Two-step application of 1. easily solvable sequence equation
Steps to solve application problems with (1) series equation
(1) find out the meaning of the problem, find out the unknown, represented by x;
(2) Find out the equal relationship between the quantities in the application problem and make an equation;
③ Solve the equation;
4 Check and write the answers.
(2) The key to solving practical problems by using equations.
After understanding the meaning of the question, find out the equal relationship between the quantities in the application question, set the unknowns appropriately, and list the equations.
(3) Using the series equation of general quantitative relation to solve practical problems.
(1) equations to solve addition and subtraction problems. For example:
The combined age of Party A and Party B is 29 years old. It is known that Party A is three years younger than Party B. How old are Party A and Party B?
Equivalence between quantities:
Age of Party A+Age of Party B = Age of Party A and Party B and
Solution: Let A be X years old, then B is (x+3) years old.
x+(x+3)=29
x+x+3=29
2x=29-3
x = 26 2
X =13 ... the age of a.
13+3= 16 (years old) ...
A: The age of A is 13, and the age of B is 16.
(2) The application of multiplication and division method in solving series equations. For example:
The school library bought 240 story books, three times as many as science and technology books. How many science and technology books did it buy?
Number of science and technology books 3 = number of story books.
Solution: Suppose you buy X science and technology books.
3x=240
x=80
I bought 80 science and technology books.
(4) Using calculation formulas, properties, numbers and counting units, the equivalence relationship between quantities is made, and equations are made to solve application problems.
① The circumference of a rectangle is 240m, and its length is 0.4 times of 65438+ width. Find the area of a rectangle.
(length+width) 2= circumference
Solution: If the width is x meters, then the length is (1.4x) meters.
( 1.4x+x) 2=240
2.4x = 240 2
x= 120 2.4
X = 50 ... the width of the rectangle.
50 1.4=70 (meters) ... the length of the rectangle.
70 50=3500 square meters
A: The rectangular area is 3,500 square meters.
② In triangle ABC, angle A is twice as big as angle B, and the sum of angle A and angle B is smaller than angle C 18 ... Find the degrees of the three angles. What triangle is this?
Angle A+ angle B+ angle C = 180 degrees.
Solution: let angle b be x degrees,
Then the angle A is (2x) degrees and the angle C is [(2x+x)+ 18] degrees.
2x+x+[(2x+x)+ 18]= 180
6x+ 18= 180
6x= 180- 18
x= 162 6
X = 27 ... degrees of angle B.
27 2 = 54 degrees ... the degree of angle A.
54+27+ 18=99 (degrees) ... the degree of angle C.
A: Angle A is 54 degrees, angle B is 27 degrees and angle C is 99 degrees.
Because: angle b
③ The sum of a two-digit number, a ten-digit number and a one-digit number is 6. If you subtract 7 from the original number, the ten digits are the same as the single digits, and the original number is found.
Ten-digit numbers, one-digit numbers.
Solution: Let the single digit of the original number be X. Then the number on the original ten digits is: 6-x; If you subtract 7 from the original number, the number in one place becomes: 10+x-7, and the number in the tenth place becomes: 6-x- 1.
6-x- 1= 10+x-7
5 x = 3+x
2x=2
X = 1 ... one digit of the original number.
6- 1 = 5 ... the tenth digit of the original number.
So the original number is: 5 1.
2. Solve the application problems of two-step and three-step calculation with column equation.
Guangshui Cinema has 32 rows of seats, with an average of 38 people in each row; After the expansion, it increased to 40 rows, 584 more people than before. How many people can sit in each row on average after the expansion?
Solution: After the expansion, X people sit in each row on average.
x 40-38 32=584
40x- 12 16=584
40x=584+ 12 16
x= 1800 40
x=45
A: After the expansion, each row can seat an average of 45 people.
3. Solve binary application problems with column equations.
A class of students together to buy a souvenir, each pay 1 yuan, more than 6 4 yuan; Everyone pays 90 cents, only 5 cents short. How much is this souvenir? How many students are there in this class?
Solution: Suppose there are X students in this class.
x-4.6=9 10 x+5 10
x-4.6=0.9x+0.5
0. 1x=5. 1
X = 5 1 ... the number of students in this class.
5 1-4.6=46.4 yuan ... unit price of souvenirs.
A: this souvenir is 46.4 yuan; There are 5/kloc-0 students in this class.
4. Comparison of solving application problems with equations and arithmetic.
What is the difference between solving application problems with equations and solving application problems with arithmetic? The main difference between them is that they have different ideas.
To solve an application problem with an equation, we should set an unknown X, put the unknown X and the known number together, analyze the quantitative relationship described in the application problem, and then list the equation according to the quantitative relationship and the meaning of the equation.
To solve practical problems by arithmetic, we should concentrate and analyze the known numbers, find out the relationship between the known numbers and the unknown numbers, and list the formulas for expressing the unknown numbers. For example:
The height of floret 160 cm is higher than that of Xiaolan 15 cm. How many centimeters is Xiaolan tall?
Solve by equation:
Solution: Make Xiaolan x cm high.
160-x= 15
x= 160- 15
x= 145
Or: x+ 15= 160.
x= 160- 15
x= 145
Solve by arithmetic:
160- 15= 145
By comparison, students can see that the main difference between the two methods lies in whether the unknown quantity participates in the formulation. The column arithmetic formula deduces the unknown from the known according to the conditions in the problem, and expresses the unknown by the relationship between the known numbers. Unknown is the result of operation, and known and unknown are separated by an equal sign. The equation is based on the order of topic narration, and the unknown is involved. The unknown and the known number are linked by operation symbols, which reflects all aspects of quantitative relationship as a whole. Therefore, the method of solving problems is flexible and diverse, and it is more convenient to answer those backward narrative questions.
Typical case analysis
Example 1 Party A and Party B have two barrels of oil, with 45 kilograms of oil in Party A and 24 kilograms in Party B. How many kilograms of oil should be poured from Party A to Party B to make the weight of oil in Party A 1.5 times that of Party B?
Analysis: According to the changed "the weight of barrel A oil is 1.5 times the weight of barrel B oil", the equivalence relation can be listed:
Now the weight of oil in barrel B is 1.5 = the weight of oil in barrel A.
Suppose that X kilograms of oil is poured from barrel A to barrel B, then the oil in barrel A is now (45-x) kilograms, and the oil in barrel B is now (24+x) kilograms.
Solution: suppose that X kilograms of oil is poured from barrel A to barrel B.
(24+x) 1.5=45-x
36+ 1.5x=45-x
36+ 1.5x+x=45
36+2.5x=45
x=(45-36) 2.5
x=3.6
Answer: Only by pouring 3.6 kilograms of oil from barrel A into barrel B can the weight of oil in barrel A be five times that of barrel B. ..
Example 2 A digit has three digits, and one digit is 5. If the number of one digit is moved to the hundredth digit, the number of the original hundredth digit is moved to the tenth digit, and the number of the original tenth digit is moved to the single digit, then the new number is smaller than the original number 108. What is the original number?
Analysis: Of the original three digits, only one digit is known, and the number of digits in the hundredth digit is unknown. If the number of the original three digits is x, the original three digits can be expressed as "10x+5" and the new number can be expressed as "5 100+x".
Solution: Let the two digits of the original three digits, which are composed of hundred digits and ten digits, be X, and the equation can be obtained:
10x+5 = 5 100+x+ 108
10x-x=500+ 108-5
9x=603
x=67
10 67+5 = 675 ... original three digits
A: The original three-digit number was 675.
The primary school affiliated to a school held two math competitions. The number of people who passed the first contest was three times that of those who failed, and the number of people who passed the second contest increased by five, which was exactly six times that of those who failed. How many people took part in the competition?
Analysis: the number of participants in this question includes those who passed and those who failed, while the number of second participants is directly related to the number of first participants, and the total number remains unchanged. So let's assume that the number of people who failed in the first game is X, then the number of people who passed the first game can be expressed as "(3x+4)", the total number is (4x+4), the number of people who passed the second game is (3x+4+5), and the number of people who failed is (x-5).
Solution: Let the number of losers in the first competition be X, and the equation can be obtained according to the meaning of the question:
3x+4+5=(x-5) 6
3x+9=6x-30
3x=39
x= 13
So 4x+4 = 134+4 = 56 ... the number of participants.
A: 56 people took part in the competition.
Examples of solutions to error-prone problems
Example 1 there are 84 hectares of grain crops in ji yang village, which is more than four times of 2 hectares of cash crops. How many hectares of cash crops are there?
Error: Assume that the cash crop has X hectares.
x=(84-2)÷4
x = 82 \4
x=20.5
A: There are 20.5 hectares of cash crops.
Analysis: The formulas listed in this question are arithmetic formulas, not equations. The mistake is not knowing the difference between equation and arithmetic. Arithmetic formulas are composed of known numbers and operation symbols, which are used to represent unknowns, such as "x = (84-2) ÷ 4" in this topic; In the equation, the unknown participates in the operation, but the "X" in this question does not participate in the operation.
Correction: Assume that there are x hectares of cash crops.
4x+2=84 (or 4x=84-2)
4x=82
x=20.5
A: There are 20.5 hectares of cash crops.
A batch of coal came from the canteen. It was originally planned to burn 2 10 kg a day, which can last for 24 days. This batch of coal can burn for 28 days after improving the furnace. Q: How many kilograms are saved on average every day compared with the original plan after improving the cooker?
Error: Suppose we save X kilograms every day than planned.
28x=2 10 24
x= 180
2 10- 180=30 (kg)
A: After improving the stove, the average daily saving is 30 kilograms compared with the original plan.
Analysis: The unknown X in the problem has different meanings from the X in the equation. The moderate "X" in the title means "the average amount of coal burned every day after improving the stove" and does not mean the amount of "saving". This problem can be solved by "indirect unknown method" or "direct unknown method"
Correction: (1) set the unknown indirectly.
Solution: If the improved furnace burns X kilograms of coal every day, it will save (2 10-x) kilograms every day compared with the original scheme.
28x=2 10 24
28x=5040
x= 180
2 10-x = 2 10- 180 = 30
(2) set the unknown number directly
Solution: After setting the improved cooker, it saves X kilograms per day on average compared with the original scheme.
(2 10-x) 28=2 10 24
2 10-x= 180
x=2 10- 180
x=30
A: After improving the stove, the average daily saving is 30 kilograms compared with the original plan.
Example 3 Wang Lan has 64 pictures, and Jiang Lei gave her 12. At this time, the number of photos of Wang Lan and Jiang Lei is equal. How many original photos of Jiang Lei are there? (Equation solving)
Error: Set X original pictures of Jiang Lei.
x- 12=64
x=76
Analysis: After Jiang Lei sent 12 pictures to Wang Lan, the number of the two pictures was equal. That is to say, after Jiang Lei reduced 12 pictures and Wang Lan increased 12 pictures, the number of their pictures was the same. This solution confuses the equivalence relation and mistakenly thinks that the number of pictures in Jiang Lei is equal to the number of original pictures restored by Wang Lan 12.
Correction: X original pictures of Jiang Lei were set.
x- 12=64+ 12
x=76+ 12
x=88
A: Here are 88 original pictures of Jiang Lei.
Guidance on problem-solving skills
1. When solving application problems, column equations are often listed as arithmetic formulas and mistaken for equations. For example, there are 84 hectares of grain crops in ji yang Village of Guangshui City, which is more than four times that of cash crops. How many hectares of cash crops are there?
Solution: Let's assume that there are x hectares of cash crops.
x=(84-2) 4
x=82 4
x=20.5
A: There are 20.5 hectares of cash crops.
"x=(84-2) 4" in this problem is an arithmetic formula. The reason for the above errors is that the difference between equation and arithmetic is not clear. Arithmetic expressions are composed of known numbers and operation symbols, which are used to represent unknowns. In the equation, the unknown quantity participates in the operation. The equation of this problem should be listed as:
4x+2=84 or 4x=84-2 or 84-4x=2.
2. According to the meaning of the question, set the unknown appropriately. For example, a batch of coal was transported from the first staff canteen. It was originally planned to burn 2 10 kg of coal every day for 24 days. After improving the furnace, this batch of coal can last for 28 days. Q: How many kilograms are saved on average every day compared with the original plan after improving the cooker?
There are generally two ways to set the unknown number: one is to set the unknown number to X directly and set the question to X; The other is to indirectly set the unknown quantity as x, and then through the relationship between this quantity and the problem to be solved, the unknown quantity needed in the application problem can be obtained.
If the unknown is directly set to x, then the equation listed in this question should be:
Solution: Suppose that X kilograms of coal is saved every day than originally planned.
(2 10-x) 28=2 10 24
2 10-x= 180
x=2 10- 180
x=30
If the unknown x is indirectly set:
Solution: If the improved furnace burns X kilograms of coal every day, it will save (2 10-x) kilograms every day compared with the original scheme.
28x=2 10 24
x= 180
2 10- 180=30 (kg)
A: It saves 30 kilograms per day than originally planned.
Old and immortal; References:
According to network collection