Meaning This is classical arithmetic problem. The problem of finding the number of chickens and rabbits each in a cage, knowing how many chickens and rabbits*** and how many feet are in the cage, is called the first chicken and rabbit in the same cage problem. Knowing the total number of chickens and rabbits and the difference between the feet of the chickens and the feet of the rabbits, the problem of finding how many chickens and rabbits each are is called the second chicken and rabbit in the same cage problem.

Quantitative Relationships First Chicken and Rabbit Problems:

If we assume all chickens, then we have

Number of Rabbits = (Actual Feet - 2×Total Number of Chickens and Rabbits) ÷ (4-2)

If we assume all rabbits, then we have

Number of Chickens = (4×Total Number of Chickens and Rabbits) ÷ (4-2) -The second problem of chickens and rabbits in the same cage:

Assuming all chickens, there are

Number of rabbits = (2×total number of chickens and rabbits - the difference between the feet of chickens and rabbits)÷(4+2)

Assuming all rabbits, there are

Number of chickens = (4 × total number of chickens and rabbits + difference between chickens and rabbits' feet) ÷ (4 + 2)

Solution Ideas and Methods To solve these questions, we generally use the hypothesis method, either by assuming that they are all chickens, or by assuming that they are all rabbits. If the first assumption is chicken, and then the rabbit for chicken; if the first assumption is rabbit, and then the chicken for rabbit. This type of problem is also called a substitution problem. The problem is solved by first assuming and then replacing.

Example 1 A long-haired rabbit and a reed-flower chicken, a chicken and a rabbit in a cage. Counting the head there are thirty-five, feet count *** there are ninety-four. Please count carefully, how many rabbits and how many chickens?

Solution Assuming that all 35 are rabbits, then

Number of chickens = (4×35-94) ÷ (4-2) = 23

Number of rabbits = 35-23 = 12

Or you can start by assuming that all 35 are chickens, then

Number of rabbits = (94-2 x 35) ÷ (4-2) = 12 (only)

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

Answer: there are 23 chickens and 12 rabbits.

Example 2: 2 acres of spinach should be fertilized with 1 kg of fertilizer, 5 acres of cabbage should be fertilized with 3 kg of fertilizer, 16 acres of both kinds of vegetables*** and 9 kg of fertilizer, how many acres of cabbage are there?

Solution This problem is actually a modified "chicken and rabbit" problem. "per acre of spinach fertilizer (1 ÷ 2) kg" and "each chicken has two feet" corresponds to, "per acre of cabbage fertilizer (3 ÷ 5) kg" and " (3÷5) kg of fertilizer per acre of cabbage" corresponds to " 4 feet per rabbit", "16 acres" corresponds to "total number of chickens and rabbits", "9 kg" corresponds to "9 kg" corresponds to "total number of feet of chickens and rabbits". Assuming that all 16 acres are spinach, there are

Cabbage acres = (9-1÷2×16) ÷ (3÷5-1÷2) = 10

Answer: There are 10 acres of cabbage.

Example 3 Mr. Lee buys the school 45 workbooks and journals*** for $69. The workbooks cost $3 .20 each and the journals cost $0.70 each. How many copies of each of the workbooks and journals were bought?

Solution This problem can be adapted to a "chicken and rabbit" problem. Assuming that all 45 books are journals, there are

Number of homework books = (69-0.70×45) ÷ (3.20-0.70) = 15

Number of journals = 45-15 = 30

Answer: There are 15 homework books and 30 journals. books and there are 30 journals.

Example 4 (Second Chicken and Rabbit Problem) There are 100 chickens and rabbits***, and there are 80 more chicken feet than rabbit feet, so how many chickens and rabbits are there?

Solution Assuming that all 100 are chickens, there are

Rabbits = (2×100-80) ÷ (4+2) = 20

Chickens = 100-20 = 80

Answer: There are 80 chickens and 20 rabbits.

Example 5 There are 100 buns for 100 monks to eat, one of the big monks eats 3 buns, and 3 of the small monks eat 1 bun, how many monks of each size are there?

Solution Assuming that all of the big monks, then *** eat buns (3 × 100), more than the actual eat (3 × 100-100), this is because the small monks are also counted as big monks, so we ensure that the total number of monks 100 unchanged in the case of the "small" for the "big". "Therefore, if we exchange a small monk for a big one, we can reduce the number of monks by (3-1/3), while keeping the total number of monks the same at 100. Therefore, *** has small monks

(3×100-100)÷(3-1/3)=75(people)

** has big monks 100-75=25(people)

Answer: *** has 25 big monks and 75 small monks. people.