1.A and b water pipes need 20 hours and 16 hours to open 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 water pipes A and B at the same time. After 5 hours, open the drain pipe C again. 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. It takes 20 days for Team A to build a canal, and 30 days for Team B to build it alone. If two teams cooperate, the work efficiency will be reduced because of the influence of each other's construction. The working efficiency of team A is four fifths of the original, while that of team B is only nine tenths of the original. Now it is planned to complete the canal in 16 days, and it is required that the two teams cooperate for as few days as possible. So how many days will the two teams cooperate?
Solution: From the meaning of the question, the work efficiency of Party A is 1/20, that of Party B is 1/30, and that of both parties is1/20 * 4/5+1/30 * 9//kloc-0. A's work efficiency >: B's ergonomics.
Because it is required that "the fewer days the two teams work together, the better", Party A should be made to work faster./kloc-If it is too late within 0/6 days, Party A should cooperate with Party B.. Only in this way can the two teams work together as little as possible.
Let the cooperation time be x days, then Party A will do it alone for (16-x) days.
1/20 *( 16-x)+7/ 100 * x = 1
x= 10
A: The shortest cooperation period between Party A and Party B is 10 day.
It takes 4 hours for Party A and Party B to do a job, and 5 hours for Party B and Party C to do a job. Now, please ask Party A and Party C to work together for 2 hours, and the remaining Party B needs to work for 6 hours. How many hours does it take to finish the work alone?
Solution:
According to the meaning of the question, 1/4 means Party A 1 hour, and Party B +0/5 means Party C 1 hour.
(1/4+1/5) × 2 = 9/10 means that Party A worked for 2 hours, Party B worked for 4 hours and Party C worked for 2 hours.
According to "After Party A and Party C work together for 2 hours, Party B needs to work for 6 hours", we can know that the workload of Party A working for 2 hours, Party B working for 6 hours and Party C working for 2 hours is 1.
So1-9/10 =110 means that B works for 6-4 = 2 hours.
110 ÷ 2 =1/20 indicates the work efficiency of Party B. ..
1 ÷ 1/20 = 20 hours means that it takes 20 hours for Party B to complete it alone.
A:B It takes 20 hours to finish it alone.
4. For a project, Party A will do it on the first day, Party B will do it on the second day, Party A will do it on the third day and Party B will do it on the fourth day, and it will be completed in integer days. If B does it on the first day, A does it on the second day, B does it on the third day and A does it alternately on the fourth day, then the completion time will be half a day longer than last time. It is known that it takes 17 days for B to do this project alone. A How many days does it take to do this project alone?
Solution: According to the meaning of the problem,
1/A+ 1/B+ 1/A+ 1/B+…+ 1/A = 1。
1/B+ 1/A+ 1/B+ 1/A+…+ 1/B+ 1/A×0.5 = 1。
(1/ A represents the working efficiency of A, 1/ B represents the working efficiency of B, and the final outcome must be as shown in the above figure, otherwise the second method will not be 0.5 days longer than the first one).
1/a =1/b+1/a× 0.5 (because the previous workload is equal).
Get 1/ A = 1/B× 2.
Because1/b =117.
So 1/ A = 2/ 17, and A = 17 ÷ 2 = 8.5 days.
Master and apprentice both process the same number of parts. When the master finishes12, the apprentice finishes 120. When the master finished the task, the apprentice finished 4/5 of this batch of parts. How many?
The answer is 300.
120 ÷ (4/5 ÷ 2) = 300
You can think of it this way: 1/2 for the first time and 1/2 for the second time are all done at once. Then after the second time, the apprentice completed 4/5, and it can be inferred that half of the 4/5 completed for the first time is 2/5, which is exactly 120.
6. A batch of saplings, if distributed to boys and girls, will average 6 saplings per person; If you plant a single copy for girls, each person will plant 10 trees on average. One for boys, how many trees per person?
The answer is 15 trees.
Formula:1÷ (1/6-110) =15 trees.
7. A pool is equipped with three water pipes. Tube A is the water inlet pipe, tube B is the water outlet pipe, which can fill the pool water in 20 minutes, and tube C is also the water outlet pipe, which can fill the pool water in 30 minutes. Now, first open the first pipe. When the water in the pool just overflows, it takes 18 minutes to open the second and third pipes. When the first pipe is filled with water, open the second pipe instead of the third one. How many minutes does it take to finish drinking water?
The answer is 45 minutes.
1÷ (1/20+1/30) =12 indicates the number of minutes required for Party B and Party C to jointly drain the full pool water.
112 * (18-12) =112 * 6 =12, that is to say, with the cooperation of Party B and Party C, the overflow pond
1/2 ÷18 =1/36 means that A enters the water once every minute.
Finally,1÷ (1/20-1/36) = 45 minutes.
8. The engineering team needs to finish it within the specified date. If Team A does it, it can be finished on schedule. If Team B does it, it will be finished three days later than the stipulated date. If Party A and Party B cooperate for two days first, and then Team B does it alone, it can be completed as scheduled. How many days is the specified date?
The answer is six days.
Solution:
From "If Team B does it, it will be completed three days later than the specified date; If Party A and Party B cooperate for two days first, and then Team B does it alone, it can be completed as scheduled. " ：
B working for three days = a working for two days.
That is, the work efficiency ratio of Party A and Party B is 3: 2.
The working time ratio of Party A and Party B is 2: 3.
The time ratio difference is 1 serving.
The actual time difference is 3 days.
So 3 ÷ (3-2) × 2 = 6 days, which is the time of A, that is, the specified date.
Equation method:
[ 1/x+ 1/(x+2)]×2+ 1/(x+2)×(x-2)= 1
The solution is x = 6.
9. For two candles with the same length, it takes 2 hours to light a thick candle and 1 hour to light a thin candle. One night, the power went out, and Xiao Fang lit two candles to read at the same time. A few minutes later, Xiao Fang put out two candles at the same time and found that the length of the thick candle was twice that of the thin candle. Q: How many minutes was the power cut off?
The answer is 40 minutes.
Solution: Suppose there is a power outage for x minutes.
According to the meaning of the problem in the equation
1- 1/ 120 * x =( 1- 1/60 * x)* 2
The solution is x = 40.
2. The problem that chickens and rabbits are in the same cage
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.