 1 ballpoint pen and 2 automatic pencils *** with 26 yuan, Xiaohua bought 1 pen, 3 ballpoint pens and 4 automatic pencils also used 26 yuan, known as the price of 1 ballpoint pen is exactly the price of 4 automatic pencils. How much is each of the pen, ballpoint pen and automatic pencils?
Example 7: Auntie Zhou bought a coat, a pair of pants and a pair of shoes at a department store for 904 yuan. It is known that the coat cost 312 yuan more than the pants, and the coat and pants a*** cost 392 yuan more than the shoes. Do you know how much each of the coat, pants and leather shoes that Auntie Zhou bought cost?
1. Mao Qi bought a stationery box, a pen and a box of watercolors for 44 yuan at Xinhua Stationery Store. It is known that the stationery box is 6 yuan more expensive than the watercolor pencils, and the stationery box and the watercolor pencils a **** more expensive than the fountain pen 6 yuan. Do you know the unit price of each of the stationery box, pen and watercolor pencil?
2. Wang spent 58 yuan to buy some pears, apples and bananas to visit the teacher, known three kinds of fruit *** weighs 2290 grams, pears than bananas less than 180 grams, pears and bananas a **** than the apples weigh 770 grams. Do you know how many grams of pears, apples and bananas that Xiao Wang bought weigh respectively?
3. Mom bought two bottles of "Seafoam" shampoo and three "Rex" soap **** 48 yuan, pay, mom and two soaps into the same bottle of shampoo, the cashier will let her 10 yuan. Please do the math, "Seafoam" shampoo and "Rex" soap unit price of how many yuan?
Four, the average problem
Example 1. math interest group held a test, four (1) class of eight students scores are 82, 75, 95, 98, 100, 80, 87, 79. find the average score of the eight students?
1. There are 8 old people in a home for the aged, their ages are 78, 76, 77, 81, 78, 76, 80, 78. Find the average age of these 8 old people.
2. The weekly temperatures measured by a meteorological team at 8:00 a.m. each day are as follows: 13, 13, 13, 14, 15, 14, 16. Find the average temperature for the week.
3. 7 students from class VI(3) participated in a math competition. Two of them scored 99, three others scored 96, and two others scored 97 and 89. What is the average score of these seven students?
Example 2.A bicycle factory produces 4500 bicycles in the first 10 days of April*** and 8400 bicycles in the last 20 days***. What is the average number of bicycles produced per day for the month?
1.Ming took a math test and scored a total of 170 on the first two tests and 270 on the last three. what was Ming's average score on the five tests?
2. Two (1) class students planted trees in three groups, the first group of eight people, **** planted 80 trees, the second group of six people **** planted 66 trees, the third group of six people, **** planted 54 trees. On average, how many trees did each person plant?
3. Handicraft class, three (4) class students in three groups to do flowers, the first group of eight people, an average of 10 flowers per person, the second group of six people, an average of 11 flowers per person, the third group of six people, an average of 9 flowers per person, three (4) class average of how many flowers per person?
Example 3. Xiao Wang for mountain climbing exercise, up the mountain he a **** walked 1200 meters, 30 meters per minute, down the mountain he returned along the original path, 60 meters per minute, find the average speed of Xiao Wang up and down the mountain.
Small king to climb the mountain exercise, up the mountain he walked 30 meters per minute, down the mountain he returned along the original path, walking 60 meters per minute, find the average speed of small king up and down the mountain.
1. The road from the top of the mountain to the foot of the mountain is 36 kilometers long, a car to go up the mountain, it takes 4 hours to reach the top of the mountain, down the mountain along the same way back to the foot of the mountain in only 2 hours. Find the average speed of this automobile for the round trip.
2. Xiao Qiang goes from home to school in the morning, walking 120 meters per minute and arriving in 10 minutes. At noon, he goes home along the same road, walking 80 meters per minute, find the average speed of Xiao Qiang's round trip to school.
3. Xiao Liang climbs a mountain at a rate of 2 kilometers per hour, and then descends immediately after reaching the top at a rate of 6 kilometers per hour. What is the average speed at which Xiao Liang goes up and down the mountain?
Example 4: Xiaojun took part in 3 math contests and his average score was 84. Knowing that the average score of the first two contests was 82, how many points did he get in the third contest?
1. Li Hua participated in the sports standardized test, the average score of the five items is 85 points, if the throwing score is not counted, the average score is 83 points, how many points Li Hua scored in throwing?
2. Xiaoli's average score for her four subjects was 92 before the math scores for the final exam were released, and his average score dropped by 1 point after the math scores were released. How many points did Xiaoli score on the math test?
3. There are seven numbers lined up in a column and their average is 31, 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 there is no one younger than 18, how old could the oldest person be?
1. If the average age of four people is 23 and none of the four is younger than 18, what is the age of the oldest person likely to be?
2. If the average age of the four people is 29 and there is no one older than 32, what is the age of the youngest person likely to be?
3. If the average age of four people is 25, and there is no one younger than 16 among the four, and the ages of these four people are not equal to each other, what is the age of the oldest person likely to be?
Example 6: Li Yun's average score on his final exam in six subjects: language, foreign language, nature, physical education, music and art is 85. The math score is 12 points higher than the average score of the seven subjects. How many points did Li Yun get in math?
1.Xiaoming's average score on the last semester's final exam in language, math, music, physical education, and art was 89, and his English score was 5 points higher than the average score on the six subjects, how many points did he get on the English exam?
2. On a foreign language test, A, B and C scored 87, 75 and 93 respectively. D's foreign language score is 6 points higher than the average score of A, B, C and D. What is D's score in the foreign language? What is the average score of the four?
3. A class of four (1) is having a rowing competition. The teacher divides these students into three groups A, B and C. Twelve boats were rented, and the cost of renting them was divided equally among the three groups. group A paid for seven boats in advance, group B paid the rest of the rent, and group C didn't pay for the time being. At the end of the rowing, the teacher settles the bill, and group C is due to pay 12 boat rentals. Ask how many dollars is due from group C to each of groups A and B?
V. Overlapping Problems
Example 1. 122 students in the fifth grade took the language and math exams, each of whom got excellent grades in at least one subject. Of these, 65 were excellent in language and 87 were excellent in math. How many students excelled in both language and math?
1. There are 48 students in a class, and the homeroom teacher asks at the class meeting, "Who has finished their language homework? Please raise your hand!" Thirty-seven people raised their hands, and then asked, "Who has finished their math homework? Please raise your hand!" Forty-two raised their hands. Finally, the question was asked, "Who has not finished their language and math homework?" No one raises their hand. Find the number of people in this class who have completed both their language and math homework.
2. There are 54 students in a fourth-grade class, and 13 of them subscribe to both Outstanding Essays for Elementary School Students and The Great World of Math. 45 of them subscribe to Outstanding Essays for Elementary School Students, and each of them subscribes to at least one kind of reading material. how many of them subscribe to The Great World of Math?
3. Each person in the school literary group will play at least one musical instrument, it is known that there are 24 people who can play the accordion, 17 people who can play the electronic organ, of which 8 people can play both instruments. How many people are in this group?
Example 2.There are 40 students in a class of five (1), of whom 25 are in the math group, 23 are in the science and technology group, and 19 are in both groups. So, how many individuals did not participate in both groups?
1.A class of 36 students took a test in which 25 answered the first question correctly, 23 answered the second question correctly, and 15 answered both questions correctly. How many answered both questions incorrectly?
2. In a class of 55 students, 32 subscribe to the Elementary School Math News, 29 subscribe to the China Youth News, and 25 subscribe to both newspapers. How many of them do not subscribe to either newspaper?
3. A school elected 50 students to participate in the district essay contest and math contest, the results of the two contests 3 people won, 27 people did not win both contests, known as the essay contest won 14 people, ask how many people won the math contest?
Example 3: There are 36 people in a travel agency, of whom 24 know Japanese, 18 know Russian, and 4 know neither.
1. There are 56 people in a class. 28 people participated in the language competition and 27 people participated in the math competition. 25 people did not participate in either of the two subjects if they did not participate in either of the two subjects. how many people participated in both the language and math competitions?
2. There are 103 people in a club, 69 of whom can play Chinese chess, 52 of whom can play chess, and 12 of whom can play neither. How many people know how to play both kinds of chess?
3. There are 40 students in a third-grade class who participate in the chorus team, 20 in the dance team, 14 in both the chorus team and the dance team, and 10 in neither team. Do the math, how many people are in this class***?
Example 4.Of all the natural numbers from 1 to 200, how many are neither multiples of 5 nor multiples of 8?
1.How many numbers are there among the natural numbers from 1 to 100 that are neither a multiple of 5 nor a multiple of 6?
2. Of all the natural numbers from 1 to 130, how many are neither multiples of 4 nor multiples of 9?
3. A class of five (1) is doing a radio drill and the class is organized into four rows, with an equal number of people in each row. Xiaohua is the 5th from the front and the 8th from the back. How many students are there in this class***?