Example 1 Two cars leave from A and B at the same time and meet five hours later. One car is 55 kilometers per hour, and the other car is 45 kilometers per hour. How many kilometers is it between A and B?

Analyze 1, first find out how many kilometers the two cars have traveled, and then find out the sum of the driving distances of the two cars, that is, how many kilometers a and b are apart.

Solution 1 How many kilometers has a car traveled?

55×5=275 km

How many kilometers has the other car traveled?

45× 5 = 225km

How many kilometers is it between A and B?

275+225=500 kilometers

Comprehensive formula: 55×5+45×5.

=275+225=500 km

Analysis 2: First find out how many kilometers two cars travel every hour, and then multiply by the meeting time to find out how many kilometers a and b are apart.

How many kilometers do two cars travel per hour?

55+45 =100km

How many kilometers is it between A and B?

100×5=500 km

Comprehensive formula: (55+45)×5

= 100×5=500 (km).

It is analyzed that the distance between 3 A and B divided by the meeting time is equal to the sum of the speeds of two cars. From this, we can list the equations and find out how many kilometers a and b are apart.

Solution 3 Let Party A and Party B be x kilometers apart.

x÷5=55+45

x= 100×5

x=500

Analyze the distance between Party A and Party B minus the distance traveled by one car equals the distance traveled by another car, and solve the equation.

Scheme 4 assumes that Party A and Party B are x kilometers apart.

x-55×5=45×5

x-275=225

x=275+225

x=500

A: A and B are 500 kilometers apart.

Explanation solution 2 and solution 1 are both arithmetic solutions, of which solution 2 is better. Solutions 3 and 4 are solutions of equations, in which solution 3 is the better solution. Comparing the above four solutions, 1 and 2 solutions can be transformed into each other by using multiplication and division methods, and the quantitative relationship between 1 and 4 solutions, 2 solutions and 3 solutions is the same respectively. By comparison, we will find that they just have different ideas and methods to solve them.

Two cars leave from two places 345 kilometers apart at the same time. One car travels 60 kilometers per hour, and the other car travels 55 kilometers per hour. In a few hours, the two cars can meet.

(Shenyang City, Liaoning Province)

Analyze 1, and first find out how many kilometers two cars travel per hour, that is, the speed sum. Then according to the formula "distance-speed sum = meeting time", it can be obtained.

Solution 1 345(60+55)

= 345/ 1 15 = 3 (hours).

Analysis 2 The sum of the distances when two cars meet is equal to the distance of 345 kilometers between the two places. From this, the solutions of the equation can be listed.

Solution 2 suppose two cars meet after x hours.

60x+55x=345

1 15x=345

x=345÷ 1 15

x=3

Analysis 3 According to the equivalence relation of "speed and × meeting time = distance between two places", solve the equation.

Solution 3 suppose two cars meet after x hours.

(60+55)×x=345

X=345 degrees (60+55)

x=345÷ 1 15

x=3

Analysis 4 The distance between two places minus the distance traveled by one car equals the distance traveled by another car. Solutions of this series of equations.

Solution 4 suppose two cars meet after x hours.

345-60 times = 55 times

60x+55x=345

1 15x=345

x=3

A: After three hours, the two cars can meet.

Explaining 1 is a good way to solve this problem because of its clear thinking and simple operation. The latter three solutions are all equation solutions. In fact, the solutions of these three equations are all the same quantitative relationship. By comparison, we will find that they are all obtained by the deformation of an equation, and solution 3 is relatively simple.

The express train and the local train leave from two cities 385 kilometers apart at the same time. Five hours later, two trains met. The local train travels 35 kilometers per hour. How many kilometers does the express train run per hour?

(nangang district, Harbin, Heilongjiang)

Analyze 1 to find out how many kilometers the local train has traveled, and then subtract the distance of the local train from the distance between the two cities, which is equivalent to how many kilometers the express train has traveled, so as to find out how many kilometers the express train can travel per hour.

How many kilometers did the local train of 1 travel?

35× 5 =175km

How many kilometers has the express train traveled?

385- 175=2 10 (km)

How many kilometers does the express run per hour?

210 ÷ 5 = 42km

Comprehensive formula: (385-35×5)÷5

=(385- 175)÷5=2 10÷5

=42 (km).

Analysis 2: divide the distance between two cities by the meeting time of two cars to get the speed sum of the two cars, and then subtract the speed of the local car from the speed sum to get the speed of the express train.

How many kilometers do two cars travel per hour?

385÷5=77 km

How many kilometers does the express train run per hour?

77-35 = 42km

Comprehensive formula: 385÷5-35=77-35=42 (km).

Analysis 3 According to the equivalence relation of "speed and × meeting time = distance between two places", solve the equation.

Option 3: set the speed of the express train to x kilometers per hour.

(35+x)×5=385

35+x=385÷5

x=385÷5-35

x=42

Analysis 4 is solved according to the equation "local travel distance+fast travel distance = distance between two places".

Option 4: set the speed of the express train to x kilometers per hour.

35×5+5x=385

5x=385-35×5

5x=2 10

x=42

Analysis 5 Assuming that the express train and the local train have the same speed, the distance between the two cities is 35×2×5=350 (km). This is 385-350=35 (km) less than the actual distance, and then divide the 35 km into 5 parts on average. The sum of each part and local speed is the speed of the express train.

Solution 5(385-35×2×5)÷5+35

=(385-350)÷5+35

= 35/5+35 = 7+35 = 42 km

A: The express train runs 42 kilometers per hour.

The commentary compares the above five schemes, and the second scheme is concise, simple and easy to think of, which is the best scheme to solve this problem.

There are four stations on an expressway in turn: A station, bilibili station, C station and D station. Xiao Ming and Xiao Hua walked out of Station A and Station D at the same time. When Xiaoming walked to bilibili in 40 minutes, Xiaohua just walked to Station C and asked them how many minutes before they met. B to C station1520m, A to D station 5320m. (Putuo District, Shanghai)

1 first find out how many meters Xiao Ming and Xiao Hua walked in 40 minutes, and then divide by 40 to get their speed sum. Divide1520m by the sum of speeds, which is equal to the meeting time when two people walk again.

1 How many meters did two people walk in 40 minutes?

5 320- 1520=3 800 meters

What is the speed sum of two people?

3 800÷40=95 (m)

How many minutes before they meet?

1520÷95= 16 (minutes)

Comprehensive formula:1520 ÷ [(5 320-1520) ÷]

= 1520÷[3 800÷40]

= 1520÷95= 16 (minutes).

Analysis 2: First find the speed sum of two people, then find out how many minutes they need to meet with * * *, and then subtract 40 minutes from * * * to get the meeting time.

Solution 2 What is the speed sum of two people?

(5 320- 1520)÷40=95 (m)

How many minutes does it take two people to walk the whole distance?

5320÷95=56 (points)

After walking for a few minutes, the two met.

56-40= 16 (minutes)

Comprehensive formula: 5320 ÷ [(5320-1520) ÷ 40]-40.

=5320÷[3800÷40]-40

=5320÷95-40=56-40= 16 (points).

Analysis 3: First find out how many times the distance traveled is the distance traveled again, and then divide it by 40 minutes to get the time required for two people to walk again.

Solution 3 How many meters did they walk?

5320- 1520=3800 meters

How many times do you travel?

3800÷ 1520=2.5 (times)

After walking for a few minutes, the two met.

40÷2.5= 16 (minutes)

Comprehensive formula: 40 ÷ [(5320-1520) ÷1520]

=40÷[3800÷ 1520]

=40÷2.5= 16 (minutes).

Analysis 4 Because the distance between the two places is equal to the speed sum, and the speed sum of two people is unchanged, the distance between the two places is directly proportional to the meeting time.

Option 4: We walk for another X minutes, and they will meet.

(5320- 1520)∶40 = 1520∶x

3800∶40= 1520∶x

x= 16

A: They met after they left 16 minutes.

The explanatory solution 1 is a general solution, which is easy to understand and master, but the calculation is more complicated. Scheme 3 is simple in thinking and uncomplicated in operation, which is a better solution to this problem. At the same time, from the idea of solution 3, we can infer the method of using fractional application problems or using the knowledge of ratio to solve problems, which readers can try.

09 Xiaoshengchu Mathematics Example Detailed Solution (2)

Example: Two cars, A and B, start from two cities respectively. Car A travels 33 kilometers per hour, while car B travels 28 kilometers per hour. Two hours after car A leaves, car B leaves, and we'll meet in three hours. How many kilometers are the two cities apart?

First, analyze the distance traveled by 1 A for two hours, and then add up the distances traveled by two cars for three hours at the same time, and the total is how many kilometers apart the two cities are.

1 How many kilometers did a car run in 2 hours?

33×2=66 km

How many kilometers does it take for Party A and Party B to drive for 3 hours?

(33+28) × 3 = 61× 3 =183 (km)

How many kilometers are the two cities apart?

66+ 183=249 km

Comprehensive formula: 33×2+(33+28)×3.

=33×2+6 1×3

=66+ 183=249 (km).

Analyze the distance traveled by car A plus the distance traveled by car B to find out how many kilometers apart the two cities are.

Solution 2 How many hours did the car drive?

2+3=5 (hours)

How many kilometers has car A traveled?

33× 5 =165km

How many kilometers has the B car traveled?

28×3=84 km

How many kilometers are the two cities apart?

165+84=249 km

Comprehensive formula: 33×(2+3)+28×3.

=33×5+28×3= 165+84=249 (km)。

Analysis 3 assumes that car A and car B start at the same time, that is, the two cars meet for 5 hours at the same time. Therefore, the distance between the two cars is 28 kilometers more than the actual distance between the two cities. From this, the actual distance between the two cities can be calculated.

Solution 3 Suppose two cars set off at the same time, how many hours did they meet?

2+3=5 (hours)

How many kilometers do two cars drive for five hours at the same time?

(33+28) × 5 = 305km

How many kilometers more than the actual calculation of B car?

28×2=56 km

How many kilometers are the two cities apart?

305-56=249 km

Comprehensive formula: (33+28)×(2+3)-28×2.

=6 1×5-28×2

=305-56=249 km

Analysis 4. A car leaves 2 hours first, which can be assumed to be later than the actual time 1 hour; The second car should be earlier than the actual departure time 1 hour. So, the original question should be: car A and car B face each other at the same time and meet after 4 hours. But the sum of the distances traveled by the two cars is 33-28=5 (km) less than the actual distance between the two cities.

Solution 4 (33+28)×(3+2÷2)+(33-28)

=6 1×4+5=244+5=249 km

A: The distance between the two cities is 249 kilometers.

Solution 1 and solution 2 are general methods, which are easy to think of, understand and master. Scheme 3 and Scheme 4 are hypothetical methods with novel ideas and troublesome formulas, but the operation is not troublesome.

09 Xiaoshengchu Mathematics Example Detailed Solution (3)

The railway between Station A and bilibili is 490 kilometers long. Two trains, A and B, start from these two stations at the same time. The speed of train A is 72 kilometers per hour, and that of train B is 68 kilometers per hour. How many kilometers did the A and B trains travel when they met?

(Shenzhen City, Guangdong Province)

Analysis 1 Calculate the meeting time of two cars according to "the sum of distances between two places ÷ speed = meeting time", and then multiply the meeting time by the speed of two cars respectively, so as to calculate how many kilometers each car has traveled.

1 how many hours did the two cars meet?

490÷(72+68)=490÷ 140=3.5 (hours)

How many kilometers did A drive?

72× 3.5 = 252km

How many kilometers has the B car traveled?

68× 3.5 = 238km

Comprehensive formula: a car: 72×[490(72+68)]

=72×[490÷ 140]

= 72× 3.5 = 252km

Bus B: 490-252=238 (km).

Analysis 2: According to the equal travel time of two trains, solve the equation.

Solution 2: Suppose that vehicle A travels x kilometers and the distance traveled by vehicle B is 490-x. 。

140x=72×490

x=

x=252

The trip of train B is: 490-252=238 (km).

Analysis 3 Because "distance ÷ speed = time", the time is certain, so the distance is proportional to the time, that is, the speed ratio of car A and car B is just the ratio of the driving distance of car A and car B, so we can first find the speed ratio of car A and car B, and then find the distance of car A and car B respectively by proportional distribution method.

Solution 3: What is the ratio of the distance traveled by a car to that traveled by a car?

72∶68= 18∶ 17

How many kilometers did Car Shop A walk?

490×490×252 km

How many kilometers has the B car traveled?

490×490×238 km

Comprehensive formula: Car A:490×252 (km)

Car B: 490×=238 (km).

Answer: when we met, the car shop was 252 kilometers and the car shop was 238 kilometers.

Comment solution 1 is a common solution, which is easy to understand and master. The third scheme is the proportional distribution scheme, which is ingenious and simple to operate, and is the best scheme to solve this problem.

09 Xiaoshengchu Mathematics Example Detailed Solution (4)

Example: Two trains, A and B, travel relatively from two places 630 kilometers apart at the same time and meet six hours later. Train A is 5 kilometers faster than train B every hour. What's the speed of these two trains?

Analysis of 1: First, find the speed sum of car A and car B, then add 5km to the speed sum, which is equal to the two-hour journey of car A, then divide by 2 to get the speed of car A, and subtract 5km from the speed of car A to get the speed of car B. 。

What is the speed sum of car 1 A and car b?

630 ÷ 6 =105km

What is the speed of car A?

(105+5) ÷ 2 =1/kloc-0 ÷ 2 = 55 (km)

B What's the speed of the train?

55-5=50 km

Comprehensive formula: a car: (630÷6+5)÷2

= (105+5) ÷ 2 =1/kloc-0 ÷ 2 = 55 (km)

B: 55-5=50 (km).

Analysis 2 Assuming that the speed of car B and car A are the same, when they meet, the sum of the distances traveled by car A and car B is 5×6=30 (km) more than the actual distance between the two places. Then divide the sum of 630 km and 30 km by 6 hours, and you can get the 2-hour journey of A car. Find the speed of B car again.

Solution 2 Assuming that the speed of car B is the same as that of car A, how many kilometers is * * *?

5×6=30 km

How many kilometers does the car travel in 2 hours?

(630+30) ÷ 6 = 660 ÷ 6 =110 (km)

How many kilometers does car A travel per hour?

110 ÷ 2 = 55km.

How many kilometers per hour is the B train?

55-5=50 km

Comprehensive formula: a car: (630+5×6)÷6÷2.

=660÷6÷2=55 km

B: 55-5=50 (km).

Analysis 3 Assuming that the speed of car A and car B is the same, the sum of the distances traveled by the two cars is 5×6=30 (km) less than the actual distance between the two places. Divide the difference between 630 km and 30 km by 6 hours, and you can get the 2-hour journey of B car. From this, the speed of car B can be obtained first, and then the speed of car A can be obtained.

Solution 3 Assuming that the speed of car A and car B is the same, how many kilometers is * * * missing?

5×6=30 km

How many kilometers did bus B travel in two hours?

(630-30)÷6=600÷6= 100 km

How many kilometers per hour is the B train?

100 ÷ 2 = 50km

How many kilometers does car A travel per hour?

50+5 = 55km

Comprehensive formula: Car B: (630-5×6)÷6÷2.

=600÷6÷2=50 km

Car A: 50+5=55 (km).

Analysis 4 is solved according to the enumerable equation of "speed and × meeting time = distance between two places".

Solution 4 Let car B travel x kilometers per hour, then car A travels (x+5) kilometers per hour.

(x+5+x)×6=630

2x+5=630÷6

2x=630÷6-5

x=(630÷6-5)÷2

x=50

x+5=50+5=55

A:A car travels 55 kilometers per hour, and B car travels 50 kilometers per hour.

Comment solution 1 is a common solution, which is easy to understand and master. Scheme 2 and Scheme 3 are both hypothetical methods, which are easy to understand and operate, and are relatively good schemes. The equation solution of solution 4 can also set the speed of armored vehicle as x, which readers can try.

09 Xiaoshengchu Mathematics Example Detailed Solution (5)

For example, buses and trucks travel in opposite directions from the midpoint between cities A and B at the same time. Three hours later, the bus arrived in City A, and the truck was 30 kilometers away from City B. It is known that the speed of the truck is 3/4 of that of the bus. How many kilometers are there between cities A and B?

From the analysis of 1, it can be seen that the freight car travels 30 kilometers less than the bus in three hours, from which the speed difference between the two cars can be found, and then divided by the corresponding score (1-3/4), the speed of the bus can be found, and then the speed of the second car can be found. Finally, according to "the sum of speed × meeting time = the distance between the two places", the distance between the two cities A and B can be calculated.

Solution 1 How many roads does the truck walk less than the bus per hour?

30÷3= 10 km

How many kilometers does this bus travel per hour?

10( 1-3/4)= 40 (km)

How many kilometers does this truck travel per hour?

40-10 = 30km

How many kilometers is it between A and B?

(40+30) × 3+30 = 240km

Comprehensive formula: 30 ÷ 3 ÷ (1-3/4) × (1+3/4 )× 3+30.

=30÷3÷ 1/4×7/4×3+30

=40×7/4×3+30=240 (km).

Analysis 2 Because "distance ÷ speed = time" and time is fixed, the distance traveled by two cars is directly proportional to the speed of two cars, that is, the speed ratio of trucks and buses is the ratio of their distance traveled. Converted to 3: 4, that is, the distance ratio between trucks and buses. It is also known that the distance between the two cars is 30 kilometers, so how many kilometers are the two cities apart?

Solution 2 30(4-3)×(3+4)+30

=30÷ 1×7+30=240 (km).

Analysis 3 According to the equation of equivalence relation "the distance traveled by bus minus the distance traveled by truck equals 30 kilometers", first calculate the speed of two cars, and then multiply the sum of the speeds by the meeting time plus 30 kilometers, then how many kilometers are the two cities apart.

Solution 3: Set the bus to travel x kilometers per hour.

3x-3(3/4)x=30

x=40

3x=30

x= 10

Distance between two cities: (40+30) × 3+30 = 240 = 240 (km).

A: The distance between the two cities is 240 kilometers.

The explanatory solution 1 is a basic solution, which is easy to understand, but the calculation is complicated. The quantitative relationship and thinking of solution 3 and solution 1 are basically the same. Scheme 2 is the best solution to this problem because of its simple thinking and simple operation.

09 Xiaoshengchu Mathematics Example Detailed Solution (6)

1 the express train takes 6 hours from city a to city B. The local train goes from city B to city A at a speed of 42.5 kilometers per hour. The distance between the two trains is 132 km. How many kilometers does the express train run per hour?

1 the whole analysis of the express train takes 6 hours, and it has been running for 2 hours. The rest of the journey, the express train will take 4 hours. In other words, the sum of the local train's 2-hour journey and 132 kilometers can be completed by the express train in 4 hours. From this, we can know how many kilometers an hour the express train is.

How many kilometers did the local train 1 travel in 2 hours?

42.5× 2 = 85km

How many kilometers can an express train travel in four hours?

85+ 132=2 17 (km)

How many kilometers does the express train run per hour?

217 (6-2) = 54.25km.

Comprehensive formula: (42.5×2+ 132)÷(6-2)

=(85+ 132)÷4

=2 17÷4=54.25 (km).

Analysis 2 Because the whole journey of the express train takes 6 hours, it has already gone for 2 hours, and the distance that the express train doesn't go is (6-2)÷2=2 (times), from which we can find out how many kilometers the express train has traveled in 2 hours, and then how many kilometers its speed is.

Option 2: How many kilometers does the express train travel?

42.5× 2+132 = 85+132 = 217 (km)

How many times is the distance that the express train has not traveled?

(6-2)÷2=2 (times)

How many kilometers did the express train travel?

2 17÷2= 108.5 (km)

How many kilometers does the express train run per hour?

108.5÷ 2 = 54.25km.

Comprehensive formula: (42.5×2+ 132)÷[(6-2)÷2]÷2.

=(85+ 132)÷[4÷]÷2

=2 17÷2÷2=54.25 (km).

Analysis 3 Because the express train runs the whole journey every hour, it takes 2 hours to run the whole journey. The distance that the express train does not run is 1-=. Divide by the distance that the express train doesn't run, and then divide by 6 hours to get the speed of the express train.

Option 3: How far is the express train?

42.5× 2+132 = 85+132 = 217 (km)

How many kilometers are the two cities apart?

217 ÷ (1-) = 217 ÷ = 325.5 (km)

How many kilometers does the express train run per hour?

325.5÷6=54.25 km

Comprehensive formula: (42.5× 2+132) ÷ (1-) ÷ 6.

=(85+ 132)÷÷6

=2 17××=54.25 (km).

Analysis 4 is solved according to the equation "the distance between two cities minus the distance traveled by the express train equals the distance traveled by the express train".

Option 4: set the speed of the express train to x kilometers per hour.

6x-2x=42.5×2+ 132

4x=2 17

x=54.25

A: The speed of the express train is 54.25 kilometers per hour.

Comment solution 3 is a general solution with complicated calculation. Solution 4 is the best solution to this problem because its equivalence is properly determined and its operation is simple. The solution 1 is simpler, more ingenious and simpler.

Example 2 A car and a truck set off from two places 432 kilometers apart at the same time and met several hours later. As we all know, the speed ratio of cars and trucks is 9: 7. How many kilometers do cars and trucks travel per hour?

(Nanning, Guangxi Zhuang Autonomous Region)

Analysis 1: divide the distance between the two places by the meeting time to get the speed sum of the car and the truck, and then use the proportional distribution method to distribute the speed sum at 9: 7, so as to find out how many kilometers the car and the truck travel per hour.

What is the speed sum of 1 two cars?

432 degrees =96 kilometers

How many kilometers does this truck travel per hour?

96×42 km

How many kilometers does this car travel per hour?

96×54 km

Comprehensive formula: automobile: 432 \u x

=432×× (km)

Truck: 432 feet.

= 432×42 km

Or: 54 ÷ 9× 7 = 42km.

Analysis 2 Because "distance-speed = time" and time are fixed, the distance traveled by two cars is directly proportional to their respective speeds. So the speed ratio of two cars is equal to the distance ratio of two cars. From this, 432 kilometers can be allocated according to 9: 7, and the speed of two cars can be calculated.

Solution 2 How many kilometers has the car traveled?

432 ÷ (9+7) × 9 = 432 ÷16 × 9 = 243 (km)

How many kilometers does this car travel per hour?

243 degrees =54 kilometers

How many kilometers has this truck traveled?

432(9+7)×7 = 189 (km)

How many kilometers does this truck travel per hour?

189 = 42km

Comprehensive formula: automobile: 432 × degrees.

= 432×54 km

Truck: 54 ÷ 9× 7 = 42km.

Analysis 3 converts 9∶7 into the speed of the car, and if the speed of the car is X, the speed of the car is X. According to the equivalence relation of "speed and × meeting time = distance between two places", the speed of the two cars is calculated by using equations.

Solution 3 Let the car run at the speed of X kilometers per hour.

(x+x)×=432

x+x=432

( 1+)x=432×

x = 96( 1+)

x=54

Truck: 54×42 km.

A: The speed of the car is 54 kilometers per hour; The truck travels 42 kilometers per hour.

Explaining this topic is an application problem that comprehensively uses the knowledge of trip sum ratio. The key to solve this kind of problem is to pay attention to the transformation and understanding of known conditions. For example, solution 3 is to understand the proportion as the number of components, which changes the thinking of solving problems. At the same time, we should pay attention to the comprehensive application of knowledge. For example, scheme 1 uses the knowledge of travel application and proportional distribution, and scheme 2 uses the knowledge of the meaning and proportional distribution. Compare the above three schemes, scheme 65440.