It takes 20 hours and 16 hours to open the two water pipes A and B and fill a pool of water respectively. It takes 10 hour for water pipe c to be opened alone. If there is no water in the pool, open both water pipes A and B at the same time, and then open drain pipe C after 5 hours. How many hours does it take to fill the swimming pool?
Solution:
1/20+116 = 9/80 indicates the working efficiency of both parties.
9/80× 5 = 45/80 indicates the water quantity after 5 hours.
1-45/80 = 35/80 indicates the required water intake.
35/80 ÷ (9/80-110) = 35 means it takes 35 hours to fill it.
A: It will take 35 hours to fill the pool after 5 hours.
2. The problem that chickens and rabbits are in the same cage
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
The sum of three digits is 17. Ten digits are larger than single digits 1. If the hundred digits and the single digits of this three-digit number are reversed, a new three-digit number is obtained, which 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.
Four. permutation and combination
There are five couples in a circle, so that the husband and wife of each couple can move next to each other. The arrangement 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.
Verb (abbreviation of verb) principle of inclusion and exclusion
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. A box of 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. How many balls must be taken out of the bag in order to ensure that the balls taken out contain at least 7 balls of the same color?
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
Seven. Distance problem
Dogs run five steps, horses run three steps, horses run four steps and dogs run 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.
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.