1. There are 100 chickens and rabbits. A chicken has 28 fewer legs than a rabbit. How many chickens and rabbits are there?
Solution:
4 *100 = 400,400-0 = 400 Suppose all rabbits have 400 rabbit feet, then the chicken feet are 0, and the chicken feet are 400 less than the rabbit feet.
400-28 = 372 The actual number of feet of chickens is only 28 less than that of rabbits, with a difference of 372. Why?
4+2 = 6 This is because as long as a rabbit is replaced by a chicken, the total number of rabbits will decrease by 4 (from 400 to 396) and the total number of chickens will increase by 2 (from 0 to 2), and the difference between them is 4+2 = 6 (that is, the original difference is 400-0 = 400, and now the difference is 396).
372 ÷ 6 = 62 indicates the number of chickens, that is to say, because 62 rabbits in 100 are supposed to be chickens, the foot difference is changed from 400 to 28, and1* * is changed to 372 rabbits.
100-62 = 38 indicates the number of rabbits.
Three. Digital problem
1. Write 2005 natural numbers from 1 to 2005, and get a multi-digit 123456789...2005. What is the remainder of this multi-digit divided by 9?
Solution:
Firstly, the characteristics of numbers divisible by 9 are studied: if the sum of numbers on each digit can be divisible by 9, then this number can also be divisible by 9; If the sum of each number is not divisible by 9, then the remainder is the remainder obtained by dividing this number by 9.
Problem solving:1+2+3+4+5+6+7+8+9 = 45; 45 is divisible by 9.
And so on: 1 to 1999 The sum of digits of these numbers can be divisible by 9.
10 ~19, 20 ~ 29 ... 90 ~ 99 All digits in the tenth place appear10 times, so the sum of digits in the tenth place is 10+20+30+...+90 = 450.
Similarly, the sum of hundreds of digits from 100 to 900 is 4500, which can also be divisible by 9.
That is to say, the sum of the digits of each of these continuous natural numbers (1~999) can be divisible by 9;
Similarly, the sum of hundreds, tens and single digits of these continuous natural numbers (1000~ 1999) can be divisible by 9 ("1" in thousands is not considered here, and we are short of 2000200 1200320042005).
The sum of a * * * 999 "1"from 1000 to 1999 is 999, which can also be divisible;
The sum of digits of 200020012002200320042005 is 27, which is exactly divisible.
The final answer is that the remainder is 0.
2.a and B are two nonzero different natural numbers less than 100. Find the minimum value of A-B in a+b ...
Solution:
(A-B)/(A+B)=(A+B-2B)/(A+B)= 1-2 * B/(A+B)
The previous 1 will not change, only the following minimum value is needed, and (A-B)/(A+B) is the maximum value.
When B/(A+B) is the minimum, (A+B)/B is the maximum.
The problem is transformed into finding the maximum value of (a+b)/b.
(A+B)/B = 1+A/B, and the maximum possibility is A/B = 99/ 1.
(A+B)/B = 100
The maximum value of (A-B)/(A+B) is 98/ 100.
3. It is known that A.B.C are all non-zero natural numbers, and the approximate value of A/2+B/4+C/ 16 is 6.4, so what is its accurate value?
The answer is 6.375 or 6.4375.
Because a/2+b/4+c/16 = 8a+4b+c/16 ≈ 6.4,
So 8A+4B+C≈ 102.4, because a, b and c are non-zero natural numbers, and 8A+4B+C is an integer, which may be 102 or 103.
When it is 102, 102/ 16 = 6.375.
When it is 103, 103/ 16 = 6.4375.
4. The sum of three digits is 17. Ten digits are greater than one digit 1. If the hundred digits of this three-digit number are switched with the single digits to get a new three-digit number, the new three-digit number is larger than the original three-digit number 198. Find the original number.
The answer is 476.
Solution: If the original digit is a, then the decimal digit is a+ 1 and the hundredth digit is 16-2a.
According to the equation100a+10a+16-2a-100 (16-2a)-10a-a =198.
A = 6, then A+ 1 = 7 16-2a = 4.
A: The original number was 476.
Write 3 in front of a two-digit number, and the three-digit number is 7 times more than the original two-digit number by 24. Find the original two digits.
The answer is 24.
Solution: let two digits be a, then three digits are 300+a.
7a+24=300+a
a=24
A: The two-digit number is 24.
6. After exchanging a two-digit unit with a ten-digit number, a new number is obtained. When it is added to the original number, the sum is exactly the square of the natural number. What's the total?
The answer is 12 1.
Solution: Let the original two digits be 10a+b, then the new two digits are10B+A.
Their sum is10a+b+10b+a =11(a+b).
Because this sum is a square number, it can be determined that A+B = 1 1.
So this sum is1/kloc-0 /×11=121.
A: Their total is 12 1.
7. The last digit of six figures is 2. If you move 2 to the first place, the original number is three times the new number. Find the original number.
The answer is 857 14.
Solution: Let the original six digits be abcde2 and the new six digits be 2abcde (you can't put a horizontal line on the letters, please treat the whole as six digits).
Let abcde (five digits) be x, then the original six digits are 10x+2, and the new six digits are 200000+x.
According to the meaning of the question, (200000+x) × 3 = 10x+2.
The solution is x = 857 14.
So the original number is 857 142.
Answer: The original number was 857 142.
8. There is a four-digit number, the sum of single digits and hundredths is 12, and the sum of tens and thousands is 9. If one digit is exchanged with a hundred digits, and thousand digits are exchanged with ten digits, the new number will increase by 2376. Find the original number.
The answer is 3963.
Solution: If the original four digits are abcd, the new digits are cdab, D+B = 12, and A+C = 9.
According to "the new number is 2376 more than the original number", it can be known that abcd+2376=cdab, and the vertical column is convenient for observation.
Speed up the receiving and delivery system
2376
cdab
According to d+b = 12, we can know that d and b may be 3 and 9; 4、8; 5、7; 6、6。
Looking at the unit of vertical position again, we can know that only when d = 3 and b = 9; Or d = 8 and b = 4.
Take d = 3, b = 9, and substitute it into the vertical hundreds, and you can determine that the tenth digit has carry.
According to A+C = 9, we can know that A and C may be 1 and 8; 2、7; 3、6; 4、5。
Looking at the ten digits in the vertical form again, we can see that it only holds when c = 6 and a = 3.
Then substitute the vertical thousand, and it will be established.
Get: abcd=3963
Then take d = 8 and b = 4 and substitute them into the vertical decimal places, so we can't find a number suitable for the vertical decimal places, so it doesn't hold.
9. There is a two-digit number. If you divide it by one digit, the quotient is 9 and the remainder is 6. If you divide two digits by the sum of one digit and ten digits, the quotient is 5 and the remainder is 3. Find this two-digit number.
Solution: Let this two-digit number be ab.
10a+b=9b+6
10a+b=5(a+b)+3
The simplified result is the same: 5a+4b = 3.
Because a and b are both one-digit integers,
Get a = 3 or 7 and b = 3 or 8.
The original number was 33 or 78.
10. If it is10,21in the morning, what time is it after 28799 ... 99 (a * * * has 20 9s)?
The answer is 10: 20.
Solution:
(28799 ... 9 (20 9s)+1)/60/24 is divisible, that is to say, just after an integer day, the time is still 10: 2 1, because the previous calculation added1minute, so now.
Four. permutation and combination
1. There are five couples in a circle, so that the husband and wife of each couple move next to each other. The arrangement method is ()
The power of A 768 kinds of B 32 kinds of C 24 kinds of D 2 in 10
Solution:
According to the multiplication principle, there are two steps:
In the first step, five couples are regarded as five whole, with 5× 4× 3× 2× 1 = 120 different arrangements. But because they form a circle end to end, there will be five repetitions, so the actual arrangement is only 120 ÷ 5 = 24.
Step 2, each couple can exchange positions with each other, that is, each couple has two permutations, totaling * * *, 2× 2× 2 = 32.
Combining these two steps, there are 24× 32 = 768 species.
If you write the letter of the English word hello wrong, the possible mistake is * * * ().
A 1 19 species B 36 species C 59 species D 48 species
Solution:
5 Full permutation 5*4*3*2* 1= 120
There are two L's, so 120/2=60
There is a correct one, so 60- 1=59.
Verb (abbreviation of verb) principle of inclusion and exclusion
1. There are 100 kinds of extreme poverty. Among them, there are 68 kinds of calcium and 43 kinds of iron. Then, the maximum and minimum values of foods containing calcium and iron are () respectively.
A 43，25 B 32，25 C32， 15 D 43， 1 1
Solution: According to the exclusion principle, the minimum value is 68+43-100 =11.
There are 43 kinds of iron at most.
2. There are only three questions in the final of the multiple intelligence competition. It is known that: (1) 25 students from a school participated in the competition, and each student solved at least one problem; (2) Among all the students who have not solved the first question, the students who have solved the second question are twice as many as the third question: (3) There are more students who have only solved the first question than the rest1; (4) Half of the students who only solved one problem did not solve the first problem, so the number of students who only solved the second problem is ()
a、5 B、6 C、7 D、8
Solution: According to "everyone answers at least one of the three questions", the answering situation is divided into seven categories: only answer question 1, only answer question 2, only answer question 3, only answer question 1 and 2, only answer question 1 and 3, only answer questions 2 and 3, only answer question 1.
The number of people in each category is a 1, a2, a3, a 12, a 13, a23, a 123.
According to (1), a1+a2+a3+a12+a13+a23+a123 = 25 ... ①.
From (2): A2+A23 = (A3+A23) × 2 ...
According to (3), a12+a13+a123 = a1-1... ③.
From (4): A 1 = A2+A3...④
From ②, A23 = A2-A3× 2...⑤
Then a12+a13+a123 = a2+a3-16 is obtained from ③ ④.
Then substitute ④ ⑤ ⑤ into ① and sort it out.
a2×4+a3=26
Since a2 and a3 both represent the number of people, we can find their integer solutions:
When A2 = 6,5,4,3,2 and 1, A3 = 2,6, 10, 14, 18 and 22.
According to A23 = A2-A3× 2...⑤, we can know: a2 & gta3.
Therefore, only A2 = 6 and A3 = 2 are eligible.
Then we can deduce A 1 = 8, a12+a13+a123 = 7, A23 = 2, and the total number of people = 8+6+2+7+2 = 25, and check that all conditions are equal.
Therefore, the number of students who only solved the second problem A2 = 6.
There are five questions in an exam. 1, 2, 3, 4 and 5, the correct rates of participants are 95%, 80%, 79%, 74% and 85% respectively. If three or more answers are qualified, what is the passing rate of this exam at least?
Answer: The pass rate is at least 7 1%.
Suppose there are 100 people taking the exam.
100-95=5
100-80=20
100-79=2 1
100-74=26
100-85= 15
5+20+2 1+26+ 15 = 87 (indicating that 1 made the most mistakes among the five questions).
87 ÷ 3 = 29 (indicating that the number of people who made mistakes in three of the five questions is the highest, that is, the number of people who failed is at most 29).
100-29 = 7 1 (the minimum number of people passing is actually ok)
The pass rate is at least 7 1%.
6. Pigeon hole principle, parity.
1. A cloth bag contains gloves of the same size and different colors, including black, red, blue and yellow. How many gloves do you have to pull out to ensure three pairs of gloves with the same color?
Solution: Four different colors can be regarded as four drawers with gloves as elements. Make sure one pair is the same color. 1 There are at least two gloves in the drawer. According to the pigeon hole principle, at least five gloves should be pulled out. At this time, take out 1 double color, and there are 3 gloves left in the last 4 drawers. According to the pigeon hole principle, as long as two gloves are pulled out, one glove can be guaranteed to be the same color, and so on.
Think of four colors as four drawers. To ensure that there are three pairs of the same color, first consider ensuring that there are 1 pair, and then you must pull out five gloves. At this time, after taking out 1 with the same color, there are still three gloves left in the four drawers. According to the pigeon hole principle, as long as two gloves are pulled out, it can be guaranteed that 1 pair is of the same color. By analogy, it is guaranteed that there are 3 pairs of the same color, and the gloves drawn out by * * * are: 5+2+2=9 (only)
Answer: At least 9 pairs of gloves must be drawn out to ensure that there are 3 pairs of gloves of the same color.
2. There are several building blocks with four colors, and each person can take 1-2 blocks at will. At least a few people can take them to ensure that three people can get exactly the same.
The answer is 2 1.
Solution:
There are four different ways for each person to take 1 piece, and there are six different ways for each person to take 2 pieces.
When there are 1 1 individuals, it can be guaranteed that at least two people will be exactly the same:
When there are 2 1 person, at least 3 people can be guaranteed to be exactly the same.
3. A box contains 50 balls, of which 10 is only red, 10 is only green, 10 is only yellow, 10 is only blue, and the rest are white balls and black balls. In order to ensure that the balls taken out contain at least 7 balls of the same color, Q: How many balls must be taken out of the bag at least?
Solution: It needs to be discussed in different situations, because it is impossible to determine the number of black balls and white balls.
When there is no black ball or white ball greater than or equal to 7, it is:
6*4+ 10+ 1=35 (pieces)
If there are seven black balls or white balls, that is:
6 * 5+3+ 1 = 34 (pieces)
If there are eight black balls or white balls, that is:
6*5+2+ 1=33
If there are nine black balls or white balls, that is:
6*5+ 1+ 1=32
4. There are four piles of stones on the ground, and the number of stones is 1, 9, 15 and 3 1 respectively. If 1 stone is taken out from three piles at the same time and then put into the fourth pile, after several calculations, can the number of stones in these four piles be the same? (If yes, please explain the specific operation; If not, please explain why. )
No
Because the total is1+9+15+31= 56.
56/4= 14
14 is an even number.
The original 1, 9, 15, 3 1 are all odd numbers, and taking out 1 and putting in 3 are also odd numbers. If you add and subtract odd numbers several times, the result must still be odd, and it is impossible to get even numbers (14).
Seven. Distance problem
1. The dog runs five steps, the horse runs three steps and the distance between the horse and the dog runs seven steps. Now that the dog has run 30 meters, the horse began to chase it. Q: How far can the dog run before the horse can catch up with it?
Solution:
According to "the distance between a horse running four steps and a dog running seven steps", it can be assumed that each step of a horse is 7x meters and each step of a dog is 4x meters.
According to "it takes five steps for a dog to run and three steps for a horse", at the same time, if the horse runs 3*7x meters = =2 1x meters, the dog runs 5 * 4x = 20m.
It can be concluded that the speed ratio of horse to dog is 2 1x: 20x = 2 1: 20.
According to "Now the dog has run 30 meters", we can know that the distance between the dog and the horse is 30 meters, and the difference between them is 2 1-20 = 1. Now, what is the distance of 2 1 of a horse, that is, 30 ÷ (2 1-20) × 265438.
2.a train and A train leave from A and B at the same time, and meet 40 kilometers from the midpoint in a few hours? It is known that it takes 8 hours for a car dealership to complete the trip, and it takes 10 hour for a car dealership to complete the trip. How many kilometers is it between A and B?
The answer is 720 kilometers.
According to "the whole journey of car A is 8 hours, and the whole journey of car B is 10 hour", when we met, there were 8 copies of car A 10 and 8 copies of car B (the whole journey 18), and the difference between the two cars was 2 copies. Because the two cars meet at the midpoint of 40 kilometers, it means that the distance difference between the two cars is (40+40) kilometers. So the formula is (40+40) ÷ (10-8) × (10+8) = 720km.
On the 3.600-meter circular runway, two brothers run clockwise from the same starting point at the same time and meet once every 12 minutes. If the speed of two people remains the same and they still start from the original starting point at the same time, and my brother runs counterclockwise, then they meet every 4 minutes. How many minutes does it take them to run once?
The answer is that two people need 6 points 12 points to run a lap.
Solution:
600÷ 12=50, indicating the speed difference between brother and brother.
600÷4= 150, indicating the sum of the speeds of the elder brother and the younger brother.
(50+ 150)÷2= 100, indicating faster speed. The method is to sum a larger number in the difference problem.
(150-50)/2=50, which means the speed is slow. The method is to sum the smaller numbers in the difference problem.
600÷ 100=6 minutes, indicating the time spent by the fastest runner.
600/50= 12 minutes, indicating the time spent by slow runners.
4. The length of the local train 125m, and the speed of the express train 17m per second. The length of the express train is 140m, and the speed is 22m per second. The local train runs ahead and the express train catches up from behind. Then, how long will it take for the express train to catch up with the rear of the local train and completely surpass the local train?
The answer is 53 seconds.
The formula is (140+125) ÷ (22-17) = 53 seconds.
It can be understood that "from catching up with the rear of the local train to completely overtaking the local train" means that the point at the rear of the local train catches up with the point at the front of the local train, so the catching distance should be the sum of the two captains.
On the 300-meter-long circular runway, both parties set off side by side in the same direction at the same time. The average speed of Party A is 5 meters per second, and that of Party B is 4.4 meters per second. How many meters before the starting line did they meet for the first time?
The answer is 100m.
300 ÷ (5-4.4) = 500 seconds, indicating the catch-up time.
5× 500 = 2500m, indicating the distance traveled when A catches up with B..
2500 ÷ 300 = 8 laps ... 100 meters, which means that the total distance of a chase is 8 laps, which is more than 100 meters, which means that we met before the original starting line 100 meters.
6. Alone by the railway. After hearing the whistle of the distant train, 57 seconds later, the train passed in front of her. It is known that the train whistle is at a distance of 1360 meters (the track is straight), and the sound travels 340 meters per second. Find the speed of the train (get the reserved integer).
The answer is 22 meters per second.
Formula:1360 ÷ (1360 ÷ 340+57) ≈ 22m/s.
Key understanding: the car arrives 57 seconds after hearing the sound, which means that the car has already traveled from the place where the sound was made when people heard the sound 1360 ÷ 340 = 4 seconds. That is, it took 4+57 = 6 1.360m * * 0 seconds.
7. The hound found a running hare in front of it at 10 meters away and immediately chased it. The hound took a big step. It ran five steps, and the rabbit ran nine steps, but the rabbit ran very fast. The hound runs two steps, but the rabbit can run three. Ask the hound how many meters it must run to catch up with the rabbit.
The correct answer is that the hound must run at least 60 meters to catch up.
Solution:
From "the hound runs five steps and the rabbit runs nine steps", we can know that for every meter the hound walks, the rabbit walks 5/9 meters. According to "the hound can run 2 steps and the rabbit can run 3 steps", at the same time, the hound can run 2 meters and the rabbit can run 5/9 a * 3 = 5/3 a meters. From this, we can know that the speed ratio between the hound and the rabbit is 2A: 5/3A = 6: 5, that is to say, the hound runs 60 meters and the rabbit runs 50 meters, and the original difference of 10 meter is just used up.
8. In AB, the ratio of time for two people to complete a cycle by bike is 4:5. If two people start exercising from AB at the same time, they will meet after 40 minutes and then move on. How many minutes will B arrive at A later than A arrive at B?
Answer: 18 minutes
Solution: let the whole journey be 1, the speed of a be x, and the speed of b be y.
Column 40x+40y= 1
x:y=5:4
X= 1/72 y= 1/90。
A takes 72 minutes and B takes 90 minutes.
So get the solution.
9. Car A and car B leave relatively from AB at the same time. After the first encounter, the two cars continued to drive and returned immediately after reaching the starting point of the other side. The distance from B to the second meeting is 1/5 of AB's whole journey. It is known that car A traveled 120km when they first met. How many kilometers is AB?
The answer is 300 kilometers.
Solution: Draw a line diagram, and you can know that when two people meet for the first time, they walked 1ab, and from the beginning to the second meeting, they walked 3 AB, so it can be calculated that the distance between A and B is three times that before the first meeting. That is, the distance A * * * walks is 120 * 3 = 360 kilometers. As can be seen from the line chart, A * * * has gone all the way (1+ 1/5).
So 360 ÷ (1+1/5) = 300 km.
It takes 4 hours and 6 hours for two people to ride bicycles from A to B respectively. Now Party A and Party B start from AB and go in opposite directions at the same time. When they meet, they are 2 kilometers away from the midpoint of AB. If two people go to B separately, they will turn back immediately after A ... There is () kilometers between the second meeting point and the first meeting point.
10. It takes 6 hours for a ship to go back and forth at the same speed and go downstream; Countercurrent for 8 hours. If the current speed is 2 kilometers per hour, what is the distance between the two places?
Solution: (1/6-1/8) ÷ 2 =1/48 indicates the fraction of water velocity.
2 ÷ 1/48 = 96 km indicates the total distance.
165438+
Solution:
Meeting is four-seventh of the whole journey, which means that the speed ratio between Party A and Party B is 4: 3.
The time ratio is 3: 4.
So the whole time of the express train is 8/4 * 3 = 6 hours.
6 * 33 =198km
12. Xiaohua travels from A to B, 1 rides, 2/3 rides; Returning from B to A, three-fifths of people rode bicycles and two-fifths rode cars, resulting in a delay of half an hour. It is known that the cycling speed is12km, and the cycling speed is 30km. Q: How many kilometers is it between A and B?
Solution:
Take the distance as 1 and get the time coefficient.
Departure time coefficient: 1/3÷ 12+2/3÷30.
Return time coefficient: 3/5÷ 12+2/5÷30.
The difference between them: (3/5 ÷12+2/5 ÷ 30)-(1/3 ÷12+2/3 ÷ 30) =1/75.
Time to go:1/2× (1/3 ÷12) ÷1/75 and1/2× (2/3 ÷ 30)/kloc-.
Distance:12× [1/2× (1/3 ÷12) ÷1/75]+30× [1/2×
Eight. balanced
1. Party A and Party B are fishing by the river. Party A caught three, and Party B caught two, and was about to eat. One person asked to eat with them, so three people shared the five fish equally. Passers-by left 10 yuan to show their gratitude. How do Party A and Party B share it? quickly
Answer: A accepts 8 yuan and B accepts 2 yuan.
Solution:
"Three people share five fish equally, and the guests take out 10 yuan", which can be understood as the total value of five fish is 30 yuan, so each fish is worth 6 yuan.
Because "A caught three" means that A voted 3 * 6 = 18 yuan before eating, and "B caught two" means that B voted 2 * 6 = 12 yuan before eating.
And the value of both parties is 10 yuan, so
A can also get back 18- 10 = 8 yuan.
B You can also get back 12- 10 = 2 yuan.
It happened to be the money paid by the guests.
2. The cost of a commodity this year has increased by 1 compared with last year, but it still maintains the original selling price, so the profit per share has decreased by two fifths. So, what is the cost of this commodity this year?
Answer 22/25
Thinking about the best line drawing;
If last year's original cost is regarded as 20 shares and profit as 5 shares, then this year's cost will increase by110, that is, 22 shares, while the profit will decrease by 2/5, and this year's profit will be only 3 shares. The increased cost of two copies is exactly the reduced profit of two copies. The price is 25 copies.
So this year's cost accounts for 22/25 of the selling price.
3. Two cars, A and B, start from A and B respectively and go in opposite directions. When they set off, the speed ratio of A to B was 5:4. After they met, A's speed dropped by 20% and B's speed increased by 20%. In this way, when A arrives at B, B is still 10 km away from A, so how many kilometers are A and B apart?
Solution:
It turns out that the speed ratio of A and B is 5:4.
Current A: 5× (1-20%) = 4.
Current B: 4× (1+20%) 4.8.
After A arrives at B, B leaves A: 5-4.8 = 0.2.
Total distance: 10 ÷ 0.2× (4+5) = 450km.
4. The circumference of the cylinder bottom is reduced by 25%, and the volume is increased by 1/3. What is the ratio of the present height to the original height?
The answer is 64: 27.
Solution: According to "the perimeter is reduced by 25%", we know that the perimeter is 3/4, then the radius is 3/4 and the area is 9/ 16.
According to "volume increase 1/3", we can know that the volume is 4/3 of the original.
Volume base area = height
The current height is 4/3 ÷ 9/ 16 = 64/27, which means that the current height is 64/27 of the original height.
Or current height: original height = 64/27: 1 = 64: 27.
Four kinds of fruits, bananas, apples, oranges and pears, were shipped from a market, including 30 tons of bananas and 45 tons of oranges and pears. Oranges account for exactly two-thirds of the total 13. How many tons of fruit did a * * * transport?
Question 2: The answer is 65 tons.
Oranges+apples = 30 tons
Banana+orange+pear = 45 tons
So orange+apple+banana+orange+pear = 75 tons.
Orange (banana+apple+orange+pear) = 2/ 13
Description: Orange is 2, banana+apple+orange+pear is 13.
Orange+banana+apple+orange+pear * * is 2+ 13 = 15.