Answer:
The speed of each bicycle is 10 miles per hour, and the two will meet at the midpoint of the distance of 2O miles after 1 hour. The speed of a fly is 15 miles per hour, so in 1 hour, it always flies 15 miles.
Many people try to solve this problem in a complicated way. They calculate the first distance between the handlebars of two bicycles, then return the distance, and so on, and calculate those shorter and shorter distances. But this will involve the so-called infinite series summation, which is very complicated advanced mathematics. It is said that at a cocktail party, someone asked John? John von neumann (1903 ~ 1957) is one of the greatest mathematicians in the 20th century. ) Put forward this question, he thought for a moment, and then gave the correct answer. The questioner seems a little depressed. He explained that most mathematicians always ignore the simple method to solve this problem and adopt the complex method of summation of infinite series.
Von Neumann had a surprised look on his face. "However, I use the method of summation of infinite series," he explained.
2. A fisherman, wearing a big straw hat, sat in a rowboat and fished in the river. The speed of the river is 3 miles per hour, and so is his rowing boat. "I must row a few miles upstream," he said to himself. "The fish here don't want to take the bait!"
Just as he started rowing upstream, a gust of wind blew his straw hat into the water beside the boat. However, our fisherman didn't notice that his straw hat was lost and rowed upstream. He didn't realize this until he rowed the boat five miles away from the straw hat. So he immediately turned around and rowed downstream, and finally caught up with his straw hat drifting in the water.
In calm water, fishermen always row at a speed of 5 miles per hour. When he rowed upstream or downstream, he kept the speed constant. Of course, this is not his speed relative to the river bank. For example, when he paddles upstream at a speed of 5 miles per hour, the river will drag him downstream at a speed of 3 miles per hour, so his speed relative to the river bank is only 2 miles per hour; When he paddles downstream, his paddle speed will interact with the flow rate of the river, making his speed relative to the river bank 8 miles per hour.
If the fisherman lost his straw hat at 2 pm, when did he get it back?
Answer:
Because the velocity of the river has the same influence on rowing boats and straw hats, we can completely ignore the velocity of the river when solving this interesting problem. Although the river is flowing and the bank remains motionless, we can imagine that the river is completely static and the bank is moving. As far as rowing boats and straw hats are concerned, this assumption is no different from the above situation.
Since the fisherman rowed five miles after leaving the straw hat, he certainly rowed five miles back to the straw hat. Therefore, compared with rivers, he always paddles 10 miles. The fisherman rowed at a speed of 5 miles per hour relative to the river, so he must have rowed 65,438+00 miles in 2 hours. So he found the straw hat that fell into the water at 4 pm.
This situation is similar to the calculation of the speed and distance of objects on the earth's surface. Although the earth rotates in space, this motion has the same effect on all objects on its surface, so most problems about speed and distance can be completely ignored.
3. An airplane flies from city A to city B, and then returns to city A. In the absence of wind, the average ground speed (relative ground speed) of the whole round-trip flight is 100 mph. Suppose there is a persistent strong wind blowing from city A to city B. If the engine speed is exactly the same as usual during the whole round-trip flight, what effect will this wind have on the average ground speed of the round-trip flight?
Mr. White argued, "This wind will not affect the average ground speed at all. In the process of flying from City A to City B, strong winds will accelerate the plane, but in the process of returning, strong winds will slow down the speed of the plane by the same amount. " "That seems reasonable," Mr. Brown agreed, "but if the wind speed is 100 miles per hour. The plane will fly from city A to city B at a speed of 200 miles per hour, but the speed will be zero when it returns! The plane can't fly back at all! " Can you explain this seemingly contradictory phenomenon?
Answer:
Mr. White said that the wind increases the speed of the plane in one direction by the same amount as it decreases the speed of the plane in the other direction. That's right. But he said that the wind had no effect on the average ground speed of the whole round-trip flight, which was wrong.
Mr. White's mistake is that he didn't consider the time taken by the plane at these two speeds.
It takes much longer to return against the wind than with the wind. In this way, it takes more time to fly when the ground speed is slow, so the average ground speed of round-trip flight is lower than when there is no wind.
The stronger the wind, the more the average ground speed drops. When the wind speed is equal to or exceeds the speed of the plane, the average ground speed of the round-trip flight becomes zero, because the plane cannot fly back.
4. Sunzi Suanjing is one of the top ten famous arithmetical classics in the early Tang Dynasty, and it is an arithmetic textbook. It has three volumes. The first volume describes the system of counting, the rules of multiplication and division, and the middle volume illustrates the method of calculating scores and Kaiping with examples, which are all important materials for understanding the ancient calculation in China. The second book collects some arithmetic problems, and the problem of "chickens and rabbits in the same cage" is one of them. The original question is as follows: let pheasant (chicken) rabbits be locked together, with 35 heads above and 94 feet below.
Male rabbit geometry?
The solution of the original book is; Let the number of heads be a and the number of feet be b, then b/2-a is the number of rabbits and a-(b/2-a) is the number of pheasants. This solution is really great. When solving this problem, the original book probably adopted the method of equation.
Let x be the pheasant number and y be the rabbit number, then there is
x+y=b，2x+4y=a
Get a solution
y=b/2-a，
x=a-(b/2-a)
According to this set of formulas, it is easy to get the answer to the original question: 12 rabbits, 22 pheasants.
Let's try to run a hotel with 80 suites and see how knowledge becomes wealth.
According to the survey, if we set the daily rent as 160 yuan, we can be full; And every time the rent goes up in 20 yuan, three guests will be lost. Daily expenses for services, maintenance, etc. Each occupied room is calculated in 40 yuan.
Question: How can we set the price to be the most profitable?
A: The daily rent is 360 yuan.
Although 200 yuan was higher than the full price, we lost 30 guests, but the remaining 50 guests still brought us 360*50= 18000 yuan. After deducting 40*50=2000 yuan for 50 rooms, the daily net profit is 16000 yuan. When the customer is full, the net profit is only 160*80-40*80=9600 yuan.
Of course, the so-called "learned through investigation" market was actually invented by myself, so I entered the market at my own risk.
6 Mathematician Weiner's age, the whole question is as follows: The cube of my age this year is four digits, and the fourth power of my age is six digits. These two numbers only use all ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. How old is Weiner? Answer: this question is difficult at first glance, but it is not. Let Wiener's age be X. First, the cube of age is four digits, which defines a range. The cube of 10 is 1000, the cube of 20 is 8000, and the cube of 2 1 is 926 1, which is a four-digit number; The cube of 22 is10648; So 10 =
A monkey picked 100 bananas in the forest and piled them up. The monkey's home is 50 meters away from the banana pile, and the monkey intends to carry the bananas home.
You can take up to 50 sticks at a time, but monkeys are greedy. He eats a banana every meter. Ask the monkey how many sticks he can take home at most.
Bananas?
25.
Recite 50 songs to 25 meters first. At this time, I ate 25 pieces and left 25 pieces. Put them down. Go back and recite the remaining 50. At 25 meters, I ate 25 more, and there are 25 more. Then pick up 25 roots on the ground, 50 of them, and continue to walk home. A ***25 meters, you have to eat 25, and there are 25 left to get home.
Wrap a piece of paper on a piece of chalk, and then cut the chalk diagonally with a knife. What is the shape of the broken edge of the paper after unfolding?
Answer: sine curve
One day after the heavy snow, Tingting and her father set out from the same point and measured the circumference of a round garden in the same direction. Tingting's stride is 54 cm, and Dad's stride is 72 cm. Because the footprints of two people overlap, only 60 footprints are left in the snow. What is the circumference of this garden?
Cause, expression
Hypothetical method
Find the least common multiple 2 16 of 54 and 72.
2 16 cm * * *, how many footprints are there?
216/54+216/72-1(because their footprints overlapped at first)
=4+3- 1
=6
60/6= 10
216 *10 = 2160 (cm)
Fifth grade Olympic competition
Inclusion and exclusion
1. There are 40 students in a class, of which 15 is in the math group, 18 is in the model airplane group, and 10 is in both groups. So how many people don't participate in both groups?
Solution: There are (15+18)-10 = 23 (people) in the two groups.
40-23= 17 (person) did not attend.
A: There are 17 people, and neither group will participate.
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There are forty-five students in a class who took the final exam. After the results were announced, 10 students got full marks in mathematics, 3 students got full marks in mathematics and Chinese, and 29 students got no full marks in both subjects. So how many people got full marks in Chinese?
Solution: 45-29- 10+3=9 (person)
A: Nine people got full marks in Chinese.
3.50 students stand in a row facing the teacher. The teacher asked everyone to press 1, 2,3, ..., 49,50 from left to right. Let the students who are calculated as multiples of 4 back off, and then let the students who are calculated as multiples of 6 back off. Q: How many students are facing the teacher now?
Solution: multiples of 4 have 50/4 quotients 12, multiples of 6 have 8 50/6 quotients, and multiples of 4 and 6 have 4 50/ 12 quotients.
Number of people turning back in multiples of 4 = 12, number of people turning back in multiples of 6 ***8, including 4 people turning back and 4 people turning back from behind.
Number of teachers =50- 12=38 (person)
A: There are still 38 students facing the teacher.
4. At the entertainment party, 100 students won lottery tickets with labels of 1 to 100 respectively. The rules for awarding prizes according to the tag number of lottery tickets are as follows: (1) If the tag number is a multiple of 2, issue 2 pencils; (2) If the tag number is a multiple of 3, 3 pencils will be awarded; (3) The tag number is not only a multiple of 2, but also a multiple of 3 to receive the prize repeatedly; (4) All other labels are awarded to 1 pencil. So how many prize pencils will the Recreation Club prepare for this activity?
Solution: 2+000/2 has 50 quotients, 3+ 100/3 has 33 quotients, and 2 and 3 people have 100/6 quotients.
* * * Preparation for receiving two branches (50- 16) * 2 = 68, * * Preparation for receiving three branches (33- 16) * 3 = 5 1, * * Preparation for repeating branches (2+).
* * * Need 68+5 1+80+33=232 (branch)
A: The club has prepared 232 prize pencils for this activity.
5. There is a rope with a length of 180 cm. Make a mark every 3 cm and 4 cm from one end, and then cut it at the marked place. How many ropes were cut?
Solution: 3 cm marker: 180/3=60, the last marker does not cross, 60- 1=59.
4cm marker: 180/4=45, 45- 1=44, repeated marker: 180/ 12= 15,15-/kloc-.
Cut it 89 times and it becomes 89+ 1=90 segments.
A: The rope was cut into 90 pieces.
6. There are many paintings on display in Donghe Primary School Art Exhibition, among which 16' s paintings are not in the sixth grade, and 15' s paintings are not in the fifth grade. Now we know that there are 25 paintings in Grade 5 and Grade 6, so how many paintings are there in other grades?
Solution: 1, 2,3,4,5 * * has 16, 1, 2,3,4,6 * * has15,5,6 * * has 25.
So * * has (16+ 15+25)/2=28 (frame), 1, 2,3,4 * * has 28-25=3 (frame).
A: There are three paintings in other grades.
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7. There are several cards, each with a number written on it, which is a multiple of 3 or 4. Among them, cards marked with multiples of 3 account for 2/3, cards marked with multiples of 4 account for 3/4 and cards marked with multiples of 12 account for 15. So, how many cards are there?
Solution: The multiple of 12 is 2/3+3/4- 1=5/ 12, 15/(5/ 12)=36 (sheets).
There are 36 cards of this kind.
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8. How many natural numbers from 1 to 1000 are divisible by neither 5 nor 7?
Solution: multiples of 5 have 200 quotients 1000/5, multiples of 7 have quotients 1000/7 142, and multiples of 5 and 7 have 28 quotients 1000/35. The multiple of 5 and 7 * * * has 200+ 142-28=3 14.
1000-3 14=686
A: There are 686 numbers that are neither divisible by 5 nor divisible by 7.
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9. Students in Class 3, Grade 5 participate in extracurricular interest groups, and each student participates in at least one item. Among them, 25 people participated in the nature interest group, 35 people participated in the art interest group, 27 people participated in the language interest group, 12 people participated in the language interest group, 8 people participated in the nature interest group, 9 people participated in the nature interest group, and 4 people participated in the language, art and nature interest groups. Ask how many students there are in this class.
Solution: 25+35+27-(8+ 12+9)+4=62 (person)
The number of students in this class is 62.
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10, as shown in Figure 8- 1, it is known that the areas of three circles A, B and C are all 30, the areas of overlapping parts of A and B, B and C, and A and C are 6, 8 and 5 respectively, and the total area covered by the three circles is 73. Find the area of the shaded part.
Solution: The overlapping area of A, B and C =73+(6+8+5)-3*30=2.
Shadow area =73-(6+8+5)+2*2=58.
A: The shaded part is 58.
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Grade four 1 class 1 1 There are 46 students taking part in three extracurricular activities. Among them, 24 students from the math group and 20 students from the Chinese group participated. The number of people who participated in the art group was 3.5 times that of those who participated in both the math group and the art group, and 7 times that of those who participated in all three activities. The number of people who participated in both the literature and art group and the Chinese group was twice that of those who participated in all three activities, and the number of people who participated in both the math group and the Chinese group was 10. The number of people seeking to join the art troupe.
Solution: Let the number of people participating in the art group be x, 24+20+x-(x/305+2/7 * x+10)+x/7 = 46, and the solution is X=2 1.
A: The number of participants in the art group is 2 1.
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12. There are 100 books in the library. The borrower needs to sign the book. It is known that 33, 44 and 55 books in 100 have the signatures of A, B and C respectively, among which 29 books have the signatures of A and B, 25 books have the signatures of A and C, and 36 books have the signatures of B and C. How many of these books have not been borrowed by any of A, B and C?
Solution: The number of books read by three people is: A+B+C-(A+B+C+C)+A, B, C =33+44+55-(29+25+36)+ A, B, C =42+ A, B, C, A, C is the most.
Three people will always read 42+25=67 (books) at most, and at least 100-67=33 (books) have never been read.
A: At least 33 books in this batch have not been borrowed by any of A, B and C.
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13, as shown in Figure 8-2, five equal-length line segments form a pentagram. If exactly 1994 points on each line segment are dyed red, how many red dots are there on this five-pointed star?
Solution: There are 5* 1994=9970 red dots on the right side of the five elements. If you put a red dot on all the intersections, then at least there are red dots. These five lines have 10 intersections, so there are at least 9970- 10=9960 red dots.
A: There are at least 9960 red dots on this five-pointed star.
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14, A, B and C are watered at the same time 100 potted flowers. It is known that A poured 78 pots, B poured 68 pots and C poured 58 pots. So how many pots were watered by three people?
Solution: A and B must have 78+68- 100=46 pots * *, and C has 100-58=42, so all three people poured at least 46-42=4 pots.
A: All three people have watered at least four pots of flowers.
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15, A, B and C are all reading the same story book. There are 100 stories in the book. Everyone starts with a story and then reads it in order. It is known that A has read 75 articles, B has read 60 articles and C has read 52 articles. So how many stories have A, B and C read together?
Solution: B and C * * * have read at least 60+52- 100= 12 stories. This 12 story A must be read no matter where it starts.
A: A, B and C have read at least 12 stories.
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15, A, B and C are all reading the same story book. There are 100 stories in the book. Everyone starts with a story and then reads it in order. It is known that A has read 75 articles, B has read 60 articles and C has read 52 articles. So how many stories have A, B and C read together?
Solution: B and C * * * have read at least 60+52- 100= 12 stories. This 12 story A must be read no matter where it starts.
A: A, B and C have read at least 12 stories.
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The following is the quotation of abc in 2004-12-1215: 42:17:
8. How many natural numbers from 1 to 1000 are divisible by neither 5 nor 7?
Solution: multiples of 5 have 200 quotients 1000/5, multiples of 7 have quotients 1000/7 142, and multiples of 5 and 7 have 28 quotients 1000/35. The multiple of 5 and 7 * * * has 200+ 142-28=3 14.
1000-3 14=686
A: There are 686 numbers that are neither divisible by 5 nor divisible by 7.
The division in the title should be exactly division.
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Grade four 1 class 1 1 There are 46 students taking part in three extracurricular activities. Among them, 24 students from the math group and 20 students from the Chinese group participated. The number of people who participated in the art group was 3.5 times that of those who participated in both the math group and the art group, and 7 times that of those who participated in all three activities. The number of people who participated in both the literature and art group and the Chinese group was twice that of those who participated in all three activities, and the number of people who participated in both the math group and the Chinese group was 10. The number of people seeking to join the art troupe.
Solution: Let the number of people participating in the art group be x, 24+20+x-(x/305+2/7 * x+10)+x/7 = 46, and the solution is X=2 1.
A: The number of participants in the art group is 2 1.
1. There are 19 people who subscribe to Juvenile Digest, 24 people subscribe to Learn and Play, and 13 people subscribe to both. Ask for a subscription "
How many people are there in Youth Digest or Learn and Play?
2. In the kindergarten, there are 58 people who learn piano, 43 people who learn painting and 37 people who learn piano and painting. How many people learn piano and painting respectively?
People?
3. Among the natural numbers from 1 to 100:
(1) How many numbers are multiples of 2 and 3?
(2) How many numbers are multiples of 2 or multiples of 3?
(3) How many numbers are multiples of 2 instead of multiples of 3?
4. The mid-term examination results of a class in mathematics and English are as follows: 12 Student English 100, 10 Student Mathematics 100, two subjects.
Three people got 100 in all courses, and 26 people didn't get 100 in all courses. How many students are there in this class?
5. There are 50 people in the class, 32 can ride a bike, 265,438+0 can skate, 8 can both, and how many can't both?
6. There are 42 students in a class, 30 students in sports teams and 25 students in literary and art teams, and each student should participate in at least one team. this
How many people are there in the two teams of the class?
Test answer
1. There are 19 people who subscribe to Juvenile Digest, 24 people subscribe to Learn and Play, and 13 people subscribe to both. Ask for a subscription "
How many people are there in Youth Digest or Learn and Play?
19+24— 13 = 30 (person)
A: There are 30 people who subscribe to Youth Digest or Learn and Play.
2. In the kindergarten, there are 58 people who learn piano, 43 people who learn painting and 37 people who learn piano and painting. How many people learn piano and painting respectively?
People?
Number of piano learners: 58-37 = 2 1 (person)
Number of people who only learn painting: 43-37 = 6 (people)
3. Among the natural numbers from 1 to 100:
(1) How many numbers are multiples of 2 and 3?
It is a multiple of 3 and 2 and must be a multiple of 6.
100÷6 = 16……4
So both 2 and 3 have multiples of 16.
(2) How many numbers are multiples of 2 or multiples of 3?
100÷2 = 50, 100÷3 = 33…… 1
50+33— 16 = 67 (piece)
Therefore, there are 67 numbers that are multiples of 2 or multiples of 3.
(3) How many numbers are multiples of 2 instead of multiples of 3?
50- 16 = 34 (piece)
A: There are 34 numbers that are multiples of 2, but not multiples of 3.
4. The mid-term examination results of a class in mathematics and English are as follows: 12 Student English 100, 10 Student Mathematics 100, two subjects.
Three people got 100 in all courses, and 26 people didn't get 100 in all courses. How many students are there in this class?
12+10-3+26 = 45 (person)
There are 45 students in this class.
5. There are 50 people in the class, 32 can ride a bike, 265,438+0 can skate, 8 can both, and how many can't both?
50-(30+2 1-8) = 7 (person)
A: There are seven people who can't do either.
6. There are 42 students in a class, 30 students in sports teams and 25 students in literary and art teams, and each student should participate in at least one team. this
How many people are there in the two teams of the class?
30+25-42 = 13 (person)
A: There are 13 students in this class.
The number of people who take the entrance examination in a class is as follows: 20 in math, 20 in Chinese, 20 in English, 8 in math English, 7 in math Chinese, 9 in Chinese English, and none of the three subjects. How many students are there in this class at most? How many people at least?
Analysis and solution as shown in Figure 6, students who get full marks in mathematics, Chinese and English are all in this class. Let's assume that there are y students in this class, which are represented by rectangles. A, B, and C stand for those who get full marks in mathematics, Chinese, and English, respectively, from A∩C=8, A∩B=7, and b ∩ c = 9.
According to the principle of inclusion and exclusion
Y=A+B+c-A∩B-A∩C-B∩C+A∩B∩C+3
That is, y = 20+20+20-7-8-9+x+3 = 39+x.
Let's look at how to find the maximum and minimum value of y.
It can be seen from y=39+x that when x takes the maximum value, y also takes the maximum value; When x takes the minimum value, y also takes the minimum value. X is the number of people who get full marks in mathematics, Chinese and English, so their number should not exceed the number of people who get full marks in two subjects, that is, x≤7, x≤8 and x≤9, from which we get x≤7. On the other hand, students who get full marks in mathematics may not get full marks in Chinese, that is to say, there are no students who get full marks in all three subjects, so x≥0.
When x takes the maximum value of 7, y takes the maximum value of 39+7 = 46, and when x takes the minimum value of 0, y takes the minimum value of 39+0 = 39.
A: There are at most 46 students and at least 39 students in this class. That's all. Welcome questions!
Be short and interesting, and be specific.
There is an urgent need for two math stories. Today is Saturday, 65438+ October, 65438+ May. My father and I went shopping in South Street.
At 8 o'clock in the morning, we arrived at South Street by bus. There happened to be an old man beside the platform, and there was a talking scale beside him.
Seeing me coming, grandpa smiled and said, "Little friend, do you weigh yourself?" ?
I asked a little curiously, "How much does it cost to weigh it once?"
Grandpa replied cheerfully: "You can only weigh 1 yuan at a time, and you can also measure your birth height!"
I thought to myself: this is really killing two birds with one stone!
So I stood firm on the scale. Grandpa turned on the switch and only felt something soft touch my head. Then, a small rectangular note is printed on the machine, which reads: "Weight: 27.0 kg, height 132.5 cm"! This half year is 4 cm taller, but what about the weight?
At this time, I remembered that the math teacher said that "kilogram" has another name called "kilogram". I didn't expect to meet you today, and I know that I have gained 2 kilograms!
I was so happy on the way back! I must exercise my body!
Compare, who uses more units? Hutangqiao Central Primary School Class 3 (2) Cao Kefei
Get up from the bed about 2 meters long in the morning;
Pick up a toothbrush about 6 cm long and start brushing your teeth;
Then, I picked up a towel 40 cm long and 20 cm wide and began to wash my face.
After washing, I took a bowl weighing about 100g and filled it with porridge.
After dinner, I came to school with a schoolbag of about 2 kilograms on my back and started a 40-minute morning reading class.
After two classes, we all stood under the flagpole about 7 meters high to do exercises.
Well, that's all I have to say. Can you speak more fluently than me?
If you want answers to interesting math questions in grade one or grade two, simply write a string of numbers.
For example 1098547566
And then turn this string of numbers into
665745890 1
Subtract the previous digit string from the new digit string.
665745890 1- 1098547566=55589 1 1335
Then add up the results.
5+5+5+8+9+ 1+ 1+3+3+5=45
Then 4+5=9, no matter what is written initially, the final result according to this process must be 9.
Another example is that the 20080808 Olympic Games fell off and became 80808002.
Subtract 80808002-20080808 = 60727194.
6+0+7+2+7+ 1+9+4=36
3+6=9
Who knows that the world-famous mathematical problems have the proof of Fermat's Last Theorem that I am interested in, and the research results of wiles for several years.
Or look for the proof of the latest Poincare conjecture. If you can understand, I admire you.
A simple and interesting math problem 1000/ ((root x)- 1) meter.
Interesting math essay