See if these are enough.

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The formula of travel problem: distance = speed × time; Distance ÷ time = speed; Distance/speed = time

The key point is to determine the position relationship in the process of traveling by using the line segment diagram.

Meeting problem: speed sum × meeting time = meeting distance (please write other formulas)

Catch-up problem: catch-up time = catch-up distance ÷ speed difference (write other formulas)

Flow problem: downstream flow range = (ship speed+current speed) × downstream time;

Upstream stroke = (ship speed-water speed) × upstream time

Downstream speed = ship speed+current speed = ship speed-current speed.

Still water velocity = (downstream velocity+upstream velocity) ÷2

Water velocity = (downstream velocity-upstream velocity) ÷2

Formula of train crossing bridge problem

(bridge length+conductor) ÷ speed = crossing time; (Bridge length+conductor) ÷ Crossing time = speed;

Speed × crossing time = sum of bridge and vehicle length.

Sum and difference problem formula (sum+difference) ÷2= larger number; (sum and difference) ÷2= smaller number.

Sum and the formula of multiple problem sum ÷ (multiple+1)= a multiple;

Multiple x multiple = another number, and-multiple = another number.

The formula of differential multiple problem is differential ÷ (multiple-1) = smaller number;

Smaller number × multiple = larger number, smaller number+difference = larger number.

Average problem formula total quantity ÷ total number of copies = average.

Engineering problem formula work efficiency × working time = total workload;

Total workload ÷ working time = working efficiency; Total amount of work ÷ efficiency = working hours.

The formula of surplus (surplus) deficit (shortage) problem

(1) One profit, one loss, (profit+loss) ÷ (difference between two distributions per person) = number of people.

(2) There are profits both times, (big profits-small profits) ÷ (the difference between each person's two distributions) = number of people.

(3) Two losses, (big loss-small loss) ÷ (the difference between two distributions per person) = number of people.

Tree planting problem formula

(1) The problem of planting trees on the unclosed line;

① Interval number+1= number of trees; (Planting trees at both ends) Road length ÷ Interval length+1= number of trees

② Interval number-1 = number of trees; (No planting at both ends) Road length ÷ Interval length-1 = number of trees;

(3) Road length ÷ number of sections = length of each section; Length of each section × number of sections = road length.

(2) The problem of planting trees on closed lines:

Road length/interval = number of trees; Road length/number of sections = road length/number of trees = length of each section;

Length of each section × number of sections = length of each section × number of trees = road length.

(3) Planar tree planting: total area ÷ area per tree = number of trees.

The number of comparisons between scores and percentages ÷ standard number = the ratio of the corresponding scores (percentages);

Number of growth ÷ standard number = growth rate; Reduction number ÷ standard number = reduction rate.

The difference between two numbers ÷ the smaller number = a few more (one percent) (increase),

The difference between two numbers ÷ the larger number = a few (hundredths) (minus).

Reciprocal formula for increasing or decreasing percentage (percentage)

Growth rate ÷( 1+ growth rate) = reduction rate; Reduction rate ÷( 1- reduction rate) = growth rate.

Solution to the application problem of comparison number

Standard number × percentage rate = comparison number corresponding to percentage rate;

Standard number × growth rate = growth number; Standard number × reduction rate = reduction number;

Standard number × (sum of dichotomy) = sum of two numbers;

Standard number × (difference of dichotomy) = difference of two numbers.

Find the comparison number of the standard number formula ÷ the percentage corresponding to the comparison number = the standard number;

Growth number ÷ growth rate = standard number; Reduction number ÷ reduction rate = standard number;

Sum of two numbers and sum of two rates = standard number; The difference between two numbers ÷ the difference between two rates = standard number;

Formula of interest rate problem

(1) Simple interest problem: principal× interest rate× term = interest, principal× (1+interest rate× term) = sum of principal and interest;

Principal and interest ÷( 1+ interest rate × term) = principal, annual interest rate ÷ 12= monthly interest rate, and monthly interest rate × 12= annual interest rate.

(2) Compound interest: principal ×( 1+ interest rate) deposit periods = sum of principal and interest.

Sales problem commodity profit = commodity price-commodity cost price;

Commodity profit rate = commodity profit/commodity cost ×100%; Total sales = sales price × sales quantity

The divisibility of a number is 1. If the last digit is 0, 2, 4, 6 or 8, it can be divisible by 2.

2. If the sum of numbers is divisible by 3, then this integer can be divisible by 3.

3. If the ten digits and two digits of an integer are divisible by 4, then the integer can be divisible by 4.

If each bit is 0 or 5, this integer can be divisible by 5.

If it is divisible by 2 and 3, then this integer can be divisible by 6.

6. If the last 3 digits of an integer are divisible by 8, then the integer can be divisible by 8.

7. If the sum of the numbers of an integer can be divisible by 9, then this integer can also be divisible by 9.

8 .. Count a number from right to left, add the odd and even numbers respectively, and find their difference. If this difference is a multiple of 1 1 (including 0), then this number can be divisible by 1 1.

I. Travel issues:

Example 1. 1.a and b practice 100 meter race. A runs 7 meters per second and B runs 6.5 meters per second. If A lets B run first 1 sec, how many seconds can A catch up with B?

2.a and B are 285 meters apart and walk in opposite directions. A walks 8 meters per second from A, and B walks 6 meters per second. If A goes first 12 meters, how many seconds will A meet B?

3. Two planes, A and B, fly in opposite directions from two airports 750km apart at the same time, and arrive at the same midway airport half an hour later. The speed of plane A is 1.5 times that of plane B. Find the speed of plane B.

4. Two trains, A and B, are144m long and180m long. Train A travels 4 meters more per second than train B, and the two trains run in opposite directions. It takes 9 seconds from meeting to staggering. What's the speed of these two trains?

From a to b, the sea is 40 kilometers closer than the land. In the morning 10, a ship sailed from A to B. In the afternoon 1, a car sailed from A to B. They arrived at B at the same time. The speed of the ship is 24 kilometers per hour, and the speed of the car is 40 kilometers per hour. So how many kilometers is the land and sea distance from A to B?

6. A group of students traveled at a speed of 5 km/h for 18 minutes. The school should send an urgent notice to the captain. The correspondent started from the school and chased the students by bike at the speed of14 km/h. How long will it take the correspondent to catch up with the students?

7. Mine blasting In order to ensure safety, the person who lit the fuse should be moved to a safe area 3000 meters away before blasting. The burning speed of the fuse is 0.8 cm/s, and the speed of people leaving is 5 m/s. How many centimeters does the fuse need at least?

8. Xiaoming and Xiaoli set out from school to watch sports games at the same time. Xiaoming walks 80 meters every minute. He walked to the playground and waited for five minutes before the game started. Xiaoli walks 60 meters every minute. When she entered the playground, the game had already started for 3 minutes. How far is it from school to the playground?

9. It takes four hours for a ship to sail between two docks, but after four and a half hours of countercurrent, it is still 8 kilometers short, and the current is 2 kilometers per hour. What's the distance between the two docks?

10.A.B is 360 kilometers apart, and the A train departs from A to B at a speed of 72 kilometers per hour. Twenty-five minutes later, the B train departs from B to A at a speed of 48 kilometers per hour. After the two cars meet, they continue to drive at the original speed, so when the distance between them is120km, one car starts from A *.

11.The distance between station A and bilibili is 5 10 km. A local train goes from Station A to bilibili at a speed of 45 kilometers per hour. After the local train had been running for two hours, another express train traveled from bilibili to Station A at a speed of 60 kilometers per hour. How many hours after the express train leaves?

12. A ship travels downstream from A for 9 hours, and it takes 1 1 hour to reach A. Given the current speed of 2 km/h, find the speed of the ship in still water.

Two. sale of goods

Example 2. 1. The purchase price of goods is 400 yuan, and the price is 600 yuan. The store requires a discount at a profit rate of not less than 5%. What is the lowest discount for selling this commodity?

2. The purchase price of a commodity is 1600 yuan, which is sold at 20% of the marked price, and the profit rate is 10%. What's its list price?

3. The purchase price of a kind of sports equipment is 1.200 yuan, and it is sold at a 10% discount on the basis of the marked price of 1.800 yuan. The purchase price of a treadmill is 2000 yuan, and the marked price is 3200 yuan, with a 20% discount. Which product has higher profit margin?

4. For a batch of goods, Party A will reduce the original price by 10 yuan, with 10% of the selling price as the capital, and Party B will reduce the original price by 20 yuan, with 20% of the selling price as the capital. If they have the same amount of capital, ask for the original price.

5. The price of a commodity is in 780 yuan. In order to make small profits but quick turnover, give a 10% discount on the selling price and then return it to 30 yuan as a gift certificate. At this time, the profit is still 65,438+00%. What is the purchase price of this commodity?

6. If a store sells a color TV at a price of 10%, it can still make a profit of 20%. If the purchase price of color TV is 2400 yuan, what is the price of color TV?

7. The price of a commodity is 165 yuan. If the price is reduced by 10% (that is, the discount is 10%), you can still make a profit by 10% (relative to the purchase price). What is the purchase price of the goods?

8. The purchase price of a commodity is 2000 yuan, and the price is 3000 yuan. The store requires a discount at the selling price, and the profit rate is not less than 5%. What's the lowest discount for salespeople to sell this product?

9. After a commodity is purchased, the retail price is set at 900 yuan per piece. In order to adapt to the market competition, the store reduced the price by 10% of the retail price and sold it in 40 yuan, with a profit of 10% (relative to the purchase price). What is the purchase price of this commodity?

10. A shop assistant sold two coats at the same time, each 135 yuan. If calculated by cost, one of them will make a profit of 25% and the other will lose 25%. Did the clerk lose money or make money this time?

1 1. Eggs in the market are priced by quantity. A vendor bought a batch of eggs at a price of 0.24 yuan each, but accidentally damaged 12 eggs in the process of selling. The remaining eggs were sold at a price of 0.28 yuan each, making a profit 1 1.2 yuan. How many eggs did the vendor buy at first?

12. A school is going to organize teachers and students to travel, including 22 teachers. There are two travel agencies, A and B, with the same pricing and preferential terms. A travel agency said that teachers are free and students charge 20% off; Travel agency b said that all teachers and students should get a 15% discount. After accounting, the actual charges of A and B are the same. How many students will take part in the tour?

13. A shareholder sells two kinds of stocks, A shares sell 1500 yuan, with a profit of 20%, and B shares also sell 1500 yuan, but with a loss of 20%. Is this shareholder a profit or a loss in this transaction? How much profit or loss?

14. A store bought a bell product of 100 and a bell product of 80 from the wholesale department of a company, and spent 2800 yuan. When the store retails, the price of each A bell product will increase by 15%, and the price of each B bell product will increase by 10%. In this way, after all products are sold, the income of * * * will be 366.

15. Due to the change of seasons, a certain commodity will be sold at a discount. If you sell at a 15% discount on the pricing, you will lose 25 yuan; if you sell at a 10% discount on the pricing, you will earn 20 yuan. What is the price of this commodity?

16. A set of furniture was sold at the cost price plus 60%. Later, under preferential conditions, the price was reduced by 72% to get 6336 yuan. What's the price of this furniture? How much money can this set of furniture make after it is sold?

17. The price of a commodity is 226 yuan. If you sell at a 30% discount, you can still make a profit 13%. What's the purchase price of this clock?

18. Xiao Zhang, a self-employed individual, can still make a profit of 20% by selling a commodity at a 10% discount on the marked price. If the purchase price of goods is 24 yuan per piece, what is the marked price of each piece?

19. The purchase price of a commodity is 3,000 yuan, and the price is 4,500 yuan.

(1) The store requires a discount on the selling price, with a profit of not less than 5%. What is the lowest discount for this product?

(2) If the market sales situation is not good, the store requires a discount at a loss. What is the lowest discount for selling this product?

(3) If the product causes a large inventory, the store requires a loss discount of no more than 5%. What is the lowest discount for selling this product?

Three. Deployment problem

Example 3. Five teachers and a group of students go to the park. The full fare for teachers is 7 yuan per person, and students only charge half price. If the ticket costs 206.50 yuan, how many students are there?

2)A has 27 people planting trees, and B has 18 people planting trees. If A has twice as many trees as B, how many people need to be transferred from B to A? .

3) Change the topic: The school organizes tree planting activities. It is known that A has 23 people planting trees, and B has 17 people planting trees. Now, 20 people are deployed to support it, so that A has twice as many people planting trees as B. How many people should A and B deploy?

4) A middle school organizes students to have a spring outing. If each car has 45 seats, 15 people have no seats. If there are 60 people in each car seat, then one car is empty and the other cars are just full. How many cars and students are there?

5) Someone bought a financing coupon of 2000 yuan, with a one-year annual interest rate of 9% and a two-year annual interest rate of 12%, which was withdrawn at the maturity of one year and two years respectively, and 450 yuan got interest. How much did he buy each of the two financing bonds?

6) There are 59 workers in a workshop, and it is known that each worker can process 15 Class A parts, 12 Class B parts, or 8 Class C parts on average every day. How to arrange daily production to make daily products complete? (3 parts A, 2 parts B, 1 part C as a set)

There are 50 students in a class. In a math exam, the pass rate of girls is 80%, boys are 75%, and the whole class is 78%. How many boys and girls are there in this class?

8) If a commodity is sold at a fixed price, everyone can make a profit in 45 yuan. Now the profit of selling 8 pieces at a 15% discount is the same as that of selling 12 pieces at a reduced price in 35 yuan. Ask about the purchase price of each item of the commodity. What is the price?

9) It is known that the original price of commodity A is 0.5 times that of commodity B+65438. Due to market changes, the percentage of price increase of commodity B is twice that of commodity A. After the price adjustment, the sum of the unit prices of commodities A and B is 2% higher than the sum of the original unit prices. Find the percentage of a commodity price decline and the percentage of b commodity price increase.

10) A commodity is made of two raw materials, A and B, of which A raw material is 50 yuan per kilogram and B raw material is 40 yuan per kilogram; After the price adjustment, the price of raw material A increased by 10% and the price of raw material B decreased by 15%, but the product cost remained unchanged after accounting. How many kilograms of raw materials A and B are needed to produce 1 1 kg of this product?

1 1) Buy cloth: The customer bought two kinds of cloth for 540 rubles 138 Russian ruler, of which 3 rubles are blue cloth and 5 rubles are black cloth. How much did each customer buy?

12) Similar variant 1: The Committee of "Hope Project" awarded 25 three-good students of the whole school a bonus of 2000 yuan, including 200 yuan for each of the three-good students at the municipal level and 50 yuan for each of the three-good students at the school level. Ask the three excellent students at the school level. How many are there in each school?

13) Similar to Variant 2: Party A and Party B jointly contribute to set up an enterprise, and agree to distribute profits in proportion to the contribution. It is known that the investment ratio of Party A and Party B is 3: 4, and the profit in the first year is 38,500 yuan. How much can Party A and Party B earn respectively?

14) A test paper * * * has 25 questions, and each question gives 4 answers, of which only one is correct. If each question is selected correctly, it will get 4 points, and if it is not selected or selected incorrectly, it will get 1 point. If a student gets 90 points, how many questions does he get right?

15) Pythagoras was asked how many students are there in his school. He replied, "Half of the students are studying math, a quarter are studying music, a seventh are resting, and there are three female students left." Ask how many students there are in Pythagoras' school.

16) There are 45 people in Class 2, Grade 7 who have signed up for a literature club or a painting and calligraphy club. It is known that there are five more people who join the literary club than the calligraphy and painting club, and there are 20 people who join both clubs. How many people have joined the painting and calligraphy club?

17) There are some cards marked with 5, 10, 15, 20, 25 ... The number on the last card is 5 larger than that on the previous card. Xiaoming got three adjacent cards, and the sum of the numbers on these cards is 240. (1) What three cards did Xiao Ming get? (2) Can you get three adjacent cards so that the sum of the numbers on these cards is 63?

18) If the sum of three consecutive integers is 72, what are these three numbers?

19) 60 students in one class went out to participate in tree planting activities. According to different tasks, they should be divided into three groups, and the ratio of the number of people in the three groups is 2: 3: 5, so as to find the number of people in each group.

20) The surface of football is surrounded by many black pentagons and white hexagons. The number ratio of black and white leather blocks is 3: 5. There are 32 leather blocks on the surface of a football. How many black leather pieces and white leather pieces are there?

2 1) The shareholders of Party A, Party B and Party C jointly run a company. The capital of Party A is half of the sum of the capital of Party B and Party C, and the capital of Party B is 530,000 yuan. What is the total capital of this company?

22) There are two fewer girls than boys in the math interest group of a class. If the number of girls increases by 3 and the number of boys decreases by 1, the number of girls is 3 more than that of the whole group. Find the original number of boys and girls.

23) There is a model TV set in the store, each set is 1.200 yuan, and the profit is 20%. At present, a merchant has bought several TV sets of this model at the total price of 1.500 yuan, so the store still has a profit of 1.5%. How many TV sets did the merchant buy?

Four. The problem of shape change

Example 4. 1. Cast three cylindrical parts with a diameter of 2 cm and a height of 16 cm with round steel with a diameter of 4 cm. How long does it take to cut round steel?

2. There is a cubic copper block with a side length of 4 cm, which should be melted and cast into a cuboid copper block with a width of 2 cm and 4 cm. What is the height of copper block after casting (excluding loss)?

3. A factory forged a cylindrical bottle with a diameter of 60 mm and a height of 20 mm, and then poured the water in the bottle into a cylindrical glass with a bottom diameter of 6 cm and a height of 10 cm. Can it be completely filled? If not, how high is the water level in the bottle? If it is not full, find the distance from the water surface in the cup to the mouth of the cup.

4. Fill all the water in a cylindrical bucket with a diameter of 40 cm and a height of 60 cm filled with water into another cylindrical bucket with a half diameter of 30 cm. What is the height of water at this time?

5. A cuboid iron block with a length of 9 cm, a width of 7 cm and a height of 3 cm and a cube iron block with a side length of 5 cm are fused into a cylinder with a base diameter of 20 cm, and the height of the cylinder is required (π is 3. 14).

6. Make a rectangle with a 5.2-meter-long iron wire, so that the length is 0.6 meters more than the width. How many meters is the length and width of this rectangle?

7. A cylindrical bucket with a diameter of 1.2 m and a height of 1.5 m is filled with water, and the water is poured into a square iron box with a bottom length of1m. When the iron box is filled with water, how many meters does the water height in the bucket drop?

8. The length-width ratio of the rectangle is 5: 3, and the length-width ratio is 12 cm. Find out the length and width of this rectangle respectively.

9. The diameter of the small cylinder is 8 cm, the height is 6 cm, the diameter of the large cylinder is 10 cm, and the volume is 2.5 times that of the small cylinder. What is the height of the big cylinder?

10. Forging a cylindrical blank with a radius of 5 cm and a height of 8 cm, how long should the round steel with a radius of 4 cm be cut?

1 1. It is known that the weight of soybean can be increased by 3.5 times after germination, and now it needs 100 kg of soybean sprouts. How many kilograms of soybeans should be used?

12. The circumference of the rectangle is 36 cm. If the length is reduced by 4 cm and the width is increased by 2 cm, the rectangle becomes a square. Find the side length of a square.

13. Use a cylindrical oil reservoir with a bottom radius of 5 cm, and soak the oil with steel balls. If 546π grams of steel balls are taken out of it, how many centimeters will the liquid level drop? (1 cubic centimeter steel ball 7.8g)

14. forging a cylindrical blank with a diameter of 70 mm and a height of 45 mm, how many millimeters should the round steel with a diameter of 50 mm be cut?

15. A machine factory wants to forge a blank with a cylinder with a diameter of 20 mm and a height of 40 mm above and a cylinder with a diameter of 60 mm and a height of 20 mm below. How long is a round steel with a diameter of 40 mm?

16. Circle a rectangle with 20cm iron wire (1), so that the length of the rectangle is 2.6cm wider. At this time, how many centimeters is the length and width of the rectangle? (2) Make the length and width of the rectangle equal. How long is the side of the square at this time?

17. There is a cylindrical iron block with a bottom diameter of 20 cm and a height of 26 cm. Forging it into a cuboid blank. If the rectangular body length is 10π cm and the width is 13 cm, find the height of the cuboid.