The Chicken-Rabbit Cage Problem is a classic puzzle. A farmer has chickens and rabbits in the same cage. We know there are 10 heads total and 24 legs total. Each chicken has 1 head and 2 legs, while each rabbit has 1 head and 4 legs. Our goal is to find how many chickens and how many rabbits are in the cage.
Let's solve this using the assumption method. First, assume all 10 animals are chickens. This would give us 20 legs total. But we actually have 24 legs, which is 4 more than expected. Since each rabbit has 2 more legs than a chicken, we can find the number of rabbits by dividing the extra legs by 2. So we have 2 rabbits and 8 chickens.
We can also solve this problem using algebra. Let c represent the number of chickens and r represent the number of rabbits. We set up two equations: c plus r equals 10 for the total heads, and 2c plus 4r equals 24 for the total legs. Solving this system, we substitute c equals 10 minus r into the second equation, which gives us r equals 2 and c equals 8.
Let's verify our answer. We found 8 chickens and 2 rabbits. Eight chickens give us 8 heads and 16 legs. Two rabbits give us 2 heads and 8 legs. Adding these together: 8 plus 2 equals 10 heads, and 16 plus 8 equals 24 legs. This matches our original problem perfectly, confirming our solution is correct.