How to solve chickens and rabbits in the same cage, ten solutions to chickens and rabbits in the sam

If you learn to solve a one-dimensional equation, you will have no problem (one is an unknown, once is a square, this kind of problem must be one-time).

Usually this kind of question will give two conditions, one about the feet and one about the head, for example.

Relatively simple, one yuan once solved) chickens and rabbits in the same cage, there are 20 heads, 10 more chicken feet than rabbit feet, how many chickens and rabbits are there?

If there are x chickens, then there are (20-x) rabbits, then according to the title:

2x-4(20-x) = 10 (2x is the number of chickens, 4(20-x) is the number of rabbits, and the difference between the two is 10).

Solution: x=15

So (20-x) = 5

A: 15 chickens and 5 rabbits!

There is also a kind that does not directly tell you how many heads there are in total, but only tells you who has more than whom.

For example, chickens and rabbits in the same cage, chickens have 10 more heads than rabbits, and 10 more chicken feet than rabbit feet.

The number of chickens is x

The rabbit is X-10

So: 2x-4*(x-10)=10

x=15, so (x-10)=5

A: 15 chickens and 5 rabbits!

Chickens and rabbits in the same cage, this question is one of the famous interesting questions in ancient China. About 1,500 years ago, this interesting question was recorded in the "Sun Tzu's Sutra". Here's how it is narrated in the book:

Today there are pheasants and rabbits in the same cage, there are thirty-five heads on the top, and there are ninety-four feet under it. The meaning of these four sentences is: There are several chickens and rabbits in the same cage, and counting from above, there are 35 heads; Counting from below, there are 94 feet.

How many chickens and rabbits are in the cage?

Let the number of chickens be x and the number of rabbits be y; Then the number of chicken feet is 2x the number of rabbit feet is 4y, and the formula is: xy=35

2x4y=94

x=35—y

Then 2(35—y).

4y=94 gives y=12

x=23, then there are 12 chickens and 23 rabbits.

The ten solutions to the same cage of chickens and rabbits are as follows:

Solution 1: List method.

1) List method one by one: It is to enumerate chickens and rabbits from 1 to 35.

Then calculate the number of legs, equal to 94, you can find the answer, but when the amount of data is large, it will be more cumbersome.

2) Jump list method: When enumerating, according to the value of the number of legs, jump the enumeration and simplify the number of enumerations.

3) Neutralization list method: first try to have the same or close number of chickens and rabbits, and then adjust according to the number of feet.

Although the above three list methods can be used to find the results, they are too cumbersome and we generally do not use them when solving problems.

Solution 2: Hypothetical method.

1) Suppose the cage is full of chickens.

Total number of feet: 35 2 = 70 (only).

Total difference: 94-70=24 (only).

Unit difference: 4-2=2 (only).

Rabbits: 24 2 = 12 (only).

Chickens: 35-12 = 23 (only).

A: There are 23 chickens and 12 rabbits.

2) Suppose it's all rabbits.

Total number of feet: 35 4 = 140 (only).

Total difference: 140-94=46 (only).

Unit difference: 4-2=2 (only).

Chickens: 46 2 = 23 (pcs).

Rabbits: 35-23 = 12 (only).

A: There are 23 chickens and 12 rabbits.

The above two hypothetical methods are used to solve the chickens and rabbits in the same cage problem in the lower grades.

often used methods.

Solution 3: Golden Rooster Independence Law.

1) Suppose the chicken lifts one leg and the rabbit lifts two.

Total number of feet on the ground: 94 2 = 47 (only).

Each additional rabbit has 1 more feet than heads

Rabbits: 47-35 = 12 (only).

Chickens: 35-12 = 23 (only).

A: There are 23 chickens and 12 rabbits.

2) Suppose both the chicken and the rabbit raise both legs.

Total number of feet on the ground: 94-2 35=24 (only).

The feet of the ground are all rabbits.

Rabbits: 24 2 = 12 (only).

Chickens: 35-12 = 23 (only).

A: There are 23 chickens and 12 rabbits.

3) Suppose the rabbit only lifts two feet.

At this time, each chicken and rabbit on the ground has 2 feet on the ground.

Total number of feet on the ground: 2 35 = 70 (only).

Total number of rabbits lifting their feet: 94-70 = 24 (only).

Rabbits: 24 2 = 12 (only).

Chickens: 35-12 = 23 (only).

A: There are 23 chickens and 12 rabbits.

Solution 4: Equation method.

1) If there are x chickens, then there are (35-x) rabbits.

According to the title: 2x+4 (35-x)=94

x=23 35-x=35-23=12

A: There are 23 chickens and 12 rabbits.

2) If there are x rabbits, then there are (35-x) chickens.

According to the title: 4x+2 (35-x)=94

x=12 35-x=35-12=23

A: There are 23 chickens and 12 rabbits.

There are 13 solutions to the chickens and rabbits in the same cage, which are:

1. List method.

2. Drawing method.

3. Golden Rooster Independence Law.

4. Whistle blowing method.

5. Hypothetical method.

6. Hypothetical method.

7. Specific function method.

8. Specific function method.

9. Specific function method.

10. Foot cutting method.

11. Rabbit tricks.

12. Equation method.

13. Equation method.

Chickens and rabbits in the same cage: Ming Pei.

Meaning: This is a classical arithmetic problem. It is known how many heads and feet there are in the cage for chickens and rabbits, and the problem of how many chickens and rabbits are each is called the first chickens and rabbits in the same cage.

It is known that the total number of chickens and rabbits and the difference between chicken feet and rabbit feet, and the problem of finding how many chickens and rabbits are called the second chicken and rabbit cage problem.

Quantitative relations. The first chickens and rabbits in the same cage problem:

Assuming that all of them are chickens, then there are rabbits (actual number of feet 2 total number of chickens and rabbits) (4 2) Assuming that all of them are rabbits, there are chickens (4 total number of chickens and rabbits and actual number of feet in the world) (4 2) The second chickens and rabbits are in the same cage problem:

Assuming that all are chickens, then there are rabbits (2 total number of chickens and rabbits The difference between chickens and rabbits' feet) (4 2) Assuming all rabbits, there are chickens (4 total number of chickens and rabbits, the difference between chickens and rabbits' feet) (4 2) <>

Ideas and methods for solving problems:

Hypothetical solutions to such problems are generally used, and you can first assume that they are all chickens or rabbits. If you assume that they are all chickens, and then exchange rabbits for chickens; If you assume that they are all rabbits, then exchange chickens for rabbits. This kind of problem is also called a displacement problem.

The problem is solved by assuming first and then substituting.

The solution of chickens and rabbits in the same cage: list method, hypothetical method, leg lifting method, and foot cutting method.

Question: There is a cage with a number of chickens and rabbits, count, there are 14 heads and 38 legs, how many chickens and rabbits are there?

1. List method.

The benefits of this approach are that it's simple, intuitive, and error-prone.

As can be seen from the table, there are 9 chickens and 5 rabbits.

2. Hypothetical method.

Suppose all 14 are chickens, 14 2 = 28, difference 38-28 = 10.

And each chicken makes up 2 thick legs to become a rabbit, and you need to make 2 legs for each of the 5 chickens.

So there are 5 rabbits, 14-5=9 chickens.

3. Leg lift method.

Have each chicken stand on one foot and each rabbit on two hind feet.

Then the total number of feet on the ground is only half of the original number, i.e. 19 feet.

The number of feet of a chicken is the same as the number of heads, and the number of feet of a rabbit is twice the number of heads of a rabbit, so subtracting the number of heads from 19 to 14 is what is left of the rabbit's coarse sock head 19 14 = 5, and the chicken has 14 5 = 9.

4. Town stool town foot cutting method.

If each one is cut off one foot and each rabbit is cut off two feet, each chicken becomes a "one-horned chicken" and each rabbit becomes a "two-legged rabbit".

In this way, the total number of chickens and rabbits has changed from 38 to 19.

If there is a rabbit in the cage, the total number of feet is 1 more than the total number of heads.

Therefore, the difference between the total number of feet 19 and the total number of heads 14 is the number of rabbits, i.e. 19 14 5 (only).

So, the number of chickens is 14 5 9 (only).

The five solutions of chickens and rabbits in the same cage include the list method, the hypothesis method, the equation method, the foot lifting method, and the foot cutting method.

First, this method is based on the fact that there are eight heads in a single situation, and then list nine different situations to calculate how many legs correspond to each situation, and then find the correct answer. The advantage of this method is that it can find out all the situations through a list, but the disadvantage is that if the number is relatively large, it is not suitable to use the list method.

The second method is to assume that it is all chickens or all rabbits. Because a chicken has two legs and a rabbit has four legs, so if all chickens are assumed, then the total number of legs will be less than the actual one, and the missing part is exactly the rabbit's legs, because a rabbit is missing two legs, so you can find the prime number of the rabbit, and then find the number of chickens.

Assuming that all rabbits are used, the same can be used to find the number of rabbits and chickens.

The third type: the equation method. You can assume that there are x chickens, then the rabbits are 35-x, and then the equation for x is based on the number of legs. In the same way, it can be assumed that there are x rabbits without first noticing.

Fourth: leg raises. The first time an animal lifts one foot, so that it lifts 35 feet, and there are 59 feet left, and the second time it continues to lift another foot, so that there are 24 feet left, so that the rest is the rabbit's foot, and then find the number of rabbits, and finally find the number of chickens.

Five: Foot chopping. I opened each house with two feet, so that 94 feet could cut 47 of them, and then 12 more than 35, which would be the number of rabbits.

Chickens and rabbits in the same cage to solve the problem of missing a good way:

Enumeration (list):

The method is very simple and the process is very complicated, that is, according to the constantly changing number of chickens and rabbits, respectively, the number of chickens and rabbit legs is filled in the **, until you know to find the correct answer, this method is only suitable for exploration in classroom teaching and the guidance of other methods, because this method is too clumsy, it takes more time, and it is generally not applicable in daily practice and exams. So this method can be understood by everyone.

Hypothetical method (contradictory method):

One of the main solutions to the problem of "chickens and rabbits in the same cage" is to make some assumptions about the problem according to the known conditions in the problem, and then reason according to the conditions, find the contradiction with the number of questions, and finally carry out the change of the joint confession to draw the correct conclusion. At the same time, the hypothetical method is also a method often encountered in Olympiad problems, and the key to this method is to find the contradiction with the quantity in the problem through hypothesis.

Let's start with the title: there are several chickens and rabbits in the same cage, counting from above, there are 35 heads; Counting from below, there are 94 feet. How many chickens and rabbits are in the cage?

Thought process: Suppose the 35 in the cage are all rabbits, then the total number of feet should be: 35 4 = 140 (only), but the actual cage only has 94 feet, which contradicts our hypothesis, there are 140-94 = 46 more feet, why is there 46 more feet?

Because the cage is not full of rabbits and chickens, we assume that a two-legged chicken is a rabbit (in reality, a rabbit has two more legs than a chicken), because of our assumption, there are 46 more legs, there is 1 chicken with 2 more legs, and there are as many chickens as there are 2s in the 46 legs, so we use 46 2 = 23 (only) to find the number of chickens, and then use 35-23 = 12 (only) to get the number of rabbits.

Our total settlement formula: number of chickens = (35 4-94) (4-2) = 23 (only).

Number of rabbits = 35-23 = 12 (only).

Inductive formula: If you assume that all rabbits are: (total number of heads Number of rabbit feet - Total number of feet) (Number of rabbit feet - Number of chicken's feet).

Of course, we can also assume that the cage is full of chickens, if it is all chickens, the total number of feet is 35 2 = 70 (only) feet, and the actual number of less 94-70 = 24 (single) feet, because socks search for chickens have always had two feet less than a rabbit, for every two feet less there is a rabbit, and 24 feet less are: 24 2 = 12 (only) rabbits, calculate the number of rabbits, the number of chickens is: 35-12 = 23 (only).

List the equation: number of rabbits = (94-35 2) (4-2) = 12 (only).

Number of chickens = 35-12 = 23 (only).

Inductive formula: If you assume that all chickens are: (total number of feet - total number of heads number of chicken feet) (number of rabbit feet - number of chicken feet).