Ancient math problem What is the original problem of chickens and rabbits in the same cage?

Chickens and rabbits in the same cage. Ancient Chinese people often used some small bamboo sticks to be arranged in different forms to represent different numbers, and to make various calculations, which is called "calculation".In the "Sun Tzu Sutra".

There are many examples of the use of the "calculation" method in the book. "Sun Tzu's Sutra" has a total of three volumes, and the book is written in Zu Chongzhi.

Before, it was about the 5th century AD. The content mainly talks about the use of mathematics, which is easy to understand, and there are many interesting mathematical problems, such as the "chicken and rabbit in the same cage" problem, which is actually a problem related to equations.

The original question was: "Today there are chickens and rabbits in the same cage, there are thirty-five heads on the top, and there are ninety-four feet on the bottom. In modern parlance, it is like this: "Today there are chickens and rabbits in the same cage, counting from the top there are 35 heads, counting from the bottom of the number of feet 94, ask how many chickens and free chickens there are?" ”

This problem is very simple to solve with the current equation, you can set the chicken to have x, and there is no y According to the problem conditions, because a chicken has 1 head and 2 legs, and a chicken is exempt from 1 head and 4 feet, so such a binary system of equations can be listed.

So there are 23 chickens in this cage and 12 free of them.

The solution on the Sun Tzu Sutra is very ingenious, it is according to the formula: rabbit number.

The number of heads is calculated, and the specific calculation is as follows: the number of rabbits.

only), the number of chickens and the number of heads are exempted from 35 12 23, and the book also gives the origin of the formula: after dividing the number of feet by 2, each chicken is left with one foot, and each rabbit is left with two legs, subtracting the number of heads, which is equivalent to subtracting one per chicken and rabbit, and the chicken feet are reduced, and each rabbit is left with only one foot, and the remaining number of feet is exactly equal to the number of rabbit heads.

Chickens and rabbits in the same cage is one of the famous interesting topics 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.

Q: How many chickens and rabbits are in the cage?

Chickens and rabbits in the same cage are easier to solve with equations, and the calculation method is:

1. If there are x chickens, y rabbits, and 35 chickens and rabbits in the same cage, then x+y=35.

2. Chickens and rabbits have a total of 94 legs, chickens have 2 feet, and rabbits have 4 legs, then 2x+4y=94.

3. Convert x+y=35 to x=35-y, and substitute it into 2x+4y=94 to get y=12.

4. Substituting y=23 into x+y=35 yields x=23.

According to the process, there are 23 chickens and 12 rabbits in the cage.

Considerations for solving equations.

1. If there is a denominator, go to the denominator first.

2. If there are parentheses, remove the brackets.

3. If you need to move the item, you will move the item.

4. Merge similar items.

5. The coefficient is reduced to 1 to obtain the value of the unknown.

6. Write "solution" at the beginning.

It is best to use a system of binary equations. If you haven't studied binary equations. It is also very good to use a system of unary equations.

For example, when chickens and rabbits are in the same cage, there are 36 legs. There are a total of 13 animals. How many chickens and rabbits are there?

Binary systems of equations to solve:

There are x chickens and y rabbits.

Rule. x+y=13

There are a total of 13 animals.

2x+4y=36

36 feet) then x=8

y=5 so chicken eight, rabbit five.

With a one-dimensional equation:

There are x chickens. then it can be known that there is a rabbit 13-x

Use the number of feet to list the equation.

2x+4(13-x)=36

x=8 so chicken eight, rabbit five.

1 Suppose it's all 3-pointers There are 33 points in total, but only 26 points, so there are 7 points less, i.e. 7 2-pointers and 4 3-pointers.

2 Assuming all 2-pointers, it would be 22 points, but he scored 26 points, four more points, or four 3-pointers.

He threw x three-pointers.

3x+2（11-x）=26

3x+22-2x=26

x=4 He made 4 three-pointers.

Hit a three-pointer x times. Two-point shot y times.

3x+2y=26

x+y=11

x=4 y=7

Hit 4 three-pointers.

Set x three-pointers and y two-pointers, 3x+2y=26, x+y=11, you can get x=4, y=7, that is, shoot 4 three-pointers.

A total of 14 shots were hit.

7 shots in the middle of the city, 22 points for the fierce roll A Xuzhichun, 26 points for B;

A has more than one shot, plus (4 + 2) 6 points, B has less than one shot in the second difference and one shot is less (5 + 3) 8 points; a difference of 14 points;

8 shots in the first and 6 shots in the first shot.

8 shots in A and 6 shots in B Detailed solution for specific consultation.