In ancient China, Sunzi Suanjing has three volumes, written in the 5th century. This book is easy to understand and has many interesting arithmetic problems, such as "the chicken and the rabbit are in the same cage":

Today, there are pheasant rabbits in the same cage, with 35 heads above and 94 feet below. Pheasant rabbit geometry?

There are 35 pheasant rabbits in the title. If the rabbit's two front legs are tied with rope as one foot and the two rear legs are tied with rope as one foot, then the rabbit becomes two feet, that is, the rabbit is regarded as a chicken with two feet first. The total number of feet of chickens and rabbits is 35×2=70 (only one), which is only 94-70=24 (only one) less than the 94 mentioned in the question.

Loosen the rope on a rabbit's foot, and the total number of feet will increase by 2, that is, 70+2=72 (one). Then loosen the rope on a rabbit's foot, and the total number of feet will increase by 2, 2, 2, 2 ... and continue until it increases by 24, so the number of rabbits is 24÷2= 12 (one).

Let's sum up the idea of solving this problem: suppose all chickens, and according to the total number of chickens and rabbits, we can calculate how many feet there are under the hypothesis. Compare the number of feet obtained by this method with the number given in the question to see how much difference there is. A difference of every two feet means 1 rabbit. Divide the foot difference by two and you can calculate how many rabbits there are. To sum up, the basic formula to solve the problem of chickens and rabbits in the same cage is: number of rabbits = (actual number of feet-number of feet per chicken × total number of chickens and rabbits) ÷ (number of feet per rabbit-number of feet per chicken). Similarly, it can be assumed that all rabbits.

think

"Chickens and rabbits in the same cage" is a famous arithmetic problem in ancient China. It first appeared in Sun Tzu's Calculations. Many elementary school arithmetic application problems can be transformed into such problems, or solved by a typical solution-"hypothesis method". So we should learn its solutions and ideas.

1 How many chickens and rabbits are there? They have 88 heads and 244 feet. How many chickens and rabbits are there respectively?

Solution: We imagine that every chicken is "golden rooster independent" and stands on one foot; And each rabbit uses two hind legs and stands on two feet like a human, and half of the total number of feet appears on the ground, that is,

244÷2= 122 (only)

In the number 122, the chicken counts once and the rabbit counts twice. So subtract 88 from 122, and the rest is the number of rabbits.

122-88=34 (only),

There are 34 rabbits and, of course, 54 chickens.

A: There are 34 rabbits and 54 chickens.

The above calculation can be summarized as the following formula:

Total number of feet ÷2- total number of heads = number of rabbits. Total number of rabbits = number of chickens.

The above solution is recorded in Sun Tzu's Art of War. Do a division and a subtraction, and the number of rabbits can be found immediately. How simple it is! This calculation can mainly make use of the fact that the number of feet of rabbits and chickens is 4 and 2 respectively, and 4 is twice that of 2. When other problems are transformed into such problems, the "number of feet" is not necessarily 4 and 2, and the above calculation method will not work. Therefore, we give a general solution to this kind of problem.

Also say the example 1.

If you imagine that 88 rabbits are all rabbits, it is 4×88 feet, which is more than 244 feet.

88×4-244= 108 (only).

Each chicken has (4-2) fewer feet than a rabbit, so * * * has chickens.

(88×4-244)÷(4-2)= 54 (only).

It shows that among the 88 "rabbits" we imagined, 54 are not rabbits. But chickens. So you can list the formulas.

Number of chickens = (number of rabbit feet × total number of heads-total number of feet) ÷ (number of rabbit feet-number of chicken feet).

Of course, we can also imagine that 88 is a "chicken", so * * * has feet 2×88= 176 (only), less than 244 feet.

244- 176=68 (only).

Each chicken has 4-2 feet less than each rabbit.

68÷2=34 (only).

Explain that 34 of the imaginary "chicken" is only a rabbit, and you can also list formulas.

Number of rabbits = (total number of feet-chicken feet × total number of heads) ÷ (number of rabbit feet-chicken feet).

You don't have to use the above two formulas at the same time. Use one to calculate the number of rabbits or chickens, and then subtract the total to know the other number.

Assuming that they are all chickens or rabbits, this is generally the solution. Some people call it the "hypothesis method".

Try the above formula with a specific problem.

Example 2 Each red pencil is 0. 19 yuan, each blue pencil is 0. 1 1 yuan, and each pencil is 2.80 yuan. How many red pencils and blue pencils do you want?

Solution: take "minutes" as the monetary unit. Let's assume that a "chicken" has 1 1 foot and a "rabbit" has 19 foot. They * * * have 16 heads, 280 feet.

Now the problem of buying pencils has turned into a problem of "chickens and rabbits in the same cage". Using the above formula to calculate the number of rabbits, there are

Number of blue pens = (19×16-280) ÷ (19-1)

=24÷8

=3 (branch).

Red pen number = 16-3= 13 (branch).

I bought 13 red pencils and 3 blue pencils.

For the calculation of this kind of problem, we can often make use of the particularity of the known number of feet. In Example 2, the sum of Feet 19 and 1 1 is 30. We can also assume that in 16, 8 is only "rabbit" and 8 is only "chicken". According to this assumption, the number of feet is

8× (11+19) = 240 (branch).

Forty is less than 280.

40 (19-11) = 5 (branch).

We know that there should be five chickens out of eight, that is, the number of chickens (blue pencils) is three.

30×8 is easier to calculate than 19× 16 or1×16. Using the particularity of the known number, the calculation is completed by mental arithmetic.

In fact, you can imagine a convenient number of rabbits or chickens at will. For example, if the number of rabbits is 16 and the number of chickens is 6, there are feet.

19× 10+ 1 1×6=256.

24 is less than 280.

24÷( 19- 1 1)=3,

You know, imagine six chickens, and you need three less.

Making imaginary numbers easy to calculate often depends on your mental arithmetic ability.

example

For a manuscript, Party A will type for 6 hours, Party B will type for 10 hours, and after a few hours, Party B will type for 7 hours. How many hours did it take to type?

Solution: We divide this manuscript into 30 copies on average (30 is the least common multiple of 6 and 10), a typing 30÷6= 5 copies per hour, and b typing 30÷ 10= 3 copies per hour.

Now, if the typing time of A is regarded as the number of "rabbits" and the typing time of B is regarded as the number of "chickens", then the total number of heads is 7. The number of feet of the rabbit is 5, the number of feet of the chicken is 3, and the total number of feet is 30, so the problem becomes the problem of "the chicken and the rabbit are in the same cage".

According to the previous formula

"Rabbit" number =(30-3×7)÷(5-3)

=4.5,

Number of chickens =7-4.5

=2.5

That is, it took 4.5 hours for A to type and 2.5 hours for B to type.

A: It took 4 hours and 30 minutes to type.

In case 4 1998, the sum of parents' ages (integer) is 78 years old, and the sum of brothers' ages is 17 years old. Four years later (2002), my father was four times as old as my brother, and my mother was three times as old as my brother. So when the father is three times as old as his brother, what year is it?

Solution: After 4 years, the total age of the two will be added by 8. At this time, the total age of brothers is 17+8=25, and the total age of parents is 78+8=86. We can regard the age of our brother as the number of chickens and the age of our brother as the number of rabbits. 25 is the "total number of heads" and 86 is the "total number of feet". According to the formula, my brother's age is

(25×4-86)÷(4-3)= 14 (years old).

1998, brother's age is?

14-4= 10 (years old).

Father's age is

(25- 14)×4+4=40 (year).

Therefore, when the father's age is three times the brother's age, the brother's age is

(40-10) ÷ (3-1) =15 (years old).

This is 2003.

A: In 2003, my father was three times older than my brother.

Spiders have 8 legs, dragonflies have 6 legs and 2 pairs of wings, and cicadas have 6 legs 1 pairs of wings. These three kinds of bugs are *** 18, with 1 18 legs and 20 pairs of wings. How many bugs are there in each species?

Solution: Because both dragonflies and cicadas have six legs, they can be divided into "eight legs" and "six legs" according to the number of legs. Using this formula, you can calculate eight legs.

Number of spiders = (118-6×18) ÷ (8-6)

=5 (only).

So you know that a six-legged bug * * *

18-5= 13 (only).

In other words, there are 13 dragonflies and cicadas, and they have 20 pairs of wings. Use this formula again.

Cicada number = (13× 2-20) ÷ (2-1) = 6 (only).

So the number of dragonflies is 13-6=7 (only).

There are five spiders, seven dragonflies and six cicadas.

Example 6 In a math exam, 52 students in the class took part in 5 questions, and * * got it right 18 1 question. As we all know, everyone answered at least 1 question correctly, seven people answered 1 question correctly, six people answered five questions correctly, and the same number of people answered two and three questions correctly. So how many people answered four questions correctly?

Solution: Some people have two, three or four problems.

52-7-6=39 (person).

They did the right thing.

181-1× 7-5× 6 =144 (road).

Because there are as many people who answered question 2 as question 3, we can regard them as those who answered question 2.5 ((2+3)÷2=2.5).

Number of rabbit feet =4, number of chicken feet =2.5,

Total number of feet = 144, total number of heads =39.

For these four questions, there are

(144-2.5× 39) ÷ (4-2.5) = 31(person).

A: 365,438+0 people answered four questions correctly.

Take 1 as an example. There are some chickens and rabbits. They have 88 heads and 244 feet. How many chickens and rabbits are there?

If we use the simple X equation to calculate, we usually use a large number as X, that is, a rabbit as X, and then the number of chickens is the total minus the number of chickens, that is, (88-X).

Solution: Let the rabbit be X, then the number of chickens is (88-X).

4X+2×(88-X)=244

The above equation is interpreted as: the number of feet of rabbits plus the number of feet of chickens is the number of feet owned by * * *. 4X is the number of feet of rabbits and 2x (88-x) is the number of feet of chickens.

4X+2×88-2X=244

2X+ 176=244

2X+ 176- 176 = 244- 176

2X=68

2x \2 = 68 \2

X=34

That is, there are 34 rabbits, and the total number is 88, so the number of chickens is 88-34=54.

A: There are 34 rabbits and 54 chickens.

Formula 1: (rabbit foot number × total foot number-total foot number) ÷ (rabbit foot number-chicken foot number) = chicken foot number.

Total number-number of chickens = number of rabbits

Formula 2: (total number of feet-chicken feet × total number of feet) ÷ (rabbit feet-chicken feet) = number of rabbits.

Total-number of rabbits = number of chickens

Formula 3: total number of feet ÷ 2-total number of heads = number of rabbits.

Total number of rabbits = number of chickens.

Formula 4: Number of chickens =(4× total number of chickens and rabbits-total number of chickens and rabbits) ÷2 Number of rabbits = total number of chickens and rabbits-number of chickens.

Formula 5: Total number of rabbits = (total number of feet of chickens and rabbits -2× total number of chickens and rabbits) ÷2 Number of chickens = total number of chickens and rabbits-total number of rabbits.

Equation 6: 4×+2 (total number -x) = total number of feet (x= rabbits, total number -x = chickens, used in the equation).

Hypothetical method

Suppose all chickens: 2×35=70 (only)

Chicken feet less than total feet: 94-70 = 24 (only)

Rabbits have more feet than chickens: 4-2=2 (only)

Number of rabbits: 24÷2= 12 (only)

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

Equation method

One-dimensional linear equation

Solution: For every X rabbits, there are (35-x) chickens.

solve

Chicken: 35- 12=23 (only)

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

solve

Rabbit: 35-23= 12 (only)

A: Rabbits 12, 23 chickens.

Note: Usually, when setting equations, animals with only a few legs are selected, which will be applied to other similar problems of chickens and rabbits in the same cage, so it is easier to calculate.

Binary linear equations

Solution: suppose there are x chickens and y rabbits.

solve

A: Rabbits 12, 23 chickens.

Leg lifting method

Method one

If the chicken raises one foot and the rabbit raises two feet, there are 94÷2=47 feet. There are more rabbits in cages than chickens 1 foot. At this time, the difference between the total number of feet and heads is 47-35= 12, which is the number of rabbits.

Method 2

If both chickens and rabbits lift their feet, there are still 94-35× 2 = 24 feet left. At this time, the chicken is sitting on the ground, only the rabbit's feet are on the ground, and each rabbit has two feet on the ground, so there are 24÷2= 12 rabbits and 35- 12 = 23 chickens.

Method 3

We can let the rabbit lift two feet first, so there are 35×2=70 feet, and the number of feet is 94-70=24 feet. These are two feet of each rabbit, one * * *, 24 feet, 24-day rabbit 12, 35- 12.