(1) A factory produced a batch of toys, and after three-fifths of the tasks were completed, 280 toys were added, so there were more toys to be made 10%. How many toys to make? (Please write down the calculation process)

Solution:

The increase is the original: 3/5+ 10%.

So it was supposed to be done: 280/(3/5+ 10%)=400 yuan.

(2) The books produced by a school-run factory this month increased by 30,000 yuan. If VAT is paid at 17% of the value-added amount, how much VAT should be paid this month? (Please write down the calculation process)

Payable: 30,000 *17% = 5100 yuan.

(3) Dad's salary this month is 2 100 yuan. According to the regulations, the income tax shall be paid on the salary1.above 600 yuan. If you pay the personal income adjustment tax at the rate of 5%, how much should Dad pay this month? How much does he actually earn? (Please write down the calculation process)

Maturity: (2 100- 1600)*5%=25 yuan.

Actual income: 2 100-25=2075 yuan.

1. Application of area calculation of parallelogram, triangle and trapezoid.

1, PLA soldiers reclaimed a parallelogram vegetable field. Its bottom is 24 meters and its height is 16 meters. What is the area of this land?

s=ah 24* 16=384

2. Trapezoidal wheat experimental field with an upper bottom of 86m, a lower bottom134m and a height of 60m. How many square meters is its area?

s =(a+b)* h/2(86+ 134)* 60/2 = 6600

3. Triangular land with a bottom height of 358m and a height of 160m. What is the area of this land?

s=ah/2 358* 160/2=28640

Second, summarize the application questions

1. The People's Liberation Army Transportation Company transports a batch of coal. If each truck is loaded with 4.5 tons, it will take 16 cars to transport it at one time. If each truck is loaded with 6 tons, how many cars will it take to transport it all at once?

4.5* 16/6= 12

2. Students put flowers, each with 9 pots, which requires 36 people; If 18 people want to put it, how many pots should each person put?

36*9/ 18= 18

Third, the application of three-step calculation method

Taiyanggou Primary School held a math knowledge contest. There are 60 participants in the third grade and 45 participants in the fourth grade. The number of participants in the fifth grade is twice that in the fourth grade. How many people took part in the competition in the third grade?

45*2+45+60= 195

Fourth, meet the application problems

1, Zhang Ming and Li Hong set out from two places at the same time, facing each other. Zhang Ming walks 50 meters per minute and Li Hong walks 40 meters per minute. 12 minutes later, they met. How many meters are they apart?

(50+40)* 12= 1080

2. The distance between Party A and Party B is 255km, and two cars leave both places at the same time. Car A travels 48 kilometers per hour and car B travels 37 kilometers per hour. Two cars meet in a few hours.

255/(48+37)=3

5. List simple equations to solve practical problems

Xiangqun Stationery Factory can produce 250 pencil boxes per hour. How many hours can you produce 10000?

Suppose: X hours can produce 10000 pieces.

250x= 10000

x=40

Answer: 40 hours 10000.

Six, about the application of cuboids and cubes, the calculation of surface area and volume.

1, a rectangular iron box, length 18cm, width 15cm and height 12cm. What is the volume of this iron box?

18* 15* 12=3240

2. The side length of the cube is15cm. What is its volume?

15* 15* 15=3375

1, fill in.

(1) There are () simplest true fractions whose denominator is 12, and their sum is ().

(2) One conductor is 45 meters long, which is 14 meters shorter than the other, and the two conductors are * * * () meters.

(3) One conductor is 45 meters long, the other conductor is shorter than it by 17 meters, and the other conductor is () meters long.

(4) Addition and subtraction of fractions with different denominators, first (), then (), and then addition and subtraction.

(5) A batch of fertilizer was shipped 13 on the first day and 25 on the second day, and the remaining batch of fertilizer () was not shipped.

(6) Make the following fractions and decimals reciprocal.

0.75=( ) 25 =( ) 3.42=( )

58 =( ) 2. 12=( ) 4 14 =( )

2. Calculation problems

5 12 +34 + 1 12 7 10 -38 - 18 4 15 +56

12 -(34 -38 ) 56 -( 13 +3 10 ) 23 +56

3. Solve the equation

17+x = 23 45-x = 14 x- 16 = 38

Step 5 solve the problem

(1) There is a piece of cloth. It takes 78 meters to make a coat and 34 meters to make a pair of trousers, leaving 1 12 meters. How many meters is this cloth?

(2) An engineering team built roads, the first week was 49 kilometers, the second week was 29 kilometers, and the third week was less than the sum of the previous two weeks 16 kilometers. How much was repaired in the third week?

(3) In class, students spend 15 hours doing experiments, teachers spend 3 10 hours explaining, and students finish their homework independently in the rest of the time. It is known that each class is 23 hours. How long does it take for students to do their homework?

fill (up) a vacancy

1.0 m is to divide 1 m into () parts and take () parts.

2. The decimal unit of is (), and it has () such decimal units.

3. Yes, there is one in the room.

4. Fill in the appropriate scores in the brackets.

24 kg = () tons 4 meters 20 cm = () meters.

360m = () km 1 hour = () days.

5.= = = =( )÷9=44÷( )

6. The maximum true score is (), the minimum false score is () and the minimum simplest fraction is ().

7. Divide the 2m-long wood into 7 sections on average, each section is 1 m long, and each section accounts for the whole length.

8.+ means () plus (), and the total is ().

9.,,,, these scores can be turned into a finite score is ().

10. Rank the scores of the following groups in descending order.

,No. () > ()> ()

、4.5()>； ()& gt( )

Second, multiple-choice questions:

1. Among the following figures, not less than ().

a、 1 B、C、

2. Put 5 kg of salt into 20 kg of water, and the weight of salt accounts for () of salt water.

A, B, C,

3. There are () simplest real scores less than.

A, 3 B, 4 C, countless

4. And these two scores ().

A, meaning the same; B, the size is equal; C, the decimal units are the same.

5. If A equals B, then A () B.

A, greater than B, equals C, less than.

Third, judge the question.

1.3 kg of water is as heavy as 1 kg of water. ( )

2. Tons of cotton = tons of iron. ( )

3. 1 is the simplest score. ( )

4. Because the ratio is very small, the decimal unit of is smaller than that of. ( )

5. The true score is always less than the false score. ( )

6. Rice is bigger than rice. ( )

7. The numerator and denominator of the simplest fraction have no common factor. ( )

Fourth, oral calculation.

+0.5 + 3.6+ +

2.4- 1 +3.6 6.43- -0.375

5. Calculate the following questions. (as simple as possible)

1+ - + - - -

2. 15-( - ) 2.85+ +2. 15+ 3.4-(0.25+ )

Sixth, solve the equation.

+x = 5.6 x-= x-( 1.4+)= 1.8

Seven, column calculation.

1.A number is 0.75 more than b number. What is the sum of the two numbers?

2. Subtract the difference of 3.25 from a number, and the result is 2.5. What's this number?

Eight, application questions.

There are 48 students in 53 classes, including 2 boys1. What percentage of girls are in the class? What is the ratio of boys to girls?

2. To produce the same parts, from 27 pieces in 12 hours to 13 pieces in 6 hours and 19 pieces in 8 hours. Who can do it fastest? Who is the slowest?

3. Build a road with a length of 1500m. If the whole project is completed in the first week and the whole project is completed in the second week, what is the score for completing the whole project?

Wang Lin read a book. On the first day, he finished reading the whole book. The next day and the third day, he read a whole book more than the first day. After three days, how many parts are left in the whole book?

5. There is a rectangle with a circumference of 68 cm, which is known to be 2 cm long and how many centimeters wide.

Responder: Folding Angel ylq- Scholar Level 3 1- 18 10:07.

What are you doing?

Respondent: Xiao's probation period is Grade 1 1-20 13: 12.

Formulas for solving the application problems of fractions and percentages

The unit "1" is known: unit "1"× corresponding fraction = corresponding quantity.

Unit 1 or unit 1 is unknown: corresponding quantity ÷ corresponding score = unit 1.

Formula for finding the fraction (or percentage) of one number and another:

One number ÷ another number = A number is a fraction (or percentage) of another number?

Find out how much one number is more than another:

Excess ÷ unit "1" = how many fractions (or percentages) one number is more than another.

Find out how much one number is less than another:

Small quantity ÷ unit "1"= how many fractions (or percentages) one number is less than another.

(Note: The words "more" and "less" here can also be replaced by words such as "increasing production" and "saving". )

(Note: Example: (1) There are 120 peach trees in the orchard, and the number of pear trees is 20% more than that of peach trees. How many pear trees are there in the orchard?

(2) There are 120 peach trees in the orchard, which is 20% less than that of pear trees. How many pear trees are there in the orchard?

Analysis idea: first find out the unit "1" and determine whether it is known or unknown. If the unit "1" is known, use multiplication; if the unit "1" is unknown, use division. "A few points more (less) than who" is "a few points of 1+(-)". )

Formula: (1)120× (1+20%)

(2) 120÷( 1-20%)

Formulas to solve the application problems of discount, profit, interest and tax.

Meaning: "20% off" means: the current price is 20% off the original price; "15% discount" means that the current price is 85% of the original price.

Formula:

Current price = original price × discount (usually written in percentage form)

Profit = selling price-cost

Interest = principal × interest rate× time

After-tax interest = principal × interest rate× time× 80% (note: national debt and education savings are not taxed)

Tax payable = tax payable × tax rate

Formulas and key sentences about the circumference and area of a circle

The ratio of the circumference to the diameter of a circle is called pi. π = C ÷ d

Find the perimeter of a known diameter: C = πd Find the diameter of a known perimeter: d = C ÷π.

Find the perimeter of a known radius: C = 2πr Find the radius of a known perimeter: r = C÷π÷2.

Find the area of known radius: S =πr

Find the area of known diameter: r = d÷2.

S = πr

Find the area of the known perimeter: r = C÷π÷2.

S = πr

Semicircle circumference = C ÷ 2+d (note: semicircle circumference = 5. 14r, suitable for filling in the blanks)

Semicircular area = S ÷ 2

Divide a circle into several parts evenly and make an approximate rectangle. (See book for pictures)

(1) Area of rectangle = area of circle.

(2) The length of the spliced rectangle = half the circumference (length =)

(3) The width of the spliced rectangle = the radius of the circle (width = r)

Fill in the blanks. (1 minute per grid, ***20 points)

(1) A number consists of three 100, two 10 and five 0.0 1. This number is written as ().

2. 7 tons and 560 kilograms = () tons, 1 hour = () minutes.

(3) decompose BaZi 80 into prime factors, (180 =)

(4), the scoring unit is (), plus a scoring table like ().

Bit gets the smallest prime number.

[5], the ratio of 2.7∶ 1 to the simplest integer is (), and the ratio is ().

[6], a triangle has at least () acute angles.

At one time, a steel cylinder could be cast into a cone with equal bottom and equal height.

(8) Remove the rice with a 5-meter cloth. How many meters are left? The formula is ().

Levies, circle is axisymmetric circle, its axis of symmetry is ().

⑽. The number of winners in the primary school mathematics competition is ***30, and the ratio of the number of first, second and third prizes is.

1∶2∶3, and the number of people who won the third prize is ().

⑾ The circumference of a circle is 18.84 cm, and the area of this circle is ().

On the map with the scale of 1: 3000000, the distance from Beijing to Guangzhou is 6.

Cm, the actual distance from Beijing to Guangzhou is about () kilometers.

Second, the judgment question. (Mark "√" correctly in brackets and "×" incorrectly) (* * 8 points)

The greatest common divisor of (1) 16 and 24 is their least common multiple. ( )

(2) The circulating decimal number is 0.5, and the rounding to two decimal places is about 0.55. ( )

(3) Fifty trees were planted in the orchard, but three trees did not survive, and the survival rate was 97%. ( )

(4) The number A is 20% less than the number B, and the number B is 25% more than the number A.. ( )

All six faces of a cube are squares. ( )

3 kg weighs as much as 1 kg. ( )

Once, a certain distance, speed and time were inversely proportional. ( )

Being, the sum of three consecutive natural numbers is m, then the largest number is (+1). ( )

Third, multiple choice questions. (Fill in the serial number of the correct answer in brackets) (65438+ 0 point for each question, ***8 points)

(1), the product of two prime numbers must not be ().

A, prime number b, composite number c, odd number d, even number

(2) If it is a false score and a true score, then ().

a、X5 C、X=5 D、X=6

(3) Xiaohong got on the bus at 9: 40 pm and got off at 8: 0012 the next morning. Her time on the train is ().

A, 10 hour 32 minutes b, 1 hour 28 minutes c, 10 minute 32 minutes.

(4), the triangle area is certain, and the bottom and height ().

A, proportional to B, inversely proportional to C, out of proportion

5], two cubes with a side length of 4 cm are combined into a cuboid, and the surface area of this cuboid is () square cm.

a、 168 B、 192 C、 160

[6] The number of base angles of an isosceles triangle is the degree of vertex angle, and the vertex angle is ().

A、 1200 B、 1350 A、300

Once, in order to clearly show the number of students in each class of grade six in our school, it was best to draw a statistical chart ().

A, horizontal bar b, dotted line c, fan-shaped

Being, the number of a is 135, (), what is the number of b? This question lacks a condition. If you calculate the number b,

The formula is: 135× (1+). Please fill in the following conditions in brackets.

A, B is B of A, A is more C than B, and B is more than A..

Fourth, the calculation problem. (***34 points)

1, a number written directly. (6 points)

0. 125+ = 0.6-0.06= 4-3 =

× = 6 ÷3= 1÷ =

2. Find the value of x below. (6 points)

x-0.3×2.4 = 1.54 1∶3.5 =

3. Off-mold calculation. (12)

72.56― 18.74―2 1.26 3.7× +63×

1375- 1702÷23 24÷ 1.6-0.8×0.9

4. Column calculation. (6 points)

(1)24 minus 3 minus 25% of 4, what is the quotient?

2. The number 2.4 is less than a number 7.6. Find this number.

5. The side length of the square in the picture below is 3 minutes. Find the area of the shaded part. (4 points)

Fifth, the application problem. (5 points for each question, ***30 points)

1, Zhangjiajie department store reduced the price by 20% to sell a sweater, and only sold 96 yuan money. What is the original price of this sweater?

2. Erjiahe Township plans to plant 1346 trees on a barren beach, which has been planted for 7 days, with an average of 103 trees per day. The rest will be planted in five days. How many trees will be planted on average every day?

The distance between the cities of Party A and Party B is 624 kilometers. A bus and a truck leave from Party A and Party B at the same time. The average speed of buses is 65 kilometers per hour, and the average speed of trucks is the average speed of buses. How many hours after the two cars left?

4. Xiaohua reads books. I originally planned to read 85 pages a day, and I can read them in 12 days. If he reads 102 pages a day, how many days can he finish it? (Use proportional solution)

5. Cast an iron block with a volume of 3 14 cubic centimeter into a cylinder. The diameter of the bottom of this cylinder is 10 cm, and the height is about how many cm?

6. A grain store sold 720 kilograms of raw rice this month. At this time, the stored rice is exactly 80% of the original rice. How many Jin of raw grain does this grain shop have?

Question 1. The clerk changed a 5 yuan RMB and a 50-cent RMB into 28 yuan RMB with face values of 1 yuan and 1 respectively. How much RMB do you want?

Question 2: There are 50 RMB * * with total face value 1 16 yuan. As we all know, one yuan is more than two yuan. How many RMB are there in three denominations?

Question 3: There are 400 movie tickets from 3 yuan, 5 yuan and 7 yuan, with a value of 1920 yuan, among which 7 yuan and 5 yuan have equal tickets. How many movie tickets are there for each of the three prices?

Question 4: Two kinds of cars are used to transport goods. Each car contains 18 boxes, and each car contains 12 boxes. Now there is a 18 car, worth 3024 yuan. If each case is cheap in 2 yuan, the goods are worth 2520 yuan. Q: How many cars are there?

Question 5. A truck can transport ore 20 times a day in sunny days and 12 times a day in rainy days. Transportation 1 12 times a day, with an average of 14 times a day. How many days are it rainy these days?

Question 6. A batch of watermelons has been delivered and will be sold in two categories, the large one in 0.4 yuan and the small one in 0.3 yuan. In this way, these watermelons are worth 290 yuan. If the price per kilogram of watermelons is reduced by 0.05 yuan, this batch of watermelons can only be sold in 250 yuan. Q: How many kilograms is the watermelon?

Question 7. In the darts competition, it is stipulated that each player gets 65,438+00 points, and each player misses the target and gets 6 points. Each player throws 10 times and scores * * * 152 points, in which player A scores more than player B 16 points. Q: How many times did each player win?

Question 8. There are 20 questions in the math contest. Every time he answers a question correctly, he gets 5 points. If he answers a wrong question, he will not only get no points, but also deduct 2 points backwards. Xiaoming got 86 points in this competition. Q: How many questions did he answer correctly?

Solution: x sheets of 1 element and (28-x) sheets of 1 angle.

x+0. 1(28-x)=5.5

0.9x=2.7

x=3

28-x=25

A: There are three Zhang Yiyuan bills and 25 dimes.

2. Solution: Let 1 element have X, 2 yuan (x-2) and 5 yuan (52-2x).

x+2(x-2)+5(52-2x)= 1 16

x+2x-4+260- 10x = 1 16

7x= 140

x=20

x-2= 18

52-2x= 12

A: There are 20 1 yuan, 8 18 in 2 yuan and 2 12 in 5 yuan.

3. Solution: 7 yuan and 5 yuan have X pieces, and 3 yuan has (400-2x) pieces.

7x+5x+3(400-2x)= 1920

12x+ 1200-6x = 1920

6x=720

x= 120

400-2x= 160

A: 3 yuan has 160 and 7 yuan and 5 yuan have 120.

4. Answer: Total cargo: (3024-2520)÷2=252 (boxes)

There are x buses and (18-x) cars.

18x+ 12( 18-x)= 252

18x+2 16- 12x = 252

6x=36

x=6

18-x= 12

Answer: Bus No.6, 12.

5. Solution: Days = 1 12÷ 14=8 days.

It rained on day X.

20(8-x)+ 12x = 1 12

160-20x+ 12x = 1 12

8x=48

x=6

There are six rainy days.

6. Solution: Watermelon number: (290-250)÷0.05=800 kg.

There is a big watermelon x kg.

0.4x+0.3(800-x)=290

0.4x+240-0.3x=290

0. 1x=50

x=500

There are 500 kilograms of big watermelons.

7. Solution: A: (152+ 16)÷2=84.

B: 152-84=68 points.

Set armor x times

10x-6( 10-x)=84

10x-60+6x=84

16x= 144

x=9

Set b to y times.

10y-6( 10-y)=68

16y= 128

y=8

A: Nine times for A and eight times for B. ..

8. Answer: Suppose he answered question X correctly.

5x-2(20-x)=86

5x-40+2x=86

7x= 126

x= 18

A: He got it right 18.

Example: 1: Unload several cases from the freighter with a total weight of 10 ton, and the weight of each case shall not exceed 1 ton. In order to ensure that these boxes can be transported at one time, how many cars with a load of 3 tons are needed at least?

[Resolution] Because the weight of each box does not exceed 1 ton, the weight of each box that a car can transport will not be less than 2 tons, otherwise another box can be put. So five cars are enough, but four cars may not be able to carry all the boxes away. For example, if there are 13 boxes, then each car can only transport 3 boxes, and 13 boxes cannot be transported by 4 cars at a time.

So in order to ensure that all the boxes can be transported at one time, at least five cars are needed.

Example 2: Intercept 100 short bamboo poles with a length of10 foot respectively. How many raw materials should be used at least? What is the most cost-effective cut?

[Analysis] A bamboo pole with a length of 10 feet should be cut in three ways:

(1) 3 feet 2 and 4 feet 1, the most economical;

(2) three feet three, more than one foot;

(3) 4 feet 2, more than 2 feet.

In order to save materials, try to use the method of (1). With 50 raw materials, 100 3-foot bamboo poles and 50 4-foot bamboo poles can be cut. If 50 4-foot bamboo poles are short, it is best to choose method (3), which requires the least raw materials, only 25, and at least 75 raw materials.

Example 3: The lengths of the three sides of an acute triangle are two digits respectively, and they are three consecutive even numbers. The sum of their numbers is a multiple of 7. What is the longest circumference of this triangle?

[Resolution] Because the three sides of a triangle are three consecutive even numbers, their unit digits can only be 0, 2, 4, 6, 8, and their sum is even, and because the sum of their unit digits is a multiple of 7, it can only be 14, and the maximum value of the three sides of a triangle can be 86, 88, 90, so the longest circumference is 86.

Example 4: Decomposition of 25 into the sum of several positive integers to maximize their products.

[Analysis] Start with a small number shape and find out its rules:

6 divided by 3+3, its product is 3×3=9.

Divide 7 by 3+2+2, and the product is 3×2×2= 12.

Divide 8 by 3+3+2, and the product is 3×3×2= 18.

9 divided by 3+3+3, its product is 3×3×3=27. ……

That is to say, in order to maximize the product of decomposition numbers, 3 should appear as much as possible. When a natural number can be expressed as the sum of several 3' s and 1, one 3' s and 1 should be taken out and then decomposed into two 2' s, so that 25 can be decomposed into 3+3+3+3+2+2.

Example 5: A and B are going to explore the desert. They go deep into the desert for 20 kilometers every day. It is known that each person can carry one person's food and water for up to 24 days. If some food is not allowed to be stored halfway, how many kilometers can one of them go deep into the desert (the last two need to return to the starting point)? What if some food can be stored on the way back?

[Analysis] Suppose A goes back after X days. When A goes back, he leaves the food he needs, and the rest is transferred to B. At this time, B*** has (48-3X) days of food, so X=8. With the remaining 24 days of food, B can only move forward for another 8 days, leaving 16 days of food for him to return, so B can go to the desert.

If the conditions are changed, the crux of the problem is the food left over in B24 days when A returns. Because 24 days of food can make B go deep into the desert alone 12 days, and the other 24 days of food will provide A and B with a walk back and forth, that is, 24÷4=6 days, so B can go deep into the desert 18 days, that is, one day.

Ex. 6: Every worker and every piece of equipment in two garment factories, A and B, can completely produce suits of the same specifications. Factory A produces tops, pants and only 900 suits a month. Factory B spends a lot of time producing tops and trousers, and just produces 65,438+0,200 suits a month. Now the two factories are jointly producing, trying to produce more suits. So how many more suits are produced each month than in the past?

[Analysis] According to the known conditions, the time ratio of producing a pair of trousers and a coat in a factory is 2: 3; Therefore, the ratio of the number of shirts and trousers produced by a factory in a unit time is 2: 3; It can also be seen that the ratio of the number of shirts and trousers produced by factory B in unit time is 3: 4; Because of this, A factory is good at producing pants, and B factory is good at producing tops. The two factories jointly produce, give full play to their respective specialties, and arrange factory B to fully produce jackets. Since factory B produces 1 0,200 jackets a month, factory B can produce 1 0,200 ÷ = 210,000 jackets a month, and factory A is arranged to fully produce pants, so factory A can produce 900 pants a month.

In order to support production, a factory first fully produced 2 100 pairs of trousers, which required 2 100÷2250 = pieces a month, and then a factory independently produced 900×=60 suits a month, so now the joint production produces more suits every month than in the past.

(2100+60)-(900+1200) = 60 sets.

There are 65,438+0,400 Weiqi pieces today. Party A and Party B play the game of taking Go. Party A takes it first, and Party B takes it later. They take turns eating once. It is stipulated that only 7P(P is 1 or any prime number not exceeding 20) can be taken at a time. Who will win the game in the end? Ask both parties who has a winning strategy.

[Analysis] Because 1400=7×200, the original question can be translated as: There are 200 chess pieces, and both parties take p pieces in turn, and whoever takes the last one wins.

[Solution] B has a winning strategy.

Since 200=4×50, p is either 2 or can be expressed in the form of 4k+ 1 or 4k+3 (k is zero or a positive integer). The strategy adopted by B is: If A takes 2,4k+1 and 4k+3, then B takes 2,3,1,so the remaining pieces are still multiples of 4. So the last remaining number is a multiple of 4, not more than 20. At this time, A can't take it all, and B can take it all and win.

[Description] (1) In this question, B is the "late Mover", so the first Mover does not necessarily have a winning strategy. The key is to look at the "situation" they face;

(2) We can analyze the solution of this problem in this way, and divide all situations-the number of remaining pieces into two categories. The first category is a multiple of 4, and the second category is others. If someone encounters the second situation when playing chess, they can go 1 or 2 or 3, so the rest is the first situation. If he is faced with the first situation when playing chess, then the second situation must be left to another person who has finished playing chess. Therefore, whoever faces the second situation first will win, and this method can be used in most double-match problems.

There is a tour group of 80 people, including 50 men and 30 women. Their hotel has three room types: 1 1, 7 and 5 people. Men and women live in different rooms. How many rooms should they live in at least?

[Analysis] In order to minimize the number of rooms, first arrange 1 1 rooms, so that 50 men arrange 3 1 1 rooms, 2 5 rooms, 1 7 rooms. 30 women should be assigned to11room, two 7 rooms, 1 5 room, and * * * has 10 room.

[practice]

If the sum of 1 and ten natural numbers is equal to 100 1, what is the maximum possible value of the greatest common divisor of these ten natural numbers? (excluding 0)

2. When the sum of two right-angled sides is constant, which right-angled triangle has the largest area? If the sum of two right-angled sides is 8, what is the maximum area of a triangle?

Five people are each holding a bucket in front of the tap, waiting for water. The time they need to fetch water is 1 minute, 2 minutes, 3 minutes, 4 minutes and 5 minutes respectively. If only one faucet arranges their water intake order reasonably, then the sum of everyone's queuing and water intake time can be minimized. How many minutes is this at least?

4. A pool can be filled with water pipes A and B, single pipe A needs to be filled in 12 hours, and single pipe B needs to be filled in 24 hours. If it takes 10 hours to fill the pool, and the less time it takes to put pipes A and B together, the better. How many hours does it take to put pipes A and B together?

5. A highway is littered with 1995 young pioneers to publicize traffic regulations. Where should they meet on the expressway after completing the task, so as to minimize the total distance from their respective propaganda posts to the meeting place along the expressway?

6. Party A and Party B take turns to write down the natural number not exceeding 10 on the blackboard. The rule is that it is forbidden to write the divisor of the number already written on the blackboard. Those who can't complete the next step are losers. Q: Will the first writer or the last writer win? How to win?

[Question reference answer and train of thought analysis]

1, ∫1001= 7×113, ∴ 7× 13 can be common divisors, so these ten positive integers can be common divisors.

2. For right triangle, isosceles right triangle has the largest area under certain conditions. If the sum of two right angles is 8, the maximum area of a triangle is ×4×4=8.

3. In order to minimize the sum of people queuing for water, there are two ways:

(1) The number of people waiting in line should be as small as possible; (2) Try to queue as little as possible. Therefore, people who draw water quickly should draw water first, so as to ensure that when there are many people in the queue, everyone will wait less, so * * * needs 5× 1+4×2+3×3+2×4+5=35 (minutes).

4. Since it is impossible for Party A and Party B to fill the pool within 10 hour under the condition of separate opening, it is necessary to have time to put it all in. In order to keep them together for the least time, we should try to open the first pipe (quickly), so that the first pipe can fill the pool of 10 hour, and the rest can only be filled by the second pipe. Therefore, it takes at least 4 hours to completely release the two test tubes.

5. We can start with the simplest problem, find the law, and thus solve the complex problem. The final meeting place should be in the middle.

6. The first author has a winning strategy. A writes 6 in the first step, and B can only write one of 4, 5, 7, 8, 9, 10, which is divided into several pairs (4, 5), (8, 10) and (7, 9). If b writes one pair,