1, using 1 can solve some seemingly difficult problems (it is very difficult for me. . . )。 For example, there are 14 cups on the desktop, with three cups facing up. Now turn four cups at a time (the cups can be turned up or down). Q: After a few turns, can the cup turn down? If not, can you turn over six at a time? How about seven o'clock?
Suppose that the cup with upward opening is+1 and the cup with downward opening is-1. Now the value of 14 cup is-1 1+3=-8, and finally-14 will be obtained.
Then there are five situations to turn four cups at a time: change +4 to -4 or -4 to +4, change +2 to -2 or -2 to +2, and change 0 to 0. In other words, there are five possible values for turning four cups at a time, plus or minus 8, plus or minus 4, 0.
-8 plus or minus eight, plus or minus four plus zero. Is it possible to get-14? Impossible, because what is missing is a -6, which can't be added up.
Similarly, there are seven possible values for turning six at a time, plus or minus 12, plus or minus 8, plus or minus 4, 0. It is also impossible. I can't get -6.
There are eight possible values for rotating seven cups at a time, namely, plus or minus 14, plus or minus 10, plus or minus 6, plus or minus 2. There is a value of -6 here, so you can turn seven cups at a time and finally make them all face down.
(I tried it with playing cards)
2. What sum of two numbers is equal to their product? We will probably think of 2+2=2*2 immediately.
(1) Can you write more of these two numbers? Can you find some patterns from it?
(2) Can you ask some similar questions? Choose one of your questions to explore.
(1) According to the requirements of the topic, it is A+B = A * B.
When a=0, substitute the above formula, and b=0.
When a= 1, b+ 1=b, b does not exist.
When a=2, b=2.
When a=3, b=3/2.
When a=4, b=4/3.
......
It can be inferred from the above that these two numbers can be written as n and n/(n- 1), where n is greater than or equal to 2.
(2) The difference between two numbers is equal to the quotient of two numbers. That is, a/b = a-B.
Denominator cannot be 0. So when b= 1 and a=a- 1, a does not exist.
When b=2, a=4.
When b=3, a=9/2.
When b=4, a= 16/3.
......
Inferred from the above, these two numbers can be written as n and n*n/(n- 1), where n is greater than or equal to 2.
Three travelers gave up the road on a snowy night and wanted to walk through a valley with a width of 4 kilometers. They walked for a long time and should have arrived at their destination according to time, but they always returned to the original starting point for no reason, and finally had to wait for dawn in the valley. Explain and calculate with mathematical knowledge: the distance between left and right feet is about 0. 1 m when people walk, and the step field of each step is about 0.7 m. Because everyone's foot strength can't be exactly the same, the stride size is different. If someone's right foot walks l meters more than his left foot (that is, the pace difference), the radius of the circle he turns is r meters, and assuming that his right foot is in the outer circle, he walks 2πR meters and his left foot walks 2π(R-0. 1) meters in the inner circle, then he walks a circle, and the distance difference between his two feet is 0.2π meters. When a person walks a circle, his left foot and his right foot take 2πR\2*0.7 steps respectively. Since step difference * number of steps = distance difference, L*2πR\2*0.7=0.2π, that is, RL=0. 14. Question: (1) If L= 1,
(2) If the diameter of the circle of three travelers is 4 kilometers, what is the difference in their steps?
This is the legendary story of seeing ghosts always spinning around in the same place. In fact, it's because the two feet are not walking the same distance, just walking in circles. )
( 1)RL=0. 14，L= 1mm=0.00 1m
R= 140m The circle radius of this person is 140m.
(2)RL=0. 14, R = 2000m, L = 0.07mm 。
The calculation is simple. Explain the topic. The topic is that a person always circles in one place because his right foot is a little bigger than his left foot, which leads to circles. The radius of this circle is r, the circumference is 2π r, and each step of 0.7m is the * * * step of 2πR\0.7, and the left and right feet are left and right, so divide by two. The step difference L between the left and right feet multiplied by this number of steps is the distance difference between the two feet in each lap.
The first part is that when walking this circle, the right foot takes a big circle, including the left foot. In fact, the trajectories of the left and right feet are two concentric circles with a radius difference of 0. 1m, so the circumference of the big circle is 2πR meters and the circumference of the small circle is 2 π (r-0. 1) meters.
You should understand by now. Actually, this question really didn't mean to embarrass you. Otherwise, RL will not be calculated for you, and there is no need to say that the diameter is 4km in the second question, because the title has already said that the valley is 4km wide.
4. Li Bai walked in the street doing nothing, carrying a pot to buy wine. When he met a shop, he doubled it. When he saw flowers, he drank a bucket (barrels were ancient containers for wine). When he met shops and flowers, he asked three times, how much wine is there in the pot?
Push back. When Li Bai saw the third flower, there was 1 fighting wine in the pot.
So when Li Bai saw the third shop, there was still 0.5 dou wine in the pot.
When Li Bai saw the second flower, there was 1.5 glasses of wine in the pot.
When Li Bai saw the second shop, there was still 0.75 doujiu in the pot.
When Li Bai saw the first flower, there was 1.75 in the pot.
So Li Bai initially had 0.875 doujiu in the pot.
If you need to write a detailed process, list the equations:
Put the original x barrels of wine in the pot. Multiply it by two when you meet a store, and subtract one when you meet flowers.
The solution of 2 [2 (2x-1)-1]-1= 0 is the same.
5. Two people are moving on a circle with a circumference of 400 meters. If they move in the same direction, they meet every 6 minutes and 40 seconds. If they move in opposite directions, then they meet every 40 seconds, and their moving speed is _ _ _ _ _.
When moving in the same direction, the meeting time is 400 divided by the speed difference between them, which is 400s, so the speed difference between them is1m/s.
When moving in opposite directions, the meeting time of 400 divided by the sum of the two people's speeds is 40s, and the sum of the two people's speeds is10m/s.
So their speed is 4.5m/S and 5.5m/S respectively.
6. In order to prepare for her college tuition in 1600 yuan, Xiaohua's parents took out a part of her education savings when she was in the seventh grade, and prepared to save a part first and borrow another part when she went to college. Xiaohua's parents have saved for 6 years (annual interest rate 2.88%) and plan to pay off the university loan in 8 years (annual interest rate 6.2 1%). 50% of the loan interest is subsidized by the government. If the interest earned by participating in education savings is equal to the interest paid by applying for loans, how much did Xiaohua's parents spend on participating in education savings? How much are you going to borrow?
We use X yuan to participate in education savings and prepare a loan of Y yuan. Equations.
x+y= 1600
6 * 2.88% * x = 50% * 8 * 6.2 1% * y 17.28 x = 24.84y x = 1.4375y
y=656.4 x=943.6
(This equation is listed in terms of one-time payment after eight years. I didn't understand the meaning of this question clearly. Besides, I don't think this child is his own. . . I am in the seventh grade, and I need to borrow more than 600 from my family. . . )
Due to the change of seasons, a commodity is ready to be sold at a discount. If it is sold at a 15% discount on the basis of pricing, 250 yuan will be compensated. If you sell at a 10% discount on the pricing, you will earn 200 yuan. Then the original price of this commodity should be _ _ yuan.
Let the original price be x yuan. Then 0.75x+250=0.9x-200 x=3000.
8. A document needs to be printed. It takes 2 hours for Xiao Ming to do it alone, 2.4 hours for Xiao Gang and 3 hours for Xiao Lan to do it alone. Now, after working with Xiao Ming and Xiao Lan for 36 minutes, Xiao Ming has something to leave and Xiao Gang works for him. How long will it take?
If the total workload is X, the efficiency of Xiao Ming is x/ 120 per minute, Xiao Gang is x/ 144 per minute, and Xiao Lan is x/ 180 per minute. Xiao Ming and Xiao Lan have completed 36 * (x/120+x/180) = 0.5x, the remaining workload is x-0.5x = 0.5x, and the efficiency after cooperation between Xiao Gang and Xiao Lan is (x/144+180.
The above is the official solution. The quick solution is that if the total workload is 12, Xiaoming's work efficiency is 6, Xiaogang's is 5, Xiaolan's is 4.36 minutes and 0.6 hours, and Xiaoming and Xiaolan's work efficiency is 10, then the workload is 6, and the remaining 6 workloads are not done. The work efficiency of Xiao Gang and Xiao Lan is 9, so it takes 6/9 hours, which is 40 minutes.
9. Furong Shopping Mall implements prize-winning sales. During the period of September 1-1 October1,shoppers will draw a lottery with the invoice, with a grand prize of1and a bonus of 10000 yuan; Two first prizes, each with a bonus of 1 1,000 yuan; The second prize will produce 10, and each person will receive a bonus of 100 yuan. The third prize will produce 200 people, and the bonus will be 5 yuan. Xinchao Commercial Building offers a 15% discount in September and1October. According to the survey. The monthly sales of these two stores are above 200,000 yuan. Which store gives customers more discounts?
The monthly discount given by Furong Shopping Mall is dead. All kinds of bonuses add up to 14000 yuan.
The monthly discount given by Xinchao Commercial Building is 5% of its own sales, which is 1 10,000 when it is 200,000. To reach 14000, its sales volume should reach 280000.
Therefore, when the sales volume of trendy shopping malls is less than 280,000, Furong Shopping Mall offers more discounts, and when it is more than 280,000, trendy shopping malls offer more discounts.
10. Put some chickens in some cages. If there are four chickens in each cage, there will be 20 chickens without cages. If you put eight in each cage, one cage is not enough. How many cages are there? How many chickens?
Suppose a * * * has x cages. 4 x +20=8(x- 1) gives x=7, and there are 48 chickens in total.
This question seems ambiguous. A dissatisfied cage is not necessarily empty.
1 1. A store bought 80 pieces of Class A goods and 80 pieces of Class B goods from a company's wholesale department, which cost 5 600 yuan. When the store retails, the price of each class A commodity increases by 15%, and the price of each class B commodity increases by 10%, so they are all sold out.
Didn't you know that you studied binary linear equation in seventh grade? . . )
Let A buy at the unit price of X yuan, and B buy at the unit price of Y yuan.
100x + 80y =5600 1
100 * 1. 15x+80 * 1. 1y = 6280 2
Subtract 15x+8y=680 from Equation 2 to get 150x+80y=6800 3.
Equation 3 subtracts 50x = 1 200 and x=24 from the equation1.
Substituting into equation 2 gives y=55.