How are chickens and rabbits in the same cage calculated?

Chickens and rabbits caged together is one of the famous anecdotes in ancient China. About 65,438+0,500 years ago, this interesting question was recorded in Sun Tzu's mathematical classics. The book wrote: "There are chickens and rabbits in the same cage, with 35 heads above and 94 feet below. What are the geometric shapes of chickens and rabbits? These four sentences mean: there are several chickens and rabbits in a cage, counting from the top, there are 35 heads; It's 94 feet from the bottom. How many chickens and rabbits are there in each cage? In the civil service examinations in Yunnan over the years, the problem of chickens and rabbits in the same cage has also appeared many times. As an interesting and frequent question in the exam, let's study with Dehong Chinese public education experts!

(A) the origin of chickens and rabbits in the same cage

Problem solving skills: geometric sketch plus basic travel formula.

Example 1. Chickens and rabbits are kept in a cage at the same time. Count them. They have 35 heads and 94 feet. Q: How many chickens and rabbits are there?

Analysis of Zhonggong:

Method 1: Suppose all 35 rabbits are 35×4= 140 feet, which is more than 94 feet 140-94=46 feet. Each chicken is 4-2=2 feet less than the rabbit, so there are 46 chickens.

Method 2: It can also be assumed that all 35 chickens have 2×35=70 feet, 94-70=24 feet less than 94 feet, and each chicken has 4-2=2 feet less than rabbits, so there are 24 rabbits.

Conclusion:

The basic formula to solve the problem of chickens and rabbits in the same cage is:

If all rabbits are assumed, then there are:

Number of chickens = (number of feet per rabbit × total number of chickens and rabbits-actual number of feet) ÷ (number of feet per rabbit-number of feet per chicken)

Number of rabbits = total number of chickens and rabbits-number of chickens

If all chickens are assumed, then there are:

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)

Number of chickens = total number of chickens and rabbits-number of rabbits

(2) the metamorphosis of chickens and rabbits

Problem-solving skills: identify the chicken and rabbit in the stem and solve it by hypothesis method.

Question characteristics: Know the index number and index headquarters of two subjects, and find the subject number.

Example 2: In a math contest, 10 test questions, 6 points for each correct answer, and 2 points for each wrong question. Xiaohong finally got 44 points, and the right questions were more than the wrong ones.

Analysis of Zhonggong:

Assuming that 10 questions are all correct, the score is 10×6=60 points, 60-44 points are more than 44 points = 16 points, and 6+2=8 points are more than wrong ones, so 16÷8=2 wrong answers in total. The correct answer is 10-2=8, and the wrong answer is 8-2=6.

Example 3: There are 18 spiders, dragonflies and cicadas, with 1 18 legs and 20 pairs of wings (spiders have 8 legs; Dragonflies have six legs and two pairs of wings; Cicada has six legs and a pair of wings. How many dragonflies are there? .

Analysis of Zhonggong:

Observing the characteristics of numbers, dragonflies and cicadas have six legs, and only spiders have eight legs. So we can start with the number of legs and find out the number of spiders. Assuming that all three animals have six legs, the total number of legs is 6× 18= 108 (strips), and the difference is118-108 =10 (strips), which is So there should be (118-108) ÷ (8-6) = 5 (only) spiders, so the remaining 18-5= 13 (only) is the number of dragonflies and cicadas. 20- 13=7 (right) less than the actual number. This is because dragonflies have two pairs of wings, and we only calculate the difference according to one pair of wings, so the number of dragonflies can only be 7÷(2- 1)=7 (only).

No matter how the "chicken" and "rabbit" are deformed, such problems can be solved quickly as long as the characteristics of the questions are grasped and the hypothesis method is used.