Solution: 7 *18-6 *19 =126-114 =12.
6* 19-5*20= 1 14- 100= 14
The two numbers removed are 12 and 14, and their product is 12* 14= 168.
10. There are seven numbers in a row, and their average value is 30, the average value of the first three numbers is 28, and the average value of the last five numbers is 33. Find the third number.
Solution: 28× 3+33× 5-30× 7 = 39.
1 1. There are two groups of numbers, the sum of the first group of nine numbers is 63, the average value of the second group is 1 1, and the average value of all the numbers in the two groups is 8. Q: How many numbers are there in the second group?
Solution: If there are X numbers in the second group, then 63+ 1 1x = 8× (9+x), and x=3.
12. Xiaoming took part in six tests, and the average score of the third and fourth tests was 2 points higher than the previous two and 2 points lower than the latter two. If the average score of the last three times is 3 points higher than the previous three times, how many points is the fourth time higher than the third time?
Solution: The third and fourth scores are 4 points more than the first two scores, and the last two scores are 4 points less. It can be inferred that the last two scores are 8 points more than the first two scores. Because the sum of the last three times is 9 points more than the sum of the first three times, the fourth time is 9-8 = 1 (points) more than the third time.
13. Mom goes to the grocery store every four days and to the department store every five days. How many times does mom go to these two stores every week on average? (expressed in decimal)
Solution: Walk 9 times every 20 days, 9÷20×7=3. 15 (times).
The ratio of the average value of 14.B and c to a is13: 7. Find the ratio of the average value of a, b and c to a. ..
Solution: If the number of A is 7, then the number of B and C is * * * 13× 2 = 26 (copies).
So the average value of a, b and c is (26+7)/3= 1 1 (copies).
So the ratio of the average of A, B, C and A is 1 1: 7.
15. The fifth-grade students participated in the pasting of cartons in the school-run factory, with an average of 76 per person. It is known that each student has at least 70 posts, and one student has 88 posts. If you don't count this classmate, then each classmate has an average of 74 posts. What is the maximum number of students who can paste the fastest?
Solution: When counting the students who posted 88 cartons, because he is 88-74 = 14 more than the average of other students, the average number of students has increased by 76-74 = 2 (one), which means that the total number of students is 14 ÷ 2 = 7 (people). So the students who posted the fastest posted the most.
74× 6-70× 5 = 94 (pieces).
16. Class A and Class B had a cross-country marching competition. Class A runs half the distance at a speed of 4.5 km/h and the other half at a speed of 5.5 km/h; During the competition, Class B travels at a speed of 4.5 km/h half the time and at a speed of 5.5 km/h the other half. Q: Who will win Class A or Class B?
Solution: The longer you walk, the shorter it takes. The fast walking distance of Class A is the same as the slow walking distance, and the fast walking distance of Class B is longer than the slow walking distance, so Class B wins.
17. It takes 3 days for the boat to go from city A to city B, and 4 days from city B to city A. How many days does it take to put an unpowered raft from city A to city B?
Solution: It takes 3 days for a boat to go downstream and 4 days for a boat to go upstream, that is to say, the boat travels in still water for 4-3 = 1 (days), which is equal to 3+4 = 7 (days) of the current, that is, the speed of the boat is 7 times of the current. Therefore, the three-day trip of the ship is equal to the 3+3× 7 = 24 (days) trip of the current, that is, it takes 24 days for the raft to drift from city A to city B.
18. Xiaohong and Xiao Qiang started from home at the same time and walked in opposite directions. Xiaohong walks 52 meters per minute and Xiao Qiang walks 70 meters per minute. They meet on the way. If Xiaohong leaves four minutes early and the speed remains the same, and Xiao Qiang walks 90 meters every minute, then the two will still meet at point A. How many meters away is Xiaohong from Xiao Qiang's home?
Solution: Because Xiaohong's speed and meeting place remain the same, Xiaohong's time from departure to meeting is the same twice. In other words, Xiao Qiang walked four points less than the first time. pass by
(70× 4) ÷ (90-70) = 14 (minutes)
It can be seen that Xiao Qiang left for the second time 14, and it is inferred that he left for the first time 18, and their homes were separated.
(52+70) × 18 = 2 196 (m)。
19. Xiaoming and Xiaojun start from A and B at the same time and go in opposite directions. If two people go at the original speed, meet at 4 o'clock; If both are faster than the original speed 1 km/h, meet at 3 o'clock. How many kilometers is it between A and B?
Solution: per hour 1 km, two people walked 6 km at 3 o'clock, which is equivalent to the distance that two people walked at the original speed 1. So the distance between A and B is 6× 4 = 24 (km).
20. Party A and Party B practice running along the 400-meter circular track, and both sides run in opposite directions from the same place on the track at the same time. After the encounter, the speed of A increased by 2m/s, and the speed of B decreased by 2m/s, so that they all returned to their original places within 24 seconds. Find the original velocity of a.
Solution: Because the speed sum of A and B is the same before and after meeting, it takes 24 seconds for them to run a circle together after meeting, so it takes 24 seconds for them to run a circle together before meeting, that is, they meet in 24 seconds.
Suppose A originally ran x meters per second, and then ran (x+2) meters per second after meeting. Because A ran back and forth for 24 seconds and * * ran 400 meters, there is 24x+24 (x+2) = 400, and the solution is x=7, 1/3 meters.
2 1. Two cars, A and B, are driving in opposite directions from two stops A and B on the expressway at the same time. It is known that the speed of car A is 1.5 times that of car B, and the time for car A and car B to arrive at station C is 5: 00 and 16: 00 respectively. When did the two cars meet?
Solution: 9: 24. Solution: It takes 16-5 = 1 1 (hours) for a car to get to station C ... When the second car is driving at 1 1, it takes1.
22. An express train goes in the opposite direction to a local train. The length of the express train is 280 meters and the length of the local train is 385 meters. The time for people sitting on the express train to see the slow train pass is 1 1 sec, so how many seconds does it take for people sitting on the slow train to see the express train pass?
Solution: The speed at which people on the express train see the local train is the same as the speed at which people on the local train see the express train, so the ratio of the length of the two cars is equal to the ratio of the time when the two cars pass by, so the required time is 1 1.
23. Party A and Party B practice running. If Party A lets Party B run 10 meter first, Party A can catch up with Party B after running for 5 seconds; If B runs 2 seconds ahead of A, A can catch up with B in 4 seconds. Q: How many meters do two people run per second?
Solution: The speed difference between Party A and Party B is 10/5=2.
The speed ratio is (4+2): 4 = 6: 4.
So A runs 6 meters per second and B runs 4 meters per second.
24.A, B and C run from A to B at the same time. When a runs to b, b is 20 meters away from b and c is 40 meters away from b; When B ran to B, C was 24 meters away from B. Q:
(1) How many meters are there between A and B?
(2) If it takes 24 seconds for C to run from A to B, what is the speed of A?
Solution: Solution: (1) When B ran the last 20 meters, C ran 40-24 = 16 (meters), the speed of C.
25. On a road, Xiaoming and Xiaoguang ride bikes in the same direction. Xiaoming rides a bike three times as fast as Xiaoguang. Every 10 minute, a bus passes by Xiaoguang, and every 20 minutes, a bus passes by Xiaoming. It is known that a bus will leave the departure station at the same time every time. Q: What is the interval between two adjacent cars?
Solution: Let the vehicle speed be A and the vehicle speed of Xiaoguang be B, then Xiaoming's cycling speed is 3b. According to the catch-up problem "catch-up time × speed difference = catch-up distance", the equation can be listed.
10(a-b)=20(a-3b)，
The solution is a = 5b, that is, the vehicle speed is five times that of low light speed. Xiaoguang walks 10 point, which is equivalent to the dealer's 2 points. Every 10 minute, a car passes through Xiaoguang, and a car is sent every 8 minutes.
26. A hare escaped 80 steps before the hounds caught up with it. Rabbits run 8 steps, hounds only need to run 3 steps, hounds run 4 steps and rabbits can run 9 steps. How many steps does a hound have to run to catch up with a hare?
Solution: The distance of dog running 12 step is equal to the distance of rabbit running 32 steps, and the time of dog running 12 step is equal to the time of rabbit running 27 steps. Therefore, every time the rabbit runs 27 steps, the dog catches up with 5 steps (rabbit step), and the dog needs to run [27× (80 ÷ 5)+80] ÷ 8× 3 = 192 (step) to catch up with 80 steps (rabbit step).
27. Party A and Party B are walking at the same speed along the railway direction, just as a train is coming. The whole train took 18 seconds to pass by Party A, and it took 15 seconds to pass by Party B two minutes later. Q:
(1) What is the speed of the train?
(2) After the train passes by B, how long will it take for A and B to meet?
Solution: (1) If the train speed is a m/s and the pedestrian speed is b m/s, the train speed is 1 1 times the pedestrian speed;
(2) From the tail of the train passing through A to the tail of the train passing through B, the train travels 135 seconds. It takes 1350×11=1485 (seconds) for one person to walk this distance, so two people walk the rest distance because A has already walked135 seconds.
28.A train from A to B can arrive 1 earlier than the original time if the speed is increased by 20%; If you drive at the original speed of 100 km and then increase the speed by 30%, you will arrive at 1 earlier than the original time. Find the distance between a and b.
29. It takes 5 days for A and 6 days for B to complete a job, or 7 days for A and 2 days for B. Q: How many days does it take for Party A and Party B to do this job alone?
Solution: A needs (7*3-5)/2=8 (days)
B demand (6*7-2*5)/2= 16 (days)
30. The pool is equipped with a drain pipe and a drain pipe. When the water pipe 5 is opened alone, the empty pool can be filled, and when the drainage pipe 7 is opened alone, the full pool can be emptied. If the drain pipe is not opened until after 2 o'clock, how long will it take for the pool to be half full?
3 1. Komatsu reads a book, and the ratio of read pages to unread pages is 3: 4. Later, he read 33 pages, and the ratio of pages read and unread became 5: 3. How many pages are there in this book?
Solution: read 3/7 at first, and then read 5/8 altogether.
33/(5/8-3/7) = 33/(11/56) = 56 * 3 =168 pages
32. An assignment can be finished at 6 o'clock, 12 at B, 8 o'clock and 6 o'clock at B. If A is done after 3 o'clock, how long will it take to finish?
Solution: A doing 2 hours is equal to B doing 6 hours, so B needs to do it alone.
6*3+ 12=30 (hours) One person needs to do it 10 hour.
So B still needs (1-3/10)/(1/30) = 21day to complete.
33. There is a batch of parts to be processed. It takes 4 days for Party A to do it alone, and 5 days for Party B to do it alone. If two people cooperate, Party A will make 20 more parts than Party B when completing the task. How many parts are there in this batch?
Solution: The working time ratio of A and B is 4: 5, so the working efficiency ratio is 5: 4.
The workload ratio is also 5: 4, with 5 for what A does and 4 for what B does.
Then A is more than B 1 serving, that is, 20 serving. So nine copies is 180.
So this batch of parts *** 180.
34. It takes team A and team B six days to dig a canal. Team a will dig for three days first, and team b will continue.
Solution: According to the conditions, Party A will dig 3/5 of this canal in 6 days, and Party B will dig it in 2 days.
So B can dig 2/5 in 4 days.
Therefore, B can dig 1 10 in1day, which means that B needs10 days to dig.
Single excavation needs1(1/6-110) =15 days.
35. It takes 40 days for Team A to build a section of road, and 24 days for Team B to complete it alone. Now the two teams start from both ends at the same time, and the result is that they meet 750 meters away from the midpoint. How long is this road?
36. A group of workers completed a project. If eight people can be added, it will be completed in 10 days; If we can add three people, it will take 20 days to finish. Only two people can be added now. How many days will it take to complete the project?
Solution: 1 The workload completed by a person in 1 day is called 1 copy. Compared with the transfer of 8 people, the transfer of 3 people takes 10 days to complete (8-3)× 10=50 (copies). These 50 jobs need to be transferred to 3 people for 0/0 day, so there are 50 workers +00-3 = 2 (people), and the whole project has (2+8) × 10 =100 (jobs). It takes 100÷(2+2)=25 (days) to transfer two people.
37.
Solution: The sum of the areas of triangle AOB and triangle DOC is 50% of that of rectangle.
So the triangle AOB accounts for 32%
16÷32%=50
38.
Solution:1/2 *1/3 =1/6.
So the area of triangle ABC is 6 times that of triangle AED.
39. In the following nine pictures, the large squares are equal in area and the small squares are equal in area. Q: Which figures have the same shadow area as the figure (1)?
Solution: (2) (4) (7) (8) (9)
40. Observe the rules of the following strings and fill in the appropriate numbers in the brackets.
2,5, 1 1,23,47,( ),……
Solution: Fill in 95 in brackets.
Law: Each term in the sequence is equal to twice the previous term minus 1.
4 1. In the table below, the upper and lower lines are arithmetic progression. What is the minimum difference between the two numbers corresponding to the upper and lower numbers?
Solution: 1000- 1=999
997-995=992
Each reduction is 7,999/7 =142 ... 5.
Therefore, the minimum value of subtracting the bottom from the top is 5.
1333- 1= 1332 1332/7= 190……2
So the minimum value of the top minus the bottom is 2.
So the minimum difference is 2.
42. If the four-digit 6 □□□□□ 8 is divisible by 73, what is its quotient?
Solution: It is estimated that the ten digits of this quotient should be 8, and you will know that it is 6 by looking at one digit.
So this quotient is 86.
43. Find the smallest natural number in which all numbers are 7 and divisible by 63.
Answer: 63=7*9
So you need at least 9 7s (because the sum of all numbers must be a multiple of 9).
44. Can1× 2× 3× …×15 be divisible by 9009?
Solution: Yes.
Decompose 9009 into prime factors
9009=3*3*7* 1 1* 13
45. Can the six digits of1,2, 3, 4, 5 and 6 form a six-digit number that is not repeated and can be divisible by 1 1? Why?
Solution: No. Because 1+2+3+4+5+6 = 2 1, if six digits can be divisible by 1 1, then the sum of odd and even digits is 16, one is 5, and the smallest sum of three digits is 65438.
46. There is a natural number, the sum of its minimum two divisors is 4, and the sum of its maximum two divisors is 100. Find this natural number.
Solution: The smallest two divisors are 1 and 3, and the largest two divisors are the quotient of the natural number itself and the natural number divided by 3. The largest divisor and the second largest divisor.
There are five natural numbers with the most divisors in 47. 100. What are they?
Solution: If there happens to be a prime factor, then the maximum divisor is 26=64, and there are 7 divisors;
If there are just two different prime factors, then the maximum approximate numbers are 23× 32 = 72 and 25× 3 = 96, each with 12 divisors;
If there are only three different prime factors, then the maximum approximate numbers are 22× 3× 5 = 60, 22× 3× 7 = 84 and 2×32×5=90, each with 12 divisors.
So the natural numbers with the most divisors within 100 are 60, 72, 84, 90, 96.
48. Write three natural numbers less than 20 so that their greatest common divisor is 1, but they are not coprime.
Solution: 6 10, 15
49. There are 336 apples, 252 oranges and 2 10 pears. How many copies of the same gift can you share with these fruits at most? How much are the three kinds of fruits in each gift?
Solution: 42 copies; There are 8 apples, 6 oranges and 5 pears in each serving.
50. The least common multiple of three continuous natural numbers is 168. Find these three numbers.
Answer: 6, 7, 8. Tip: Two adjacent natural numbers must be coprime, and the least common multiple is equal to the product of these two numbers. If there is only one even number among three adjacent natural numbers, the least common multiple is equal to the product of these three numbers; If there are two even numbers, the least common multiple is equal to half of the product of these three numbers.
5 1. A deck of playing cards ***54, the top card is the king of hearts. If you move the top 12 cards to the bottom at a time without changing their order and direction, how many times will it take for the K of hearts to appear at the top again?
Solution: Because [54, 12] = 108, every time the card of 108 is moved, it returns to the original situation. Because every time you move 12 cards, at least move 108÷ 12=9 (times).
Grandpa said to Xiaoming, "I am seven times your age now, six times your age in a few years, five times, four times, three times and two times your age in a few years." Do you know the age of Grandpa and Xiaoming now?
Solution: Grandpa is 70 years old and Xiaoming 10 years old. Tip: The age difference between Grandpa and Xiaoming is the common multiple of 6, 5, 4, 3 and 2. Considering the actual situation of age, take the least common multiple. (60 years old)
53. The number obtained by adding or subtracting 6 from a prime number is still a prime number. How many such prime numbers can you find within 50? And write them down.
Solutions: 1 1, 13,17,23,37,47.
In August of the summer vacation, Xiaoming stayed at his grandmother's house for five days. The dates of these five days are prime numbers, and only one day is a composite number. These four prime numbers are composite number minus 1, composite number plus 1, composite number multiplied by 2 minus 1, and composite number multiplied by 2 plus 1. Q: When was Xiaoming with his grandmother?
Solution: Let this composite number be a, then these four prime numbers are (a- 1), (a+ 1), (2a- 1) and (2a+ 1) respectively. Because (A- 1) and (A+ 1) are prime numbers with a difference of 2, there are five groups of1~ 31:3,5; 5,7; 1 1, 13; 17, 19; 2 1,3 1。 After trial calculation, the meaning of the question can only be satisfied when a = 6, so these five days are August 5, 6, 7, 1 1, 13.
55. There are two integers, and their sum happens to be two numbers with the same number, and their product happens to be three numbers with the same number. Find these two integers.
Solution: 3, 74; 18,37。
Tip: Three digits with the same sign must have a factor of11. Because11= 3× 37, one of these two integers is a multiple of 37 (only 37 or 74) and the other is a multiple of 3.
56. On a wooden stick with a length of 100 cm, dye a red dot every 6 cm from left to right, and dye a red dot every 5 cm from right to left at the same time, and then saw the wooden stick section by section along the red dot. Q: How many short sticks are there with the length of 1 cm?
Solution: Because 100 is divisible by 5, it can be seen that the whole is colored from left to right. Because the least common multiple of 6 and 5 is 30, that is, the red dot is dyed at 30 cm at the same time, so the dyeing occurs in a period of 30 cm. The situation of a cycle is shown in the following figure:
As can be seen from the above picture, there are two wooden sticks of 1 cm in a period of time. So there are six of the three periods, that is, 90 cm, and finally there are 1 in1and 7 in * *.
If a commodity is sold at a fixed price, it will make a profit. If you sell it at 80% of the fixed price, you will lose 832 yuan. Q: What is the purchase price of the goods?
Solution: 8000 yuan. The difference between the two prices is 960+832 = 1792 (yuan), which is 20% of the fixed price selling income, so the fixed price selling income is 1792 ÷ 20% = 8960 (yuan), including the profit 960 yuan, so the purchase price is 8000 yuan.
58. The water in barrel A is 20% more than that in barrel B, and the water in barrel C is 20% less than that in barrel A. Which barrel has more water, B or C?
Solution: more barrels.
59. In the school math contest, three questions, A, B and C, were given. At least 25 people answered correctly, including 10, 13 and 15. If only 1 person answers both questions correctly, how many people answer both questions correctly and one question correctly?
Solution: The number of people who only do two questions correctly is (10+13+15)-25-2×1=1(people).
The number of people who can only answer one question correctly is 25-11-1=13 (people).
60. The school holds a chess competition, which consists of three events: chess, Weiqi and military chess. Each person can participate in at most two events. According to the number of applicants, the school decided to award prizes to the top six players in chess, the top four players in Go and the top three players in military chess. Q: How many people won the prize at most? How many people won the prize at least?
Solution: * * * has 13 winners, so there are at most 13 winners. Everyone can participate in at most two events, that is, at most two prizes, so at least seven people will win the grand prize.
6 1. Among the first 1000 natural numbers, how many natural numbers are neither square nor cubic?
Solution: Because 3 12 < 1000 < 322, 103 = 1000, there are 3 1 squares in the first 1000 natural numbers,/kloc-0. The natural number * * * is1000-(31+10)+3 = 962.
62. How many different three digits can be formed by using the numbers 0, 1, 2, 3 and 4 (numbers are allowed to be repeated)?
Solution: 4*5*5= 100.
63. Choose an advanced group in learning, physical education and health from six classes in grade five. How many different results are there?
Solution: 6*6*6=2 16 species.
64. Known 15 120=24×33×5×7. Q: How many different divisors are there in 15 120 * *?
Solution: The divisors of 15 120 can all be expressed in the form of 2a×3b×5c×7d, where A = 0, 1, 2,3,4, B = 0, 1, 2,3 and C = 0.
65. There are no more than 50 cartoons of Dalin and Kobayashi. What are the possibilities of the number of comic books they own?
Solution: One of them may have 0 ~ 50 books. If they have n books, Dalin may have 0 ~ n books, which means that the distribution of these n books between two people is (n+ 1). Therefore, there are1+2+3 ...+51=1326 (species) under all possible distributions of no more than 50 books.
66. In the picture on the right, take the shortest route from point A to point B, step by step. How many different ways are there? (Note: The same route but different steps are considered as different ways. )
Solution: 80 kinds. Tip: There are 10 different routes from A to B * *, and each route is 5 line segments long. Take one or two line segments at a time, and each route has eight ways, so the different ways are 8× 10=80 (species).
67. There are five different books, which are lent to three students, and each student borrows one. How many different ways are there?
Solution: 5*4*3=60 kinds.
68. Five students borrowed three different books, and each student borrowed at most one. How many different ways are there?
Solution: 5*4*3=60 kinds.
69. How many three digits * * * have exactly two digits?
Solution: Among the 900 three-digit numbers, there are 9× 9× 8 = 648 (pieces) with different three digits, 9 with the same three digits and 900—648—9=243 (pieces) with the same two digits.
70. Take two numbers from 1, 3,5 and two numbers from 2,4,6. * * * How many four digits can you make up?
Solution: There are three ways to choose two from odd numbers and three ways to choose two from even numbers. * * * There are 3×3×4! =2 16 (pieces).
7 1. How many acute angles are there in the left picture?
Solution: C( 1 1, 2)=55.
72. A circle 10 people, choose two people who are not adjacent. How many different methods are there?
Solution: C c( 10/0,2)-10 = 35 kinds.
The grass on the pasture grows at a constant speed every day. This kind of grass can feed 27 cows for 6 weeks or 23 cows for 9 weeks. So how many weeks can it feed 2 1 cow?
Solution: If 1 cow eats 1 week of grass is 1 serving, then 27 cows eat 162 for 6 weeks, and 23 cows eat 162 for 9 weeks = 45 (serving). 2 1 cow, 15 cows eat the new grass, and the other 6 cows eat the original grass. It takes 72 ÷ 6 = 12 weeks to finish eating.
74. There is a pool, the bottom of which is constantly gushing with spring water. To drain the water in the pool, 10 pumps need to pump for 8 hours, and 8 pumps need to pump for 12 hours. How many hours will it take if six pumps are used?
Solution: Take the water pumped by 1 pump as 1. The amount of spring water per hour is
(8×12-10× 8) ÷ (12-8) = 4 (copies).
The raw water in the pool is (10-4) × 8 = 48 (parts), and the pumping capacity of 6 pumps is 48÷(6-4)=24 (hours).
75. stipulate that a * b = (b+a) × b, and find (2*3)*5.
Solution: 2*3=(3+2)*3= 15.
15*5=( 15+5)*5= 100
76. 1! +2! +3! +…+99! What is the unit number?
Solution: 1! +2! +3! +4! = 1+2+6+24=33
From 5! At first, the single digits of each item will be 0.
So 1! +2! +3! +…+99! The number of units is 3.
77 (1). There are many small flags of four colors, three of which are randomly taken out and arranged in a line to represent various signals. How many of the 200 signals are exactly the same?
Solution: 4*4*4=64
200÷64=3……8
So at least four signals are exactly the same.
77.(2) More than 370 freshmen enrolled this year were all born in the same year. Try to explain: at least two of them were born on the same day.
Solution: Because there are at most 366 days in a year, it can be regarded as 366 drawers.
Because of 370>, according to the pigeon hole principle, at least two people were born on the same day.
78. Take six random natural numbers from the previous 1 1, and prove that two of them are certain coprime.
Proof: Divide the first 1 1 natural numbers into the following five groups.
( 1,2,3)(4,5)(6,7)(8,9)( 10, 1 1)
When six numbers are put into five groups, there must be two numbers in the same group, so these two numbers must be prime numbers.
79. Xiaoming went to climb the mountain. He walks 2.5 kilometers per hour when going up the mountain, 4 kilometers per hour when going down the mountain, and it takes 3.9 hours to go back and forth. How many kilometers did Xiaoming walk back and forth?
80. There are two wharves A and B along the Yangtze River. It is known that passenger ships sail 500 kilometers from A to B and 400 kilometers from B to A every day. If it takes 18 days for a passenger ship to sail five times between Pier A and Pier B, what is the distance between the two piers?
Solution: 800 kilometers. Tip: the speed ratio of a to b and b to a is 5: 4, which is used in a to B.