1:
Three brothers work in other places, the eldest brother goes home once every six days, the second brother once every eight days, the younger brother once every 12 days, and the three brothers go home at the same time 1 1 day. How many days will it take for the three of them to meet next time?
A: We can guess, that is, the process of pushing.
The three brothers set out at the same time in one day, that is, they went home at the same time in one day.
Next time:
Big Brother went home for the first time after 6 days, the second time after 12 days, the third time after 18 days, and the fourth time after 24 days, that is, the fourth time after 24 days.
The second brother went home for the first time after 8 days, the second time after 16 days and the third time after 24 days, that is, the second brother went home for the third time after 24 days;
12 days later, my brother went home for the first time, and 24 days later, that is, my brother went home for the second time after 24 days;
No matter how many times the eldest brother, the second brother and the younger brother go home, they will get together again in 24 days.
This method is not suitable for examples with a large amount of data, and it is not clear enough to explain the process as an application problem, which is a bit inappropriate.
(2): Three brothers go home at the same time on 1 1. The days after their next meeting should be multiples of 6 and 8, and also multiples of 12. In other words, the number of days after their next meeting is the common multiple of 6,8, 12, and only the least common multiple is needed.
The least common multiple of 6, 8 and 12 is 24.
The three brothers go home at the same time on 1 1, and it will take 24 days for the three to meet next time.
Note: "1 1 day" in the question "Three brothers1day go home at the same time" has nothing to do with the time and days of the next meeting. It is just a narrative way, a narrative way to express integrity.
2:
A rectangular iron sheet with a length of 105 cm and a width of 75 cm should be divided into square iron sheets with exactly the same size. How many square iron pieces can this rectangular iron piece be divided into at least?
Analysis:
It should be divided into square iron sheets with the same size and no excess, that is, the side length of a square is a divisor of the length of the original rectangle and the width of the original rectangle.
That is, the side length of a square is the common divisor of the length and width of the original rectangle;
And because it is to find out how many square iron pieces this rectangular iron piece can be divided into at least, the number of squares is the least, that is, the greater the side length of the square, and the side length of the square just analyzed is the common divisor of the length and width of the original rectangle.
Now is the time to find the square with the largest side length, that is, to find the greatest common divisor of the length and width of the original rectangle as the side length of the square.
The greatest common divisor of 105 and 75 is 15.
Namely:
Side length of a square: 15cm.
Number of squares:
(105× 75) ÷ (15×15) = 35 (piece)
The square number can also be obtained by using the divisor and quotient in the short division formula of the decomposition prime factor.
The divisors of 105 and 75 are both 15, that is, the greatest common divisor of 105 and 75 is 15, and the quotient of 105 is 7 (indicating that 105 can be expressed as1. The quotient of 75 is 5 (indicating that 75 can be divided into 5 segments according to 15).
It is divided into 7 segments and 5 segments. The number of squares is 7×5=35.
3:
There is a basket of apples, whether it is distributed to 8 people or 10 people, there are still 3 apples left. How many apples are there at least in this basket?
Analysis:
If you subtract 3 from the total number of apples, you will get a new total, whether it is distributed to 8 people or 10 people, there is nothing left, just finished eating.
That is to say, the new total number of apples is not only a multiple of 8, but also a multiple of 10, which is the common multiple of 8 and 10. How many apples does this basket need? So only 8 and the least common multiple of 10 are needed.
The least common multiple of 8 and 10 is 40.
That is to say, the total number of new apples is 40, and the total number of apples is obtained by subtracting 3 from the total number of apples:
That is 40+3=43 (pieces)
Note: sometimes it's not easy to understand because of formulas. For example, there is a basket of apples in the above question. Whether it is distributed to 8 people or 10 people, there are still 3 left.
Can be expressed as:
? 8= quotient ... 3
? 10 = quotient ... 3
? -3=A
A is divisible by 8, and A is divisible by 10. In other words: A is a multiple of 8, and A is a multiple of 10. That is, a is the common multiple of 8 and 10.
Then go down and analyze it.
Problem sets of cuboid and cube units
1:
Combine three cuboids with a length of 3cm, a width of 2cm and a height of 1cm into a cuboid with the smallest surface area. What is the surface area and volume of this cuboid?
Analysis:
According to the meaning of the question, how to spell out a cuboid with the smallest surface area is the key. There are many spelling methods. To find the smallest surface area of a spliced cuboid, we can find this spelling with the help of objects or drawings, that is, to make a cuboid with the smallest surface area, and try to hide the larger surface in the original small cuboid, that is, the surface that overlaps the most when spelling.
Obviously, the 3×2 surface is the largest, and the surface to be overlapped is this surface.
Large cuboids assembled together:
3 cm long, 2 cm wide and 3 cm high.
With the length, width and height, we can calculate the surface area and volume.
Surface area:
(3× 2+3× 3+2× 3 )× 2 = 42 (square centimeter)
Volume:
3×2×3= 18 (cubic centimeter)
You can also find 3 times of the original small cuboid, because the volume of the inverse spelling is unchanged, that is, the volume of the current large cuboid is equal to the volume of the original three small cuboids. 3×2× 1×3= 18 (cubic centimeter)
2:
Divide a cuboid with a surface area of 80 square decimeters into two completely equal cubes. What is the surface area of each cube?
Analysis:
According to the meaning of the question, the cuboid is divided into two completely equal cubes, and it can be concluded that this cuboid is a special cuboid (first, a group of faces are squares, and the other four faces are rectangles with equal sizes, and two square faces can be combined into a rectangular face. )
So each rectangular surface can be divided into two squares on average, and four rectangular surfaces can be divided into eight squares.
The six faces of the original cuboid can be divided into 10 squares with equal size, and the square is any face of the cube that is now divided.
Namely:
The area of one face of a divided cube is
80÷ 10=8 (square decimeter)
The surface area of a cube is 6 faces.
8×6=48 (square decimeter)
It is suggested to draw pictures or use physical objects to help analysis.
3:
After the cuboid wood with a length of 1 m was sawed into two sections on average, the surface area increased by 80 square centimeters. What is the volume of this piece of wood in cubic centimeters?
Analysis:
The key understanding is that the average surface area has increased by 80 square centimeters after sawing into two parts. Two additional surfaces after sawing, namely two bottom surfaces. The area of the bottom surface can be obtained by dividing the added 80 square centimeters by 2.
Bottom area:
80÷2=40 (square centimeter)
The area of the bottom multiplied by the height is greater than the volume.
Volume:
1 m = 100 cm
40× 100=4000 (square centimeter)
4:
After a cube is divided into eight identical cubes, the surface area is increased by 320 square centimeters. What is the surface area of this cube?
Analysis:
Divide the big cube into eight small cubes on average. On the surface, one face of each big cube is divided into four small squares on average, and the small squares are just one face of the small cube. As far as appearance is concerned, we can see three faces of the small cube, which are only seen after being divided, that is, three faces are added after each small cube is divided, and the original big cube has three faces of the small cube without time division. By analogy, every small cube is like this.
So the increased area is actually equal to the surface area of the original large cube. In this way, the increased 320 square centimeters are replaced by the surface area of this big cube.
It is also recommended to draw pictures or find relationships with objects. It is not recommended to use 320 to remove the (3×8) plane contained in eight small cubes and multiply it by the (3×8) plane contained in one large cube. Because 320 ÷ (3× 8) = 13.33 ... At this time, there is a circular decimal, and then how to calculate it. This problem only needs to be explained, and there is no need to express the process with any formula.
Analysis of classic good questions in primary school mathematics review
answer the question
1, both parties built1875m expressway at the same time, which took 25 days. When completed, Team B repaired125m less than Team A, and Team B repaired 35m on average every day. How many meters does Team A repair on average every day?
Parsing 1:
Use (total number of meters-total number of meters repaired by Team B) ÷25= number of meters repaired by Party A every day. 125m in the question is redundant.
Equation: (1875-35×25)÷25=40 (m)
Analysis 2:
There is no need to use the average number of meters repaired by team B+the number of meters repaired by team B is less than that of team A = the number of meters repaired by team A every day. The total length in the question is 1875 meters.
Equation: 35+ 125÷25=40 (m)
2. It takes 8 hours for the express train to reach bilibili, and 12 hours for the local train to reach Station A. If the express train and the local train leave from Station A and bilibili at the same time, the express train will travel 180 kilometers more than the local train. How many kilometers will these two stations meet?
Parsing 1:
According to the known conditions, the speed of the express train is 1/8 and the speed of the local train is112. First find out the meeting time, then find out the fraction of the total length where the express train meets the local train, and finally divide it by the relative quantity to get the total length.
Column type:
1÷ (1/8+112) = 24/5 (hours)
( 1/8- 1/ 12)×24/5= 1/5
180 ÷1/5 = 900km
Analysis 2:
It can also be solved by the method of "proportional distribution"
1/8: 1/ 12=3:2
3+2=5
180(3/5-2/5)= 900 (km)