Chickens and rabbits in the same cage to solve the problem, respectively using equations, lists, hyp

Example 1 There are several chickens and rabbits in a cage with a total of 50 heads and 140 legs.

Solution 1 Hypothetical method.

Assuming that an unknown number is known, for example, if 50 heads are all rabbits, then there are a total of feet (4 50 =) 200 (only), which is inconsistent with the known 140 in the question, more (200-140 =) 60 (only), the reason for more is that each chicken counts 2 more legs after the chicken is a rabbit, so the number of chickens is (60 2=) 30 (only), then the number of rabbits is (50-30) 20 (only).

This solution has a clear idea, but it is more complex and inconvenient to operate. Can you draw a picture vividly? Let's try.

Solution 2: Graphical method.

From the figure, the area of ACDF is 4 50 200 (feet), which is more than the actual area of GHEF 200-140 60 (feet), ab=gh=60 2=30 (chickens), bc=ac-ab=50-30 20 (rabbits).

Solution 2 is more advanced than Solution 1, and the arithmetic is the same. The answer here is calculated in the diagram, and obviously both solutions require paper and pen. Without paper and pen, it is sure to use mantras or easy-to-remember formulas, which are heirlooms of my husband.

Solution 3 Formula method.

My husband said: Just blow with a whistle and shout a command: "All stand solemnly".

At this time, each chicken is in the form of a golden rooster independently, each rabbit is in the shape of a jade rabbit worshipping the moon, the sum of the number of feet on the ground has (140 2) 70 (only), wherein the number of heads and feet of the chicken is equal, because the feet of each rabbit are more than the number of heads 1, so the number of heads of the rabbit is (70 50) 20 (pieces), that is, the rabbit has 20, then the chicken has (50 20) 30 (only). In this story, my husband actually used the following formula.

The number of feet and the number of 2-heads = the number of rabbits.

The little grandchildren were greatly interested in hearing this, and they asked their husbands to ask a few more questions. The old man is out again.

1) 30 heads, 80 feet ....... Rabbit 10, chicken 20).

2) 100 feet, 40 heads ....... Rabbit 10, chicken 30).

3) 80 heads, 200 feet ....... Rabbit 20, chicken 60).

The little grandchildren all answered happily.

This formula is simple and easy to use, is it passed down from the ancestors or did the husband come up with it? Our Chinese culture is vast and profound, and both possibilities exist. Does this formula happen to be right or is it arithmetic?

This is very important. The mathematician Gauss once said: "Many methods and theorems in mathematics are discovered by induction, and proof is only a procedure for making up the line."

Now let's make up for it.

2 chicken heads = chicken feet.

4 Rabbit head = rabbit foot.

Delet: rabbit feet + chicken feet = 2 chicken heads + 4 rabbit heads.

2 (Chicken Head + 2 Rabbit Head).

Question: Chickens and rabbits have a total of 20 heads and 60 legs.

Equation: Let the chicken x, the rabbit y x+y=20 2x+4y=60 get x=10 y=10

List method: chicken: 0 3 5 7 9 10

Rabbit: 20 17 15 13 11 10

Feet: 80 74 70 66 62 60

Hypothesis: Suppose all rabbits have 4 * 20 = 80 legs, but there are actually 60 more than 80-60 = 20 legs, that's because the chickens are also rabbits, so each chicken has 2 more legs, so there are 20 2 = 10 chickens, so there are 20-10 = 10 rabbits.

The column equation now has x chickens or rabbits, and the rabbits or chickens are the total number of heads -xThen use the number of legs of the chicken + the number of legs of the rabbit = the total number of legs.

The list is all listed.

The assumption is that it is assumed that it is all rabbits, and the number of chickens is calculated. Assuming that they are all rabbits, there will be more legs, and the number of chickens will be obtained by dividing the extra by 2 (the difference between rabbits and chickens).

1.It can be said that the hypothetical method:

For example, if you assume that all rabbits are there, it will be less than the actual number of legs.

2.It can also be said that the substitution method:

For example, replace a chicken with a rabbit and add two legs ......

There are 50 chickens and rabbits in the same cage, 102 legs, how many chickens and rabbits are there?

Assuming all chickens, 50 chickens share legs:

2 50 = 100 entries.

Then less than the total number of legs:

102-100 = 2 articles.

A rabbit has two more legs than a chicken, so the extra two legs are a rabbit.

Therefore: a rabbit.

Chickens 50-1 = 49.

1.Suppose it's all chickens or all rabbits2With a quadratic equation.

3.If there are 35 heads and 100 feet, it can be seen that if all the rabbits are rabbits, the rabbits have 4 legs, multiply 35 by the rabbit's 4 feet, there are 140 feet, use 140 to subtract the actual 100 feet, 40 more feet, divide 40 by 2, calculate the number of rabbits to supply chickens, there are 15 rabbits, and 20 chickens.

Chicken and Rabbit Problem Formula].

1) Know the total number of heads and the total number of feet, find the number of chickens and rabbits

Total number of feet - number of feet per chicken Total number of heads) (number of feet per rabbit - number of feet per chicken) = number of rabbits;

Total number of heads - number of rabbits = number of chickens.

or (number of feet per rabbit total number of heads - total number of feet) (number of feet per rabbit - number of feet per chicken) = number of chickens;

Total number of heads - number of chickens = number of rabbits.

Hypothetical method. 1. Suppose it is all 5 yuan 5x100 = 500 yuan.

800-500=300 yuan.

300 (The formula for 10-5 is:.)

Number of rabbits = (actual number of feet - number of feet per chicken Total number of chickens and rabbits).

The number of feet per one.

The little red lips have two yuan and five yuan, a total of two small red lips, two yuan and five yuan, a total of 25 pieces, a total of 80 yuan. How many pieces of each of these two renminbi are in Xiaohong's deposit.

It can be solved with equations and assumptions.

Generally, chickens and rabbits have the total number of legs and the total number of birds in the same cage, and the unary equation generally sets the number of one of the oak banquet Liangxi animals as x, and the number of the other animal as (total number of animals - x). In this way, multiplying the number of legs set by one animal, and multiplying the number of legs by the number of legs of another animal, equals the total number of legs. For example:

A total of 100 legs, 40 animals, among which there are chickens and rabbits.

Set x chickens, (40-x) rabbits.

2x+4（40-x）=100

2x+160-4x=100

2x+160=100+4x

2x+60=4x

60=2xx=30

40-x=40-30=10

So a total of 30 chickens, 10 rabbits.

Note that when 2x+160-4x, move the item and move 160 or 4x to the right.

Hypothesis method: Ideas: Assuming that all of them are rabbits, in this example, there are a total of 4*40=160 legs, but there are 60 more legs than the actual one, because some chickens are counted as rabbit legs.

Whereas, rabbits have 2 more legs per than chickens. There are a total of 60 more legs, each with 2 more legs, so there are a total of 60 2 = 30 chickens, 40-30 = 10 chickens. The equation is:

Chicken: (40*4-100) (4-2)=60 2=30 (only).

Rabbits: 40-30 = 10 (only).

Note: The number of legs of the "chicken" and "rabbit" can be changed, for example, to become a tricycle and a car, but the number can be calculated by replacing the number with the number of the equation above.

You should be a primary school or junior high school student, just teach you the unary equation and the hypothetical method, read the translated version of "Sun Tzu's Sutra", and you will understand it all.

Summary. For example, there are several chickens and rabbits in the same cage, counting from above, there are 35 heads, and counting from the bottom, there are 94 legs. Q: How many chickens and rabbits are in each cage?

Let the total number of chickens be x heads, and the total number of rabbits be (35-x) heads, according to the equivalence relationship of the number of feet, the equation 2x+4(35-x)=94 can be listed, if the system of equations can be set to have x rabbits and y chickens, two equations of x+y=35 and 4x+2y=94 can be obtained, and the system of equations can be solved simultaneously.

The chickens and rabbits in the same cage assumption method and the equation method assume that all rabbits are rabbits, (the total number of heads of each rabbit of the number of feet - the total number of feet) (the number of feet of each rabbit - the number of feet of each chicken) = the number of chickens assumes that all are chickens, (the total number of feet of the stool - the total number of heads of each rabbit the number of feet of each rabbit) (the number of feet per rabbit - the number of feet of each chicken) = the number of rabbits.

Column equation method, Gaozhou can be listed as a one-dimensional liquid sail one-time equation, and can also be listed as a two-element one-dimensional equation. The chickens and rabbits in the same cage problem contain two equal relations of hail: (1) the total number of chicken feet + the total number of rabbit feet = the total number of feet, (2) the total number of chickens + the total number of rabbits = the total number of heads.

For example, Sun Min has several chickens and rabbits in the same cage, counting from above, there are 35 heads, and counting from below, there are 94 legs. Q: How many chickens and rabbits are in the cage? Let the total number of chickens be x, the total number of rabbits is (35-x) heads, according to the equal relationship of the number of feet, the equation 2x+4(35-x)=94 can be listed, if the system of equations can be set to have x rabbits and y, two equations of x+y=35 and 4x+2y=94 can be obtained, and the system of equations can be solved by intermodal transportation.

The solution of the equation for chickens and rabbits in the same cage is as follows:

1. Equation method 1: unary linear equation.

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

Column equation: 4x+2(35-x)=94.

Solution equation: 4x+2*35-2x=94;2x+70=94；2x=94-70；2x=24；Solution: x=12.

Then there are: 35 - 12 = 23 chickens.

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

Column equation: 2x+4(35-x)=94.

Solve the equation: 2x+4*35-4x=94;140-2x=94；2x=140-94；2x=46；Solution: x=23.

Then rabbits have: 35 - 23 = 12 (only).

A: There are 12 rabbits and 23 chickens.

2. Equation method 2: a system of binary linear equations.

Solution: There are x chickens and y rabbits.

Column equations: x+y=35;2x+4y=94。

Solution: x=12. y=23。

A: There are 12 rabbits and 23 chickens.

1. Suppose all chickens: 2 35 = 70 (pieces); Chicken feet are less than the total number of feet: 94 - 70 = 24 (only).

The number of feet that rabbits have more than chickens: 4 - 2 = 2 (only); Number of rabbits: 24 2 = 12 (only); Number of chickens:

35 - 12 = 23 (only).

2. Suppose all rabbits: 4 35 = 140 (only); More rabbit feet than the total: 140 - 94 = 46 (only).

The number of feet that rabbits have more than chickens: 4 - 2 = 2 (only); Number of chickens: 46 2 = 23 (only); The rabbit cleverly called only the number of filial piety:

35 - 23 = 12 (only).

The solution to the problem is as follows:

Example: The chicken and rabbit are in the same cage is one of the famous mathematical problems 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, counting from above, there are 35 heads, and counting from below, there are 94 legs. Q: How many chickens and rabbits are in each cage?

Here's a simpler calculation:

Total number of feet - total number of heads Number of chicken's feet) (Number of rabbit's feet - Number of chicken's feet) = Number of rabbits.

94-35 2) 2 = 12 (number of rabbits) Total number of heads (35) - number of rabbit barrages (12) = number of chickens (23).

Explanation: Let the rabbit and chicken lift both feet at the same time, so that the total number of feet in the cage is reduced by 2, since the chicken only has 2 feet, so there are only two feet left in the cage of the rabbit, and then 2 is the number of rabbits.

Solution 1. Total number of feet: 2 - total number of heads = number of rabbits.

Total number of rabbits = number of chickens.

Solution 2: The number of rabbit's feet x the total number of feet - the total number of feet) (the number of rabbit's feet and the chicken's feet) = the number of chickens.

Total number of chickens = number of free chickens.

Chickens are exempt from the same cage equation to solve the problem.

If there are chickens, then there are no sakura (total -x) because each rabbit has 4 legs and each chicken has 2 legs. Therefore, there are 2x chicken feet and 4 free feet (total-x). So we can get the equation: 2x + 4 (total - x) = total full number.

Chickens are free of cages is one of the famous mathematical problems in ancient China. About 1,500 years ago, this interesting question was recorded in the Sutra of Sun Tzu. Here's how it is narrated in the book:

There are a number of late chickens in the same cage, counting from above, with 35 heads, and counting from below, with 94 legs.

Q: How many chickens are in each cage?

The simplest algorithm for chickens to be free of the same cage: (total number of feet - total number of heads x number of chickens) two (number of rabbit feet - number of chickens) the number of two free animals, that is, (94-35x2) code surplus search - 2 = 12 (number of free children). Total.

Number of heads (35) - number of children (12) = number of chickens (23).