2。 Party A, Party B and Party C leave at the same time. Among them, Party C rode from Town B to Town A, while both Party A and Party B rode from Town A to Town B (Party A drove slowly at a speed of 24 kilometers per hour and Party B walked at a speed of 4 kilometers per hour). On the way, Party C and Party A meet in Town D and ride back to Town B. Party A turns around to pick up Party B, then Party A picks up Party B and goes back to DB. After Party A picks up Party B (Party B gets on the bus), it goes to Town B at a speed of 88 kilometers per hour. As a result, three people arrived at town B at the same time. So what is the speed of Party C's cycling per hour?
Answer: 1. Solution: Let the distance of AB be S, the meeting time of A and C be T 1, and A and B be T2. Later, the time when three people arrived at B at the same time was T3! The speed is X.
Get (24+x) t 1 = S 1.
(24+4)T2=(24-4)T 1 ②
4(T 1+T2)88+T3=S ③
X(T2+T3)=XT 1 ④
From ②, T2=5/7T 1 ⑤.
From ④, T3=2/7T 1 ⑥.
Substitute ⑤ and ⑤ into ⑤ to get.
224/7T 1=S ⑦
Substitute ⑦ into ⑦ to get.
X=8
2. Answer and process: Solution: It takes X days to set up and Y days to work, where X is greater than 30. The equation is:
16X = 12(82-X-Y)+2 * 2+ 14 * 4
16X = 984- 12X- 12Y+60
28X+ 12Y= 1044
7X+3Y=26 1
It says that x must be greater than 30, so it is concluded that only X=33, Y= 10, X=36 and Y=3 can satisfy the problem. The result of 1 group is brought into the equation, and the number of days is less than 30, so the solution is wrong, and the answer is the second group of solutions.
So I worked for three days.
This is something else I found.
1. Let a, b and c be real numbers, and | A |+A = 0, | AB | = AB, | C |-C = 0, and find the value of the algebraic formula | B |-| A+B |-C-B |+| A-C |.
2. If m < 0, n > 0, | m |
3. Let (3x-1) 7 = A7X7+A6X6+…+A1X+A0, and try to find the value of A0+A2+A4+A6.
5. Solve equation 2 | x+ 1 |+x-3 | = 6.
6. Solve the inequality || x+3 |-x- 1 || > 2.
7. Compare the following two figures:
8.x, Y and Z are all nonnegative real numbers and satisfy:
x+3y+2z=3，3x+3y+z=4，
Find the maximum and minimum values of u = 3x-2y+4z.
9. Find the quotient and remainder of x4-2x3+x2+2x- 1 divided by x2+x+ 1.
10. As shown in figure 1-88, Zhu Xiao lives in village A and grandma lives in village B. On Sunday, Zhu Xiao went to visit grandma, first cutting a bundle of grass on the north slope, and then cutting a bundle of firewood on the south slope to send grandma. Excuse me, which route should Zhu Xiao take for the shortest journey?
1 1. as shown in figure 1-89. AOB is a straight line, OC and OE are bisectors of ∠AOD and ∠DOB, respectively, and ∠ COD = 55. Find the complementary angle of ∠DOE.
12. As shown in figure 1-90, the bisected line ∠ABC, ∠ CBF = ∠ CFB = 55, ∠ EDF = 70. Verification: BC ∠ AE.
13. As shown in figure 1-9 1. In △ABC, EF⊥AB, CD⊥AB, ∠ CDG = ∠ BEF. Verification: ∠ AGD = ∠ ACB.
14. As shown in figure 1-92. In △ABC, ∠B=∠C, BD⊥AC is in D.
15. As shown in figure 1-93. In △ABC, e is the midpoint of AC, d is on BC, BD∶DC= 1∶2, and AD and BE intersect at F. Find the ratio of the area of △BDF to the area of quadrilateral FDCE.
16. As shown in figure 1-94, two opposite sides of quadrilateral ABCD extend and intersect at K and L, and diagonal AC‖KL and BD extension lines intersect with KL at F. Verification: KF = FL.
17. Can the sum of the number obtained by arbitrarily changing the order of a three-digit number and the original number be 999? Explain why.
18. There is a piece of grid paper with 8 rows and 8 columns, in which 32 squares are randomly painted black and the remaining 32 squares are painted white. Next, the color grid paper is operated, and each operation changes the color of each square in any horizontal or vertical column at the same time. Can you finally get a grid paper with only one black square?
19. If both positive integers p and p+2 are prime numbers greater than 3, then verify: 6 | (p+ 1).
20. Let n be the smallest positive integer satisfying the following conditions, which is a multiple of 75 and has exactly
2 1. There are several stools and chairs in the room. Each stool has three legs and each chair has four legs. When they are all seated, * * * has 43 legs (including everyone's two legs). How many people are there in the room?
22. Find the integer solution of the indefinite equation 49x-56y+ 14z=35.
23. Eight men and eight women dance in groups.
(1) If there are two substations, male and female;
(2) If men and women stand in two rows, in no particular order, only consider how men and women form a partner.
How many different situations are there?
24. How many of the five numbers1,2, 3, 4 and 5 are greater than 34 152?
25.A train is 92 meters long and B train is 84 meters long. If they travel in opposite directions, they will miss each other after 1.5 seconds. If they travel in the same direction, they will miss each other in six seconds. Find the speed of two trains.
26. The two production teams of Party A and Party B grow the same vegetables. After planting for four days, Team A will finish the rest alone, and it will take two more days. If Party A finishes all the tasks by itself three days faster than Party B, how many days does it take to ask Party A to finish it by itself?
27. A ship starts from a port 240 nautical miles apart, and its speed decreases by 65,438+00 nautical miles per hour before reaching its destination 48 nautical miles. The total time it takes after its arrival is equal to the time it takes for the whole voyage when its original speed is reduced by 4 nautical miles per hour, so that we can find out the original speed.
28. Last year, two workshops A and B of a factory planned to complete tax profits of 7.5 million yuan. As a result, workshop A exceeded the plan 15%, workshop B exceeded the plan 10%, and two workshops * * * completed tax profits of 8.45 million yuan. How many million yuan of tax profits did these two workshops complete last year?
29. It is known that the sum of the original prices of commodities A and B is 150 yuan. Due to market changes, the price of commodity A decreased by 65,438+00% and the price of commodity B increased by 20%. After the price adjustment, the sum of the unit prices of commodities A and B decreases by 1%. What are the original unit prices of goods A and B respectively?
Xiaohong bought two children's toothbrushes and three toothpastes in the shop last summer vacation, and just ran out of money with her. It is known that each toothpaste is more than each toothbrush 1 yuan. This summer, she took the same money to the store and bought the same toothbrush and toothpaste. Because each toothbrush rose to 1.68 yuan this year and the price of each toothpaste rose by 30%, Xiaohong had to buy two toothbrushes and two toothpastes, and she got back 40 cents. How much is each toothpaste?
3 1. If a shopping mall sells goods with a unit price of 8 yuan at 12 yuan, it can sell 400 pieces every day. According to experience, if each piece is sold for less than 1 yuan, more than 200 pieces can be sold every day. How much should each piece be reduced to get the best benefit?
32. The distance from Town A to Town B is 28 kilometers. Today, A rode his bike at a speed of 0.4km/min, and set out from Town A to Town B. After 25 minutes, B rode his bike to catch up with A at a speed of 0.6km/min. How many minutes does it take to catch up with A?
33. There are three kinds of alloys: the first contains 60% copper and 40% manganese; The second type contains manganese 10% and nickel 90%; The third alloy contains 20% copper, 50% manganese and 30% nickel. Now a new alloy containing 45% nickel is composed of these three alloys, and its weight is 1 kg.
(1) Try to express the weight of the second alloy by the weight of the first alloy in the new alloy;
(2) Find out the weight range of the second alloy in the new alloy;
(3) Find out the weight range of manganese in the new alloy.
Answer: a≤0 because | A | =-A, b≤0 because | AB | = AB, C ≥ 0 because | C | = C Therefore, A+B ≤ 0, c-b≥0 and A-C ≤ 0.
The original formula =-b+(a+b)-(c-b)-(a-c) = b.
3. Because m < 0, n > 0, so | m | =-m, | n | = n So | m | 0. When x+m≥0, | x+m | = x+m; When x-n≤0, | x-n | = n-X. Therefore, when -m≤x≤n,
|x+m|+|x-n|=x+m-x+n=m+n。
4. Let x= 1 and x=- 1 respectively, and substitute them into the known equation to obtain.
a0+a2+a4+a6=-8 128。
5.②+③ Finishing
x=-6y，④
(k-5) y = 0 when substituting ①.
When k=5, y has infinite solutions, so the original equations have infinite groups of solutions; When k≠5, y=0. If it is substituted into ②, (1-k) x = 1+k is obtained. Because x=-6y=0, 1+k = 0, so k =-/kloc-0.
Therefore, when k=5 or k=- 1, the original equations have solutions.
When < x ≤ 3, 2 (x+ 1)-(x-3) = 6, so x =1; When x > 3, there are
, so you should give up.
7. From | x-y | = 2
X-y=2, or x-y=-2,
therefore
From the previous equations.
|2+y|+|y|=4。
When y
Similarly, it can be solved by the latter equations.
So the solution is
X of solution ① is ≤-3; Solve ②
-3 < x
③ x > 1 of the solution.
So the original inequality solution is x 0.9. Let a = 9999111,then
therefore
Obviously there is a > 1, so a-b > 0, that is, a > b.
10.y and z can be obtained by known.
Because y and z are non-negative real numbers, there are
u=3x-2y+4z
1 1.
So the quotient is x2-3x+3 and the remainder is 2x-4.
12. The route of the small cylinder is a broken line consisting of three line segments (as shown in Figure 1-97).
We use the method of "symmetry" to transform the line of this broken line of a small cylinder into a "connecting line" (a line segment) between two points. The symmetry point of the north hillside of Shijiacun (the hillside is regarded as a straight line) is a'; The symmetry point of village B on the south hillside is B', which connects A' B'. If the intersection points of the line segment connected by A' B' and the north hillside and the south hillside are A and B respectively, the route of A →A→B→ B is the best choice (that is, the shortest route).
Obviously, the length of route A →A→B→ B is exactly equal to the length of line segment A ′ B ′. Using the above symmetry method, any other route from village A to village B can be transformed into a broken line connecting A' and B'. They are all longer than the line segment A'B'. So the distance from A to A → B → B is the shortest.