Honghua shirt factory wants to produce a batch of shirts. It originally planned to produce 400 shirts a day and finish them in 60 days. The actual number of pieces produced per day is 65438+ 0.5 times of the original planned number of pieces produced per day. How many days did it actually take to complete the task of making these shirts? To analyze and understand how many days it takes to complete the task of making these shirts, we must know the total number of these shirts and the actual number of them produced every day. Knowing that the original plan was to produce 400 pieces a day and complete them in 60 days, we can find out the total number of these shirts; Knowing that the number of pieces actually produced every day is 65438+ 0.5 times of the original planned number, we can find out the number of pieces actually produced every day. The actual number of days to complete this batch of shirts is 400× 60 ÷ (400×1.5) = 24,000 ÷ 600 = 40 (days). It can also be considered that the total number of shirts to be produced is certain, so the number of days required to complete this batch of shirts is inversely proportional to the number of shirts produced every day. It can be concluded that the number of days to actually finish the task of making these shirts is 1.5 times, which is exactly 60 days, so it is concluded that the number of days to actually make these shirts is 60÷ 1.5 = 40 (days). A: It actually took 40 days to complete the task of making these shirts. Example 2: Dongfeng Machinery Factory originally planned to produce 240 parts per day, which was completed in 18 days. It was actually finished three days ahead of schedule. How many more parts are actually produced every day than originally planned? To analyze and find out how many parts are actually produced every day than originally planned, we must first find out how many parts are actually produced every day, and then subtract the number of parts planned to be produced every day: 240×18 ÷ (18-3)-240 = 4320 ÷15-240 = According to the meaning of inverse proportion, the number of parts produced every day is inversely proportional to the number of days required to complete the production of these parts. Therefore, the ratio of the number of days originally planned to complete the task to the number of days actually completed the task is 18: (18-3), that is, 6: 5, that is, the ratio of the number of parts actually produced every day to the number of parts originally planned to produce every day. Of course, the actual number of parts produced every day is 6/5 of the original planned number of parts produced every day. So it is found that the number of parts actually produced every day =48 (pieces). It can also be considered that the total number of parts produced is 240× 18=4320 (pieces); This number is decomposed into prime factors, and then the decomposed prime factors are appropriately grouped to represent the product of the original planned daily production number and the number of completed days and the product of the actual daily production number and the actual number of completed days respectively. 4320 = 25× 33× 5 = (24× 3× 5 )× (2× 32) ... The product of the original planned daily output and the days of completion = (25× 32 )× (3× 5) ... The product of the actual daily output and the days of completion, plus the number of days when the actual daily output is more than the original planned daily output, is: 25×. Example 3: In the "Creation Cup" exhibition of Chunguang Primary School, 36 items are not in the sixth grade, 37 items are not in the fifth grade, and 45 items are known to be in the fifth and sixth grades. So how many exhibits are there in grades five and six? Analysis and solutions According to the known information, 36 items are not in Grade VI, that is to say, there are 36 items 1 ~ 4-level exhibits plus five-level exhibits. There are 37 items that are not in the fifth grade, that is to say, there are 37 items 1 ~ 4 grade exhibits plus 6 grade exhibits. Comparing the above two situations, it can be concluded that there are 37-36= 1 exhibit in Grade 6 compared with Grade 5. I also know that there are 45 exhibits in grade five and six, so I found that the exhibit in grade five is (45- 1)÷2=44÷2=22 (pieces), and the exhibit in grade six is (45+ 1) ÷ 2 = 46 ÷. Example 4: Parts Processing Group of Machinery Factory 1 Master, 6 apprentices, 7 people. Each apprentice can process 50 parts every day, and the number of parts processed by the master every day is 24 more than the average of 7 people in the whole group every day. How many parts does the master process every day? The number of parts processed by the analysis and solution master every day is 24 more than the average number of parts processed by the whole group of 7 people every day. Divide these 24 apprentices into 6 apprentices on average, and add 50 apprentices to process them every day, which is exactly the average number of people processing them every day. This number plus 24 is the number of parts processed by the master every day. 24÷6+50+24 =4+50+24 =54+24 =78 (pieces) A: The master processes 78 pieces every day. Example 5: The children's clothing factory produces red coats and yellow coats. Each red coat needs 2 buttons, and each yellow coat needs 4 buttons. Made of coats of two colors, 30 pieces are packed in a box, and each box of clothes needs 72 buttons. How many red coats and yellow coats are there in each box? Analysis and solution It is known that each yellow coat needs 4 buttons and each red coat needs 2 buttons. If you divide the yellow coat in two, it becomes a "semi-yellow coat". At this time, both the red coat and the "half yellow coat" need two buttons. It is known that each box of two-color shirts has 72 buttons, so it can be found that there are 72÷2=36 red shirts and "semi-yellow shirts". In fact, there are 30 pieces of two colors in each box, and 36 pieces are 6 pieces more than 30 pieces, which means that 6 pieces are split in two, so there are 6 pieces in each box. Furthermore, the number of red shirts per box is 30-6=24 (pieces), and the formula is: 72÷2-30=36-30=6 (pieces) 30-6=24 (pieces). You can also think of it this way: remove two buttons from each of the 30 shirts in each box, so that there are no buttons on the red shirt. At this time, there are 72-60= 12 buttons left on the coat in the box. Because there are only two buttons left on each yellow coat, 12÷2=6 (pieces) is the number of yellow coats per box. Then, the number of red shirts per box is 30-6=24 (pieces). The formula is: (72-2× 30) ÷ (4-2) = (72-60) ÷ 2 =12 ÷ 2 = 6 (pieces) 30-6=24 (pieces) A: There are 24 red shirts in each box. There are apples and peaches in the master's basket. There are three times as many apples as peaches. A group of naughty little monkeys, while the owner was not looking, each little monkey took 8 apples and 3 peaches. When the owner found out, the peach had been taken away by the little monkey, leaving 10 apples. How many naughty little monkeys are there? Analysis and Solution The number of apples in the basket is three times that of peaches. Each little monkey takes three peaches and takes them all. Then if each little monkey takes nine apples, he can also take all the apples (because the number of apples is exactly three times that of peaches). However, each little monkey only took eight apples, leaving 10 apples, which just shows that the little monkey has 10. A: There are 10 naughty monkeys. Example 7: bright primary school originally planned to burn 9 1800kg coal in 92 days. If the coal saved by the original scheme can be burned for a few more days every day, it is necessary to know how much coal is saved by a * *, and the amount of coal burned every day after saving. How many kilograms of coal has a * * * saved? How many days can the saved coal burn? 5400÷450= 12 (days) You can also think like this: 17 units, then the actual daily coal consumption is 1 unit, and the actual daily coal consumption is 16 units. Originally planned to burn coal 192 days, one * * can save coal 192 units, and can burn: 192÷ 16 = 12 (days). Answer: The saved coal can be burned again 12. Example 8: There are 1993 people and flour 1993 kg. 1 took all the flour 1/2, the second person took the remaining flour 1/3, the third person took the remaining 1/4, ... the remaining 1992. So 1993 How many Jin of flour did this man take? It is not appropriate to analyze and solve this problem by step-by-step calculation. 1.993 Jin of flour was taken away by 1/2, and the rest was of course 1/2. This is a decimal, and it is more complicated how many kilograms of flour are left after the second person takes it away. Therefore, considering the whole, it is much easier to solve it comprehensively. Answer: 1993 people took 1 kg flour. Example 9: I bought a batch of flour in the canteen. On the first day, I ate the total amount of flour, the next day I ate the remaining amount of flour, and every day for the next seven days. Finally, on the tenth day, I ate 4 bags and just finished eating. How many bags of flour are there in this batch? According to the meaning of the problem, the 1 pile was uniform from the day 10 to the ninth day, after the noodles were finished on the day 1. Therefore, this batch of flour * * * has 4× 10=40 (bags). A: This batch of flour was originally * * *. There are two containers. The first container contains 1 liter of water, and the second container is empty. Pour 1/2 of the water in the first container into the second container, then pour 1/3 of the water in the second container back into the first container, and then pour 1/4 of the water in the first container into the second container, and so on, pour 1993. It is not difficult to see from the above table that the water in the first container is 1, 3, 5, ... poured after the odd number. Of course, after pouring 1993 times, the water in the first container is also 1/2 liters. It can also be calculated continuously: for example, 1 1, the kindergarten children celebrate Children's Day on June 1st, and the aunt gives the children three apples at first. Results 15 people only gave two; Later I bought 40 apples and distributed them to the children. As a result, I got four apples each. How many children are there in kindergarten? Analysis and problem solving tell us that at first, each child was divided into three parts, and as a result, 15 children were only divided into two parts, that is to say, each child was divided into three parts, and 15 apples were missing. Later I bought 40 apples for the children. As a result, everyone got four apples. Take out 15 of these 40 apples and give it to the children who only gave two apples at first. At this time, there are still 25 apples left, and each person gives 1, which is exactly 4 apples. So it is concluded that there are 25 children in kindergarten. (40-15) ÷ (4-3) = 25 ÷1= 25 (person) A: Kindergarten 1 * * has 25 children. Example 12. A box full of solid balls weighs * * * 12kg. After taking out the solid ball of 1/4 from the box, the remaining solid ball weighs 9.5 kg. How much does this box weigh? Analysis and solution A box is filled with solid balls, and the box weighs * * *12 kg; After taking out 1/4 solid balls from the box, the remaining 3/4 solid balls weigh 9.5 kg. It can be concluded that the weight of the solid ball is 1/4 (12-9.5) kg, so the total weight of the solid ball is = 10 (kg) and the box weight is:12-10 = 2 (kg). Example 13: Record the depth with rope. Fold the rope into three strands and leave 1 m outside the well; Fold the rope into four strands to measure, there is more than one meter outside the well. How deep is this well? Analysis and solution take the total length of the rope as "1" and fold the rope into three strands, that is, use1/3 of the rope length; When the rope is folded into four strands, it is measured by the rope length of 1/4. The difference of rope length outside the well is the difference of rope length 1/3 and rope length 1/4. So the total length of the rope is: you can also think of it this way: it is exactly the length of the rope. Exactly the length of the rope. The depth of the well is good. So the depth of the well is: for example 14, students engage in camping activities. A classmate went to ask the teacher in charge of logistics for a bowl. The teacher asked him how much he had taken in the exam, and he said 55. Ask again, "How many people eat?" He said, "One person has 1 rice bowl, two people have 1 vegetable bowl and three people have 1 soup bowl." Please count how many bowls this classmate took to the camping activities. Analysis and solution: first calculate how many bowls 1 person needs on average, and then calculate how many people need 55 bowls. The formula can also be answered in this way: when eating, everyone has 1 rice bowls, and how many rice bowls are needed means how many people take part in camping activities. The title also said that two people 1 vegetable bowl, three people 1 soup bowl. As we know, the least common multiple of 2 and 3 is 6, that is to say, when six people eat, they need six rice bowls, three vegetable bowls and two soup bowls. So it is concluded that when six people eat, * * * needs 6+3+2= 1 1 bowl. So we divide the people who take part in camping activities into groups of 6 people, and each group needs to eat with 1 1 bowl. From 55÷ 1 1=5, we can know that people who eat are divided into five groups, so we can find that this classmate should be the leader bowl of 6×5=30 people. A: This classmate brought bowls to 30 people who took part in camping activities. Example 15. The age of the son is the age of the mother, and the age of the father is two years older than the mother. How old is father? How old is mom? How old is the son? At that time, my father was older than my mother 1 year. The title tells us that my father is two years older than my mother, so we can see that my mother is 40 years old. My father replied: my father is 42 years old, my mother is 40 years old, and my son is 12 years old. There are some boys and some girls in the classroom. The teacher asked them the number. A boy told us in the old analysis and problem solving that the number of boys is the number of girls except 1, and the number of girls is 3/5 of the number of boys except 1. Shows the number of girls, excluding 1 girls, which is exactly 9 girls. The denominator of 15 just represents the number of boys. Excluding 1 boys is exactly 14 boys. It is concluded that there are 15 boys and 10 girls in the classroom. A: Boys in the classroom 15, girls 10. Example 17. There are some books in the bookstore. On the first day, 1/2 were all sold, and on the second day, 900 books were entered. On the third day, 40 books were sold more than the existing books, leaving 800 books. How many books are there in the bookstore? Analysis and Solution According to the conditions given in the question, we can go back and find out how many books there are in the bookstore. Assuming that the books sold on the third day are 40 fewer than the existing books (that is, 40 fewer books are sold), we can find out how many books were in the bookstore before the books were sold on the third day. Suppose 900 books were not delivered the next day, and the books in the bookstore were sold out on the first day. Check the number of original books in the bookstore. A: There are 720 books in the bookstore. Example: 18, 7 bags of rice, weighing 12kg, 15kg, 17kg, 20kg, 22kg, 24kg and 26kg respectively. A takes one bag first, and B, C and D take the rest. It is known that B and C take the same weight, which is twice as much as D, so what is the weight of the bag that A takes first? Analysis and solution tell us that A takes one package first, and then B, C and D take the rest. As is known to all, B and C weigh as much as D, so B, C and D weigh five times as much as D.. The total weight of 7 bags of rice is12+15+17+20+22+24+26 = 136 (kg) minus 5 times of136, and the rest is the weight of A ... In order to divide a number in 136 by 5, one digit of this number must be 1 or 6. The weight of the seven bags of rice listed in the title is only 26 kilograms, and the unit number is 6, so the weight of the bag of rice that A took first is 26 kilograms. A: The weight of the bag of rice that A took first was 26 kilograms. Example 19: There are several stacks of Weiqi pieces, and the number of Weiqi pieces in each stack is the same, with 28% of albinos in each stack. Yao Ming took half of the pieces from the first pile, all black pieces. Now, among all the chess pieces, white chess pieces account for 32%. So how many piles of Go are there? According to the meaning of the question, the number of white pieces has not changed before and after the pieces were explicitly taken away. Because the black pieces were removed, the total number of pieces changed, so the percentage of white pieces in the total number of pieces also changed. The original white parts accounted for 28%, and later accounted for 32%. So, A: It turns out that * * * has four piles of Go. On Arbor Day, the school distributed a batch of saplings to some students in grades three to six for planting. If some students in grade three plant alone, they will plant 6 plants per person on average; If some fourth-grade students plant alone, each person will plant 12 trees on average; If some fifth-grade students plant alone, they will plant an average of 20 trees per person; If some sixth-grade students plant alone, an average of 30 trees will be planted. At present, there are some students in grade 3456 who are going to plant them. How many trees does each person plant? No matter what level of trees are planted, the total number of saplings is certain. Assuming that all the saplings to be planted will plant trees, the average number of trees per race can be thought of as follows: According to the average number of trees planted by each person in grades 3-6 given in the question, it can be inferred that the total number of trees planted must be the common multiple of the four numbers 6, 12, 20 and 30. The least common multiple of these four numbers is 60. Assuming that 60 trees are to be planted, it is not difficult to calculate that the number of students in grades 3-6 is 10, 5, 3 and 2 respectively. Then when some students in grades 3-6 plant trees, the average number of trees planted per person is: when students in grades A:3, 4, 5 and 64 plant trees, the average number of trees planted per person is 3. Example 2 1, a project, if A did it alone for 12 days, and then B did it alone for 9 days, it was just completed; If B does it for 2 1 day and A does it for 8 days, it's just over. If A does this project alone, how many days can it be completed? See Figure 49 for the conditions given in the analysis and problem solving. It is not difficult to see from Figure 49 that Party A needs 12-8=4 (days) and Party B needs 2 1-9= 12 (days) to complete the same workload, so it is found that the time ratio between Party A and Party B is 4∶2. Therefore, it is necessary for Party A to complete the project alone: the project will be completed by Party A alone within 15 days. A pool can be filled with water from two water pipes, A and B. It takes 10 hour to fill the empty pool when the pipe is opened; It takes 20 hours to fill the empty pool by opening the B tube alone. Now it is required to fill the empty pool in 8 hours, and the time to open pipes A and B should be as little as possible. How many hours will it take for pipes A and B to close? Analysis and solution: Because the first pipe injects water quickly, the first pipe should always be open, and the time for opening the second pipe to inject water into the empty pool in 8 hours is: that is, the shortest time for opening the first pipe and the second pipe is 4 hours. You can also think of it this way: because the nail tube is filled with water faster, it should always be open. Because it only takes 10 hour to fill the empty pool, it only takes 8 hours to open the nail tube and 2 hours to fill the empty pool. It is known that it takes 10 hour to open the first pipe, and it takes 20 hours to open the second pipe to fill the pool. Therefore, the water volume of the first pipe for 2 hours is the water volume of the second pipe for 4 hours, that is, the second pipe needs to be opened for 4 hours, that is, the shortest time for opening the two pipes is 4 hours. A: It takes at least 4 hours for two pipes to open together. Example 23: A project can be completed in 20 days by itself; B one person can do it in 30 days. Now Party A and Party B do it together, because Party B took a few days off on the road and it took 14 days to complete the task. So how many days did B rest on the way? Analysis and problem solving tell us that it took 14 days for B to complete the task, because B had a few days off in the whole project. Assuming that there is no rest on the way to B, then 14 days of both parties will exceed the whole project quantity, and the excess part happens to be that Party B didn't do it because of rest, so the number of days of rest on the way to B =5 (days). A: Party B rested for five days on the way. Example 24: A project, which is jointly completed by the three teams of Party A, Party B and Party C, takes 8 days to complete. It is known that the daily work efficiency of Team A is equal to the sum of the daily work efficiency of Team B and Team C, and the daily work efficiency of Team C is equal to the sum of the daily work efficiency of Team A and Team B 1/5, so how many days will it take for Team B to complete this project alone? Analysis and problem solving tell us that the daily work efficiency of Team A is equal to the sum of the daily work efficiency of Team B and Team C, and that the daily work efficiency of Team C is equal to the sum of the daily work efficiency of Team A and Team B. It also tells us that if Team A, Team B and Team C work together, the project can be completed in eight days, while the daily work efficiency of Team A is equal to the sum of the daily work efficiency of Team B and Team C. So if this project is completed by Team A alone, it is concluded that Party B will complete this project alone. A: If this project is completed by Party B alone, it will take 24 days to complete. Example 25: A project will take 12 days to complete if the first, second and third teams cooperate; If the first, third and fifth teams work together, it will take 7 days to complete; If the second, fourth and fifth teams work together, it will take 8 days to complete; If the first, third and fourth teams work together, it will take 42 days to complete. Now these five teams are working together on this project. How many days will it take to finish it? When these five teams need to work together, analyze and understand the number of days required to complete the project. First, seek the sum of the work efficiency of these five teams. Let the work efficiency of these five teams be A, B, C, D and E respectively. According to the known results, the above four formulas can be added to get 3(A+B+C+D+E)= 1/2, so A+B+C+D+E= 1/6. Therefore, the first, second, third, fourth and fifth teams should cooperate on this project. A normally open drain pipe should be used at the bottom of the pool, and several water inlet pipes with the same thickness should be used at the top. Four water inlet pipes are opened, and it takes 5 hours to fill a pool of water; It takes 15 hours to open two water inlet pipes and fill a pool with water. How many inlet pipes do you need to open to fill a pool of water in 2 hours? Analysis and Solution Assuming that the water intake of each water inlet pipe is 1 per hour, then open four water inlets and the water intake for five hours is 4×5=20. Open two water inlet pipes, and the water inflow for 15 hours is 2× 15=30. Comparing the above results, it is not difficult to find that the hourly flow rate of the drain pipe is (30-20)÷( 15-5)= 1, and the water volume of the full pond is 20- 1×5= 15 or 30-/kloc-0. At least the water inlet pipes to be opened are: (15+ 1×2)÷2=8.5≈9 (pieces) A: At least 9 water inlet pipes should be opened. Example 27: Two people, A and B, leave from A to B at the same time along the same road. The speed of A will never change, but when A walks the first 1/5 distance between AB, the speed of B is twice that of A, and after walking, AB has less time, so A arrives at B first. A: A gets to B first. From city A to city B, Party A needs 2 hours, and Party B needs 1 hour and 40 minutes ... If A leads 10 minutes, how many minutes after B leaves, where will it catch up with A? Analysis and Solution It is known that from City A to City B, it takes 60×2-(60+40)=20 minutes for A to spend more than B. That is to say, if A leaves 20 minutes earlier than B, two people can arrive at City B at the same time. Now Party A leaves earlier than Party B 10 minute, which means that Party A leaves earlier than Party B 10 minute, so both of them will reach the midpoint of the two cities. It takes 50 minutes for B to reach the midpoint of two cities, that is to say, it takes 50 minutes for B to set out and catch up with A at the midpoint of A and B. Answer: 50 minutes after B sets out, catch up with A at the midpoint of two cities. Example 29: A bus and a truck set off from A and B at the same time. When the bus traveled 3/5 of the way between A and B, it happened to meet the truck ... After the encounter, the truck still traveled to a place at the original speed of 40 kilometers per hour, and it took 18 hours to reach a place. Find the speed of the bus. If the speed of the bus is needed in the analysis and solution, then we first require the distance traveled by the bus and the time required to travel this distance. In the title, it is known that passenger cars and trucks travel in opposite directions from A and B at the same time. The bus meets the truck in 3/5 of the whole journey between A and B, and the truck travels 2/5 of the whole journey between A and B. The truck still travels at the original speed (40 kilometers per hour) for 18 hours before it reaches A, that is, it takes 18 hours to travel 3/5 of the whole journey. Therefore, the distance between A and B is = 1, 200 (km). The truck travels 40 kilometers per hour, which is 2/5 of the whole journey and 3 of the bus journey. =480÷40 = 12 (hours) =720÷ 12 =60 (kilometers). According to the fact that a truck needs 18 hours for the whole journey, it can be concluded that it needs several hours for the whole journey. Therefore, the speed of the bus is 40× 1.5=60 (km). It can also be considered that the bus and the truck start from A and B at the same time, and the driving time is the same, so the distance ratio of the bus and the truck is the speed ratio of the bus and the truck. So the speed of the bus is: A: The speed of the bus is 60 kilometers per hour. Example 30: It takes a car 13.5 hours to transport a batch of goods from Jiangcheng to Xianghai and a batch of goods from Xianghai to Jiangcheng. It takes 1.25 times as long as going back, and the speed of going back is 6 kilometers slower than coming back. How many kilometers did this car run back and forth? Analysis and solution It is known that the round trip time of this car is 13.5 hours, and the time to return is 1.25 times, that is, the round trip time ratio is 1.25: 1, that is, 5∶4. Obviously, the time to go is 7.5 hours, because the round-trip distance is equal and the round-trip time ratio is 5: 4, so the round-trip speed ratio is 4: 5. It is known that the speed of the journey is 6 kilometers slower than that of the return journey, so we can find out that the speed of the journey is 6÷(5-4)×4 =6÷ 1×4 =24 (km), and then we can find out the round-trip distance of the car. This car has traveled 24×7.5×2= 360 (km). Answer: This car has traveled 360 kilometers.