Today we'll solve the classic chickens and rabbits in cage problem. In this problem, we have chickens and rabbits in a cage. We know there are 35 heads total and 94 feet total. Each chicken has 1 head and 2 feet, while each rabbit has 1 head and 4 feet. Our goal is to find how many chickens and how many rabbits are in the cage.
Now let's understand the constraints of this problem. We know the total number of heads is 35 and the total number of feet is 94. Each chicken contributes 1 head and 2 feet, while each rabbit contributes 1 head and 4 feet. We can define variables: let C equal the number of chickens and R equal the number of rabbits. This gives us two fundamental equations: C plus R equals 35 for the total heads, and 2C plus 4R equals 94 for the total feet.
Let's solve this system using algebra. We start with our two equations: C plus R equals 35, and 2C plus 4R equals 94. From the first equation, we can express C as 35 minus R. Substituting this into the second equation gives us 2 times 35 minus R plus 4R equals 94. Expanding this, we get 70 minus 2R plus 4R equals 94, which simplifies to 70 plus 2R equals 94. Solving for R, we get 2R equals 24, so R equals 12. Then C equals 35 minus 12, which is 23. Let's verify: 23 plus 12 equals 35 heads, and 23 times 2 plus 12 times 4 equals 46 plus 48, which is 94 feet.
Now let's try the assumption method, which uses logical reasoning instead of algebra. First, assume all 35 animals are chickens. This would give us 35 times 2 equals 70 feet. But we actually have 94 feet, so we need 94 minus 70 equals 24 extra feet. Each rabbit has 2 more feet than a chicken, since 4 minus 2 equals 2. To get 24 extra feet, we need 24 divided by 2 equals 12 rabbits. Therefore, we have 12 rabbits and 35 minus 12 equals 23 chickens. This visual transformation shows how we convert chickens to rabbits to account for the extra feet.
Let's verify our solution visually. We found that there are 23 chickens and 12 rabbits in the cage. Let's count the heads: 23 plus 12 equals 35 heads, which matches our constraint. Now let's count the feet: 23 chickens times 2 feet each gives 46 feet, plus 12 rabbits times 4 feet each gives 48 feet, for a total of 46 plus 48 equals 94 feet. This also matches our constraint perfectly. Both the algebraic method and the assumption method gave us the same correct answer, demonstrating the reliability and consistency of mathematical problem-solving techniques.