Topic 1: a green vegetable is produced in a certain place and sold directly in the market, with a profit of 1000 yuan per ton; After rough machining, the profit per ton can reach 4500 yuan; After fine processing and sales, the profit per ton rose to 7500 yuan. A local agricultural and industrial company harvested 140 tons of this vegetable. The processing capacity of this company is: if vegetables are roughly processed, they can process16t every day; If fine processing is carried out, 6t can be processed every day, but the two processing methods cannot be carried out at the same time. Due to seasonal conditions, the company must sell or process all vegetables within fifteen days. To this end, the company has formulated three schemes: scheme one: rough machining all vegetables; Option 2: Refine the vegetables as much as possible, and sell the unprocessed vegetables directly in the market; Scheme 3: Sort out some vegetables and roughly process the remaining vegetables in exactly 15 days. How much profit can you make by processing vegetables with these three schemes? Which scheme is the most profitable? Question 2: There are 10 vegetable farmers, and each vegetable farmer can grow 3 hectares of vegetable A or 2 hectares of vegetable B. It is known that vegetable A can earn 0.5 million yuan per hectare and vegetable B can earn 0.8 million yuan per hectare. If the total income is not less than 1.56 million yuan, how many people can you arrange to grow vegetables at most? Question 3: Take any point A on a straight line, the intercept AB= 12cm, and then the intercept AC=38cm, where DE is the midpoint of AB and AC respectively, and find the distance from D to E 1, and the scheme1:16 = 140 total roughing profit = 4,500 *140 = 630,000. Scheme 2: 6 * 15 = 90 {buy a bottle of 30 yuan for Huamei shampoo, get one for five bottles, get two for eight bottles, and get one for five bottles. How much is a bottle? Mom and her colleagues bought 12 bottles in partnership. How to get a good deal? A factory has made a production plan of 20 1 1 year. The available data are as follows: (1) 400 workers; (2) The annual working time of each worker is 1 100 hours. The estimated annual sales volume is 8-654.38 million cases. Each box will be produced for 2 hours, and the materials used will be 10 kg. At present, the stock is 300 tons, and 900 tons can be replenished by the end of the year. According to the data, determine the annual output and the number of workers. Solution: 1 The available working hours resources of this factory are: 400 x1100 = 440,000 hours. 2. Available material resources are 300+900= 1200 tons =1200,000 kg. 3.38+0) When the working time is 160000-200000, the required number of workers is 146- 182 (2) Material: 800000- 1000000 kg, so the maximum forecast year is adopted. Answer: It can be determined that the annual output is 100000 boxes and the number of workers is 182. Example: 1: Unload several cases from the freighter with a total weight of 10 ton, and the weight of each case shall not exceed 1 ton. In order to ensure that these boxes can be transported at one time, how many cars with a load of 3 tons are needed at least? [Analysis and Solution] Because the weight of each box is not more than 1 ton, the weight of each box that a car can transport is not less than 2 tons, otherwise another box can be put. So five cars are enough, but four cars may not be able to carry all the boxes away. For example, if there are 13 boxes, then each car can only transport 3 boxes, and 13 boxes cannot be transported by 4 cars at a time. So in order to ensure that all the boxes can be transported at one time, at least five cars are needed. Example 2: Intercept 100 short bamboo poles with a length of10 foot respectively. How many raw materials should be used at least? What is the most cost-effective cut? [Analysis and solution] A bamboo pole with a length of 10 feet should be cut in three ways: (1) 3 feet 2 and 4 feet 1, which are the most economical; (2) three feet three, more than one foot; (3) 4 feet 2, more than 2 feet. In order to save materials, try to use the method of (1). With 50 raw materials, 100 3-foot bamboo poles and 50 4-foot bamboo poles can be cut. If 50 4-foot bamboo poles are short, it is best to choose method (3), which requires the least raw materials, only 25, and at least 75 raw materials. Example 3: The lengths of the three sides of an acute triangle are two digits respectively, and they are three consecutive even numbers. The sum of their numbers is a multiple of 7. What is the longest circumference of this triangle? [Analysis and Solution] Because the three sides of a triangle are three consecutive even numbers, their unit digits can only be 0, 2, 4, 6, 8, and their sum is even, and because the sum of their unit digits is a multiple of 7, it can only be 14, and the maximum value of the three sides of a triangle can be 86, 88, 90, so the longest circumference is 86+. Example 4: Decomposition of 25 into the sum of several positive integers to maximize their products. [Analysis and Solution] Start with a small number and find its law: 6 divided by 3+3, its product is 3× 3 = 9; Divide 7 by 3+2+2, and the product is 3×2×2= 12. Divide 8 by 3+3+2, and the product is 3×3×2= 18. 9 divided by 3+3+3, its product is 3×3×3=27. That is to say, if you want to maximize the product of decomposed numbers, you should appear as many as possible, and when a natural number can be expressed as the sum of several 3 sums 1, you should take out a 3 sum 1 and split it into two 2 sums, so that 25 can be split into 3+3+3+3+3. Example 5: A and B are going to explore the desert. They go deep into the desert for 20 kilometers every day. It is known that each person can carry one person's food and water for up to 24 days. If some food is not allowed to be stored halfway, how many kilometers can one of them go deep into the desert (the last two need to return to the starting point)? What if some food can be stored on the way back? [Analysis and Solution] Suppose A goes back after X days, A leaves the food he needs when he goes back, and the rest is transferred to B. At this time, B*** has (48-3X) days of food, so X=8. For the remaining 24 days of food, B can only move forward for 8 days, leaving 16 days of food for him to return, so B can. If the conditions are changed, the crux of the problem is the food left over in B24 days when A returns. Because 24 days of food can make B go deep into the desert alone 12 days, and the other 24 days of food will provide A and B with a walk back and forth, that is, 24÷4=6 days, so B can go deep into the desert 18 days, that is, one day. Ex. 6: Every worker and every piece of equipment in two garment factories, A and B, can completely produce suits of the same specifications. Factory A produces tops, pants and only 900 suits a month. Factory B spends a lot of time producing tops and trousers, and just produces 65,438+0,200 suits a month. Now the two factories are jointly producing, trying to produce more suits. So how many more suits are produced each month than in the past? [Analysis and solution] According to the known conditions, the time ratio of producing a pair of trousers and a coat in a factory is 2: 3; Therefore, the ratio of the number of shirts and trousers produced by a factory in a unit time is 2: 3; It can also be seen that the ratio of the number of shirts and trousers produced by factory B in unit time is 3: 4; Because of this, A factory is good at producing pants, and B factory is good at producing tops. The two factories jointly produce, give full play to their respective specialties, and arrange factory B to fully produce jackets. Since factory B produces 1 0,200 jackets a month, factory B can produce 1 0,200 ÷ = 210,000 jackets a month, and factory A is arranged to fully produce pants, so factory A can produce 900 pants a month. In order to support the production, a factory first produced 2 100 pairs of trousers, which required 2 100÷2250 = pairs a month, and then a factory independently produced 900×60 suits a month, so now the joint production produces more than in the past (2100+60)- Party A and Party B play Go. Party A takes it first, and Party B takes it later. They each take turns eating once. It is stipulated that only 7P(P is 1 or any prime number not exceeding 20) can be taken at a time. Who will win the game in the end? Ask both parties who has a winning strategy. [Analysis] Because 1400=7×200, the original question can be translated as: There are 200 chess pieces, and both parties take p pieces in turn, and whoever takes the last one wins. [Solution] B has a winning strategy. Since 200=4×50, p is either 2 or can be expressed in the form of 4k+ 1 or 4k+3 (k is zero or a positive integer). The strategy adopted by B is: If A takes 2,4k+1 and 4k+3, then B takes 2,3,1,so the remaining pieces are still multiples of 4. So the last remaining number is a multiple of 4, not more than 20. At this time, A can't take it all, and B can take it all and win. [Description] (1) In this question, B is the "late Mover", so the first Mover does not necessarily have a winning strategy. The key is to look at the "situation" they face; (2) We can analyze the solution of this problem in this way, and divide all situations-the number of remaining pieces into two categories. The first category is a multiple of 4, and the second category is others. If someone encounters the second situation when playing chess, they can go 1 or 2 or 3, so the rest is the first situation. If he is faced with the first situation when playing chess, then the second situation must be left to another person who has finished playing chess. Therefore, whoever faces the second situation first will win, and this method can be used in most double-match problems. There is a tour group of 80 people, including 50 men and 30 women. Their hotel has three room types: 1 1, 7 and 5 people. Men and women live in different rooms. How many rooms should they live in at least? 【 Analysis and Solution 】 In order to minimize the number of rooms, first arrange 1 1 rooms, so that 50 men will arrange 3 1 1 rooms, 2 5 rooms and 1 7 rooms. 30 women should be assigned to11room, two 7 rooms, 1 5 room, and * * * has 10 room. Example 9 has a 3×3 chessboard and 9 cards of square size. Write a number on each card at will. Party A and Party B play games, choose a card in turn and put it in one of the nine squares. For Party A, calculate the sum of six numbers in the upper and lower columns, and for Party B, calculate the sum of six numbers in the left and right columns. The person with the larger amount wins. Proof: No matter what number is written on the card, if A goes first, there can always be a strategy to make B impossible to win. There are three situations: (1) When A 1+A9 > A2+A8, A will win. A's strategy is: firstly, choose a9 to put it in the A grid, and then choose a number as small as possible to put it in the B or D grid, so that the sum of the numbers in the A and C grids is not less than a 1+a9, and the sum of the numbers in the B and D grids is not more than a2+a8, so that A wins. (2) When A 1+A9 < A2+A8, A will also win. A takes a 1 and puts it into B grid for the first time, and a8 or a9 is put into A or C grid for the second time, so that the sum of the numbers of A and C grids is not less than a2+a8, and the sum of the numbers of B and D grids is not more than a 1+a9, and A wins. (3) When a 1+a9 = a2+a8, if A wins or draws, A can adopt any of the above strategies. Say hello is not good. I am in primary school. Too many answers 1. The distance between B and B is 6 kilometers. Someone walked from A to B, with an average of 80 meters per minute in the first half and 70 meters per minute in the second half. How many minutes did it take him to walk the last half of the journey? Xiaoming has two roads with the same length from home to school, one is a flat road and the other is a half slope. Xiao Ming takes as much time to go to school by two roads. It is known that the downhill speed is 1.5 times that of the flat road, so how many times is the uphill speed? The ship went back and forth from place A to place B for two hours. When it came back, it was very smooth, walking 8 kilometers per hour more than when it went, so the second hour walked 6 kilometers more than the first hour. So how many kilometers is the distance between A and B? 4. The starting station and the terminal of the tram line are Station A and bilibili respectively. There is a tram from Station A to bilibili every five minutes, and the whole journey takes 15 minutes. A man started from bilibili and rode to Station A along the tram line ... When he started, a tram just arrived in bilibili and met the oncoming tram 10 on the way. When we arrived at Station A, another tram had just left from Station A. How many minutes did it take him to get from bilibili to Station A? A and B are swimming in the river, starting from a certain place and swimming in the same direction at the same speed. Now A is in front of B, and B is 20 meters away from the starting point. When B swims to where A is now, A will be 98 meters away from the starting point. Q: How many meters is A from the starting point now? 6. Two cars, A and B, 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? 7. Li Hua walks from school to the winter camp 20.4 kilometers away at a speed of 4 kilometers per hour. 0.5 hours later, the camp teacher went to meet him when he heard the news, which was 1.2 kilometers more than Li Hua's every hour. After another 1.5 hours, Zhang Ming reported to the camp by bike from school. As a result, three people met somewhere on the road at the same time. Q: How many kilometers do cyclists travel per hour? The express train and the local train leave from A and B respectively at the same time, in opposite directions, and meet five hours later. It is known that it takes 12.5 hours for the local train to travel from B to A, 0.5 hours for the local train to stop at A, and 1 hour for the express train to stop at B, so how long does it take for the two cars to meet for the first time? There is a road between the school and the factory. The school sent a car to the factory at 2 pm to pick up a model worker to give a report. Round trip takes 1 hour. The model worker leaves the factory in the afternoon 1 and walks to school. On the way, I met the car that picked him up, got on the bus and drove to school immediately, and arrived at 2: 40 pm. Q: How many times faster is the speed of a car than that of a model worker?