2. Two cars, A and B, start from C and D respectively, and keep going back and forth between A and B. It is known that the speed of car A is 30 kilometers per hour and that of car B is 70 kilometers per hour. The third meeting place and the fourth meeting place of A and B are just100km apart. So how many kilometers is the distance between C and D?
1. In stock trading, stamp duty and commission shall be paid at 0. 1% and 0.3% of the transaction amount respectively for each stock bought and sold. Yao Yao's father bought 3000 shares one day at 10.65 yuan per share, and then sold them at 2.86 yuan per share. In this transaction, he * * *.
13.86*3000*( 1-0. 1%-0.3%)- 10.65*3000*( 1+0. 1%+0.3%)
3. The "Farmer Orchard" fruit supermarket buys winter jujube from its producing area, and the purchase price is per kilogram 1.20 yuan. The distance from the place of origin to the fruit supermarket is 400km, and the freight is per ton of goods 1.50 yuan. If the loss is not considered, the fruit supermarket should achieve a profit of 25%, and the price per kilogram of winter jujube is X yuan.
x- 1.20-400 * 1 * 1.50/ 1000 = 25% x
x=2.4
Jingshan Primary School organizes spring outing for students. If 45 buses are rented, 15 people have no seats. If you rent the same number of 60 buses, one bus is empty. It is known that 45 passenger cars are rented in 220 yuan and 60 passenger cars are rented in 300 yuan. Q: (1) How many students are there in this school? (2) What is the most economical way to rent a car?
There are x students.
(x-15)/45-1= x/60240/60 = 44 * 300 =1200 yuan.
X = 240 220 * 6 =1four 60-seater cars in 320 yuan.
5. The payment method of the manuscript fee is as follows: ① The manuscript fee is not higher than that of 800 yuan; (2) If the payment is not higher than 4,000 yuan in 800 yuan, the tax above 800 yuan 14% shall be paid; (3) If the contribution fee is more than 4,000 yuan, the tax of 1 1% of the total contribution fee shall be paid. Recently, Mr. Wang received a manuscript fee. According to the regulations, he should pay 434 yuan personal income tax. How much is this manuscript fee?
Pay 434 yuan personal income tax, so the manuscript fee exceeds that of 800 yuan.
(4000-800)* 14%=448 No more than 4000 yuan.
434/ 14%=3 100 yuan
800+3 100=3900 yuan
6. The store priced a certain VCD at 140% of the purchase price, and then implemented the preferential policy of "10% discount for guests and taxi fare to 50 yuan". Results Each VCD made a profit 145 yuan. So what's the purchase price of each VCD?
The purchase price is x yuan.
140%x*90%-50-x= 145
X=750
7. The head length of the deep-sea great white shark is 4 meters, the body length is equal to the head length plus the tail length, and the tail length is equal to the head length plus half the body length. What is the total length of this great white shark? Body length =4+ tail length
Tail length =4+ body length /2=4+(4+ tail length) /2
Tail length = 12m
Length = 16m
8. Party A, Party B, Party C and Party D each have 60 copies. If A increases 4 books, B decreases 1 book, C increases 5 times and D decreases by half, then the books of four people are equal. How many books does each of them have?
60/4= 15 copies
A has 15-4= 1 1 serving.
B has 15+ 1= 16 copies.
C has 15/5=3 copies.
Ding You 15*2=30 copies.
9. Master and apprentice produce the same number of parts. The master's pass rate is 94% and the apprentice's pass rate is 80%. Two people have 130 unqualified parts. How many parts do they have?
Let * * * have x parts.
( 1-94%)x+( 1-80%)x = 130
X=500
500*2- 130=870
1. There are two trains, one is102m long, and it runs 20m per second. The length of the train is120m, and it runs at the speed of17m per second. Two cars are driving in the same direction. How many seconds does it take from the first train to catch up with the second train to the departure of the two cars?
2. Someone walks at a speed of 2 meters per second. A train came behind, which took 10 seconds longer than him. As we all know, this train is 90 meters long. Find the speed of the train.
At present, two trains run in the same direction at the same time. 12 seconds later, the express train overtook the local train. The express train runs18m per second, and the local train runs10m per second. If the tails of two trains are flush and driving in the same direction at the same time, the express train will overtake the local train after 9 seconds. Find the body length of two trains.
4. It takes 40 seconds for a train to cross a 440m bridge and 30 seconds for a 3 10/0m tunnel at the same speed. What is the speed and body length of this train?
Xiaoying and Xiao Min took two stopwatches to measure the speed and length of the passing train. Xiaoying used her watch to record that the time that the train passed in front of her was 15 seconds. Xiao Min used another watch to record that it took him 20 seconds to cross the first telephone pole in front and the second telephone pole in the back. It is known that the distance between two poles is 100 meters. Can you help Xiaoying and Xiao Min calculate the total length and speed of the train?
6. It takes 40 seconds for a train to cross a 530-meter bridge and 30 seconds to cross a 380-meter cave at the same speed. Find the speed and body length of this train.
7. The two started from two places along the path next to the railway line and walked at the same speed. A train came, 10 seconds. The whole train passed by A. Three minutes later, B met the train, and the whole train only took 9 seconds to pass by B. How long did the train leave B before they met?
8. Two trains, one with a length of120m and a speed of 20m per second; The other train is160m long and runs at a speed of15m per second. The two cars are driving in opposite directions. How many seconds does it take from the front meeting to the back leaving?
9. Someone walks at a speed of 2 meters per second. The train overtook him from behind 10 seconds. As we all know, the length of this train is 90 meters. Find the speed of the train.
10. Party A and Party B walk along the railway at the same speed. It took 8 seconds for a train to pass by Party A, and only 7 seconds to pass by Party B after leaving Party A for 5 minutes. How many minutes after Party B met the train?
Second, answer the question.
1 1. Fast train length182m, traveling 20m per second, slow train length1034m, traveling 8m per second. The two cars are parallel in the same direction. How long does it take for the express train to cross the local train when the rear of the express train meets the rear of the local train?
12. The length of the express train is182m, the length of the local train is1034m, and the speed of the local train is18m per second. The two cars are parallel in the same direction. How many seconds can an express train pass through the local train when the heads of the two cars are aligned?
13. A person is running along the railway at a speed of 120 meters per minute. A 288-meter-long train came from the opposite side. It took him 8 seconds to find the speed of the train.
14. A train is 600 meters long. It passes through a 200-meter-long tunnel at a speed of 10 meter per second. How long does it take to leave the tunnel from the front to the rear?
———————————— Answer the case ————————
Fill in the blanks
120m
102m
17x meter
20x meters
tail
tail
head
head
1. This problem is the catch-up problem of "two trains". Here "catching up" means that the head of the first train catches up with the tail of the second train, and "leaving" means that the tail of the first train leaves the head of the second train. Draw a line segment as follows:
Suppose it takes x seconds from the first train catching up with the second train to the departure of the two trains, and the equation is:
102+ 120+ 17x = 20x
x =74。
2. Draw a paragraph as follows:
head
90 meters
tail
10x
Assuming that the speed of the train is x meters per second, the equation is obtained.
10 x =90+2× 10
x = 1 1。
head
tail
express train
head
tail
slow train
head
tail
express train
head
tail
slow train
3.( 1) The locomotive heads are in a straight line and driving in the same direction at the same time. Draw a line segment as follows:
Then the express delivery length:18×12-10×12 = 96 (meters).
(2) Align the rear of the car and drive in the same direction. Draw a line segment as follows:
head
tail
express train
head
tail
slow train
head
tail
express train
head
tail
slow train
Then the length of the local train is 18×9- 10×9=72 (meters).
4.( 1) The train speed is: (440-310) ÷ (40-30) =13 (m/s).
(2) The body length is: 13×30-3 10=80 (m).
5.( 1) The train speed is:100 ÷ (20-15) × 60× 60 = 72,000 (m/h).
(2) The body length is 20× 15=300 (m).
6. Set the train body to be x meters long and y meters long.
①②
solve
7. Let the train body be x meters long, A and B each walk y meters per second, and the train travels z meters per second. According to the meaning of the question, list the equations and get.
①②
①-②, so:
After the train left B, they met:
(seconds) (minutes)
8. Solution: The sum of the distances traveled by two cars is exactly the sum of the distances of two train conductors, so the time required to encounter the problem is: (120+60)? (15+20)=8 (seconds).
9. Think of it this way: when the train passes by people, their distance difference is the conductor. Divide the distance difference (90 meters) by the crossing time (10 second) to get the speed difference between the train and people. This speed difference plus the walking speed of people is the speed of the train.
90÷10+2 = 9+2 =11(m)
A: The speed of the train is 1 1 meter per second.
10. It is required that when A and B meet in a few minutes, the relationship between the distance between A and B and their speed must be found, which is related to the movement of the train. The distance between a and b can only be found by the movement of the train. The running time of the train is known, so it is necessary to find out its speed, at least the proportional relationship between it and the speeds of A and B. Because this question is more difficult.
① Find the relationship between the train speed and the speed of Party A and Party B, and let the train length be L, then:
(I) It takes 8 seconds for a train to pass through A, and this process is to catch up with the problem:
Therefore; ( 1)
(i i) It takes seven seconds for the train to pass through B. This process is a meeting problem:
Therefore. (2)
From (1) and (2),
So ...
(2) The distance between locomotive encounter A and train encounter B is:
.
③ Find the distance between A and B when the locomotive meets B. 。
After the locomotive meets A, it takes (8+5×60) seconds for the locomotive to meet B. Therefore, when the locomotive meets B, the distance between A and B is:
(4) How many minutes will A and B meet?
(seconds) (minutes)
A: In another minute, Party A and Party B will meet.
Second, answer the question.
11.1034÷ (20-18) = 91(seconds)
12.182 ÷ (20-18) = 91(seconds)
13.288 ÷ 8-120 ÷ 60 = 36-2 = 34 (m/s)
The speed of the train is 34 meters per second.
14. (600+200)10 = 80 (seconds)
Answer: It takes 80 seconds from the front of the car entering the tunnel to the rear leaving the tunnel.
Average problem
1. In the final exam, Cai Chen scored 89 points in political language, mathematics and English biology, 9 points in political mathematics, 84 points in Chinese English, 86 points in political English, and 65,438 points more in English than in Chinese.
2. Two cotton fields, A and B, with an average yield per mu 1.85 kg. A cotton field is 5 mu, with an average yield of 203 kg per mu. B Average yield per mu of seed cotton in cotton field170kg. B How many acres of cotton fields are there?
3. Given that the sum of eight consecutive odd numbers is 144, find these eight consecutive odd numbers.
4. 8.8 yuan per kilogram of sugar, 7.2 yuan per kilogram of sugar B, and how much sugar B is mixed into 5 kilograms of sugar A to make the price of sugar per kilogram reach 8.2 yuan?
I bought five sheep in the canteen, weighed two sheep at a time, and got ten different weights (kilograms): 47, 50, 5 1, 52, 53, 54, 55, 57, 58, 59. How much do these five sheep weigh?
arithmetic series
1, here is a series of numbers arranged according to the law. What number is 1995?
Answer: 2,5,8, 1 1, 14, ... According to the law, this is a arithmetic progression, the first term is 2, and the tolerance is 3, so the term 1995 = 2+3× (/kloc-0.
2. Of the natural numbers starting from 1, what is the100th number that cannot be divisible by 3?
Answer: We found that every three numbers in 1, 2, 3, 4, 5, 6, 7, ... are grouped from 1, and the first two of each group cannot be divisible by 3. If there are two groups, 100 will have 100 ÷ 2.
3. If 1988 is expressed as the sum of 28 consecutive even numbers, what is the largest even number?
Answer: 28 even numbers are grouped into 14 groups, and 2 symmetrical numbers are grouped into groups, that is, the minimum number and the maximum number. The sum of each group is:1988 ÷14 =142, and the difference between the minimum number and the maximum number is 28- 1=27 tolerance.
4. In the integer greater than 1000, find all the numbers with equal quotient and remainder after division by 34, so what is the sum of these numbers?
Answer: Because 34× 28+28 = 35× 28 = 980 < 1000, there are only the following figures:
34×29+29=35×29
34×30+30=35×30
34×3 1+3 1=35×3 1
34×32+32=35×32
34×33+33=35×33
The sum of the above numbers is 35× (29+30+31+32+33) = 5425.
5. There is a 1, 2, 3, ... which says 134 and 135 respectively. Take some cards out of the box at random, calculate the sum of the numbers on these cards and divide by 17, then write the remainder on another yellowcard and put it back in the box.
Answer: It is difficult to grasp several times at a time. It is better to consider the whole situation and take a step back to a simple situation analysis: suppose there are two numbers, 20 and 30, and divide their sum by 17 to get the number of yellow cards. If calculated separately, it is 3 and 13, and then the sum of 3 and 13 is divided by 65436. That is to say, no matter how many numbers are added, the remainder of the sum divided by 17 remains unchanged, and we return to the topic1+2+3+...+134+135 =136×136. The remainder is 0, and19+97 =1116 ÷17 = 6 ... 65438+.
6. The following formulas are arranged regularly:
1+ 1, 2+3, 3+5, 4+7, 1+9, 2+ 1 1, 3+ 13, 4+ 15, 65438
Solution: First find out the law: each formula is added by two numbers, the first number is the cycle of 1, 2, 3, 4, and the second number is the continuous odd number starting from 1 Because 1992 is even, the second of the two addends must be odd, so the first one must be odd, so it is 1 or 3. If it is 1: then the second number is1992-1=191. 199 1 is (1 991+1) ÷ 2 = 996, and1is always odd and inconsistent, so this formula is 3+/kloc-0.
7. As shown in the figure, the upper and lower rows in the table are arithmetic progression, so what is the minimum difference (reduction of large numbers) between two numbers in the same column?
Answer: From left to right, their differences are: 999, 992, 985, ... 12, 5. From right to left, their differences are: 1332, 1325,13/kloc.
8. There are 19 formulas:
So what are the results on the left and right sides of equation 19?
Answer: Because the left and right sides are equal, we might as well just consider the situation on the left and solve two problems: How many are used in the first 18 formula? All numbers are 5, 7, 9, ..., 18, and 5+2× 17 = 39, 5+7+9+...+39 = 396, so the equation of 19 starts from 397; How many numbers are added to the formula 19? The numbers on the left are 3, 4, 5, ... and 19 should be 3+ 1 × 18 = 2 1, so the result of 19 formula is 397+398+399+...+.
9. The number of two known columns: 2, 5, 8, 1 1, …, 2+(200-1) × 3; 5、9、 13、 17、……、5+(200- 1)×4。 They are all 200 projects. How many pairs of the same number of items are there in these two columns?
A: It is easy to know that the first such number is 5. Note that in the first series, the tolerance is 3, and in the second series, the tolerance is 4. That is to say, the second logarithm minus 5 is a multiple of 3, which is a multiple of 4, so the number converted into arithmetic numbers is calculated with the tolerance of12,5,17,29, ... The maximum number of the second sequence is 5+(200-1) × 4 = 80. The maximum number of new series cannot exceed 599, because 5+ 12× 49 = 593, 5+ 12× 50 = 605, so there are 50 pairs of * *.
10, as shown in the figure, has a lower triangle with a side length of 1 m. Starting from the vertex of each side, take a point every 2 cm, and then take these points as the endpoint to divide the large regular triangle into many small regular triangles with a side length of 2 cm as parallel lines. Find (1) the number of small regular triangles with a side length of 2 cm, and (2) the total length of parallel lines.
Answer: (1) From top to bottom, * * has 100÷2=50 lines, the first line has 1, the second line has 3, the third line has 5, ... and the last line has 99, so * * * has (/kloc. There are three parallel lines, which are the same. There are 49 lines * * * in the horizontal direction, the first one is 2cm, the second one is 4cm, the third one is 6cm, ..., and the last one is 98 cm, so the length of * * * is (2+98)×49÷2×3=7350 cm.
1 1, a factory 1 1 is busy in October and does not rest on Sundays. And since the first day, the same number of workers have been sent from the general factory to the branch factory every day. By the end of the month, there were 240 workers left in the main factory. If at the end of the month, the workload of workers in the general factory is 8070 working days (one person works 1 working day) and no one is absent, how many workers will the general factory send to work in the branch factory this month?
Answer: 165438+ 10 has 30 days. According to the meaning of the question, the number of people in general factories is decreasing every day, and finally it is 240. The number of people every day constitutes arithmetic progression. According to the nature of arithmetic progression, the sum of the number of people on the first day and the last day is equivalent to 8070÷ 15=538, that is to say, there are 538-240=298 workers on the first day, and (298-240) II is scheduled every day.
12, Xiao Ming read an English book. When he first read it, he read 35 pages on the first day, and then read 5 more pages every day than the day before. As a result, he only read 35 pages on the last day. The second time, I read 45 pages on the first day, and then I read 5 more pages every day than the day before. As a result, I only need to read 40 pages on the last day. How many pages are there in this book?
Answer: The first scheme: 35, 40, 45, 50, 55, ... 35 The second scheme: 45, 50, 55, 60, 65, ... 40 The secondary scheme is adjusted as follows: the primary scheme: 40, 45, 50, 55, ... 35+35. The second plan: 40, 45, 50, 55, ... (The last day is on the first day) So the second plan must be 40, 45, 50, 55, 60, 65, 70, ***385 pages.
13 and 7 teams * * * planted trees 100, and the number of trees in each team is different. Among them, the team with the most trees planted 18 trees, and the team with the least trees planted at least how many trees?
Answer: We know that the other six teams planted 100- 18=82 trees in order to make Li Ru-Nan's "What's the value of neon" stamp? As many as possible, including:17+16+15+14+13 = 75 trees, so the minimum team should plant at least 82-75=7 trees.
14. Line up 14 different natural numbers from small to large. It is known that their total number is 170. If the maximum and minimum numbers are removed, the remaining total is 150. What is the second number in the original arrangement order?
A: The sum of the maximum and minimum numbers is 170- 150 = 20, so the maximum number is 20- 1 = 19. When the maximum number is 19, there are19+18+17+16+15+14+13+. Yes18+17+16+15+13+12+/kloc-0.
Periodic problem
Basic exercises
1, (/kloc-0 1)○□□□□□□△□□…… The 20th number is (□).
(2) The thirty-ninth chess piece is (sunspot).
2. Xiaoyu practices calligraphy. She wrote the sentence "I love the great motherland" over and over again, and the 60th word should be written as (big).
Class 3.2 (1) participated in the tug-of-war competition of the school. Their participating teams are arranged in a row according to "three men and two women", and the 26th student is (male).
4. There is a column number: 1, 3,5, 1, 3,5, 1, 3,5 ... The 20th number is (3), and the sum of these 20 numbers is (58).
5. There are three kinds of beads with the same size *** 100, which are discharged continuously according to the requirements of 3 red and 2 white 1 black.
……
(1) The 52nd one is a (white) bead.
(2) There are (17) white beads in the first 52 beads.
6.a Q B: Today is Friday, and it will be Sunday in 30 days.
B Q A: If 16 is Monday, then the 3 1 day of this month is Tuesday.
May 1 2006 is Monday, so the 28th of this month is Sunday.
A, B, C and D play poker. Party A puts "Wang" in the middle of 54 cards, and the 37th card is counted from top to bottom. Party C thought about it, confidently grabbed the card first, and finally caught the "king". Do you know how C is worked out? ※? (37 ÷ 4 = 9 ... 1 The first person to get the card must catch the "king". )
answer
1、( 1)□。
(2) sunspots.
2. It's very big.
3. Male students.
The 20th number is (3), and the sum of these 20 numbers is (58).
5、
(1) The 52nd one is a (white) bead.
(2) There are (17) white beads in the first 52 beads.
6. (days) (2). (days).
(37 ÷ 4 = 9 ... 1) The first person to get the card will definitely catch the "king". ※
Improve practice
1, (/kloc-0 1)○□□□□□□△□□…… The 20th number is (□).
(2) ○○○○○○○○○○○○○○○○○○○○○○○○○○○○967
2. There is a row of colorful flags on the sports field, with 34 faces, arranged according to "three reds, one green and two yellows", and the last face is (green flag).
3. "Love mathematics since childhood, love mathematics since childhood ..." The 33rd word is (love).
4. The class (1) participated in the school tug-of-war. The teams in their competition are arranged in the order of "three men and two women", and the 26th student is (male).
5. There is a column number: 1, 3,5, 1, 3,5, 1, 3,5 ... The 20th number is (3), and the sum of these 20 numbers is (58).
6.a Q B: Today is Friday, and it will be Sunday in 30 days.
B Q A: If 16 is Monday, then the 3 1 day of this month is Tuesday.
May 1 2006 is Monday, so the 28th of this month is Sunday.
A, B, C and D play poker. Party A puts "Wang" in the middle of 54 cards, and the 37th card is counted from top to bottom. Party C thought about it, confidently grabbed the card first, and finally caught the "king". Do you know how C is worked out? ※?
37 ÷ 4 = 9 ... 1 (The first person to get the card will definitely catch the "king"). ※
answer
1、( 1)□。
(2)○。
2. Green flag.
3. Love.
4.( 1) male students.
5. The 20th number is (3), and the sum of these 20 numbers is (58).
6. (days) (2). (days).
37 ÷ 4 = 9 ... 1 (The first person to get the card will definitely catch the "king"). ※
1. Two children start from the same point A and move in opposite directions on a circular track. Their speeds are 5m/s and 9m/s respectively. If they start and end at the same time when they first meet at point A, how many times have they met from the starting point to the finishing point?
2. 16 ball is divided into 8, 3 and 5 * * * piles. Move according to the following rules, take any two piles A and B. If the number of balls in pile A is not less than that in pile B, move from pile A to pile B with the same number of balls. After doing this for several times, all the balls are in a pile, as shown in the figure. If the ball of 16 is divided into seven, six and three * * * piles, then all the balls can be made into a pile by moving at least a few times according to the above rules.
Figure: (8, 3, 5,) ——& gt；; (8，6，2)——& gt； (8，4，4)——& gt； (8，8，0)——& gt； ( 16,0,0)
3.2 power of 19+2 power of 20+2 power of1+40 power of ... 2 =
4. Group garden, 50 people per person 12 yuan, 5 100 people per person 10 yuan, 8 yuan per person 100 yuan. There are two tour groups, A and B. If you buy tickets separately, the two groups have to pay 16544 in total.
5.1/3+1(2 times of 3)+...1/(02 times of 65438+3)
6. Comparative dimension A = 2005 * 2006/2007 * 2008b = 2006 * 2007/2008 * 2009c = 2007 * 2008/2008 * 2010/6/7.
7. The least common multiple of two numbers is 1650. When these two numbers are divided by their greatest common divisor, the sum of their quotients is 13. What are these two numbers?
.
Let the two children be called A and B.
The speed ratio of Party A to Party B is 5:9.
The distance ratio in the same time is also 5:9.
That is, A can run 5 laps and B can run 9 laps.
Finally, A and B ran 5+9= 14 laps.
Party A and Party B meet once every 1 circle.
So from beginning to end, the two met 14 times (including the last time).
From the beginning to the end, a * * * satisfies 13 times (not including the last one).
2.
(7，6，3)->(4，6，6)->(4， 12，0)->(8，8，0)->( 16,0,0)
At least 4 times
3.
Geometric series summation, never learned geometric series, using the following methods:
Let m = 219+220+221+...+240.
Multiply both sides by 2 to get:
2m=2^20+2^2 1+2^22+...+2^40+2^4 1
Subtraction, get:
m=(2^20+2^2 1+...+2^4 1)-(2^ 19+2^20+...+2^40)
=2^4 1-2^ 19
(0 of 465438+2 minus 09 of 65438+2)
4.
864÷8 = 108 & gt； 100
So it adds up to 108 people.
The number of people in both groups cannot be less than 50.
Nor can one be at 5 1- 100, and one be above 100.
1 142 cannot be divisible by 10, so the number of people in the two groups cannot be between 5 1- 100.
If it is less than 50 people and more than 100 people respectively,
If purchased separately, the maximum fare is12× 7+101× 8 = 892.
Then the number of people in the two groups is less than 50 and 5 1- 100 respectively.
Now it is the basic cage problem of chickens and rabbits.
If this 108 person buys a ticket with 10 yuan.
I need:108×10 =1080 yuan.
Missing:1142-1080 = 62 yuan.
Under 50 people, the fare per person is more: 12- 10=2 yuan.
So the number of people under 50 is: 62÷2=3 1.
5 1- 100: 108-3 1=77.
The two tour groups are 3 1 and 77 respectively.
5.
Or a geometric series, using the method of the third question just now:
Let m =1/3+1/(3 2)+1(3 3)+...+1/(312).
Multiply it by 1/3 and you get:
1/3m= 1/(3^2)+ 1/(3^3)+...+ 1/(3^ 13)
Subtraction, get:
( 1- 1/3)m= 1/3- 1/(3^ 13)
2/3m= 1/3- 1/(3^ 13)
m=[ 1/3- 1/(3^ 13)]*3/2
= 1/2[ 1- 1/(3^ 12)]
6.
a/B = 2005 * 2006/(2007 * 2008)×2008 * 2009/(2006 * 2007)
=2005*2009/2007^2
=(2007+2)(2007-2)/(2007^2)
=(2007^2-4)/(2007^2)<； 1
So a < B
What exactly is C?
Please check the data or compare it with the above method.
Please call me if you need help.
7.
1650=2*3*5*5* 1 1
When the least common multiple of two numbers is divided by their greatest common divisor, the quotient of these two numbers is coprime.
13 is decomposed into the sum of two prime numbers, which are divisors of 1650.
13=2+ 1 1=3+ 10
1)
The two quotients are 2, 1 1 respectively.
The greatest common divisor is 75.
These two figures are:
1650÷2=825
1650÷ 1 1= 150
Or:
The two quotients are 3 10 respectively.
The greatest common divisor is 55.
These two figures are:
1650÷3=550
1650÷ 10= 165