The Chicken-Rabbit Cage Problem is a classic Chinese mathematical puzzle. In this problem, chickens and rabbits are kept together in the same cage. We are given the total number of heads and the total number of legs, and we need to figure out how many chickens and how many rabbits there are.
Let's work through a specific example. We have a cage with 35 heads total and 94 legs total. We need to find how many chickens and how many rabbits are in the cage. Remember, each chicken has 1 head and 2 legs, while each rabbit has 1 head and 4 legs.
We can solve this using algebra. Let x equal the number of chickens and y equal the number of rabbits. From the given information, we set up two equations: x plus y equals 35 for the heads, and 2x plus 4y equals 94 for the legs. Solving by substitution, we get x equals 23 chickens and y equals 12 rabbits.
Here's an intuitive method called the assumption approach. First, assume all 35 animals are chickens. This gives us 70 legs. But we actually have 94 legs, so there are 24 extra legs. Since each rabbit has 2 more legs than a chicken, we divide 24 by 2 to get 12 rabbits. Therefore, there are 23 chickens and 12 rabbits.
Our final answer is 23 chickens and 12 rabbits. Let's verify this solution. For heads: 23 plus 12 equals 35, which matches our given total. For legs: 23 chickens times 2 legs each gives 46 legs, plus 12 rabbits times 4 legs each gives 48 legs, totaling 94 legs. This confirms our solution is correct.