The Chicken and Rabbit in a Cage problem is a classic mathematical puzzle. We have chickens and rabbits in a cage, and we know the total number of heads and the total number of legs. Each chicken has one head and two legs, while each rabbit has one head and four legs. Our goal is to determine exactly how many chickens and how many rabbits are in the cage.
The first method is the assumption method. Let's work through an example where we have 8 heads and 22 legs total. Step one: assume all animals are chickens. Step two: if all 8 animals were chickens, we would have 8 times 2 equals 16 legs. Step three: find the difference between actual legs and calculated legs: 22 minus 16 equals 6 extra legs. Step four: since each rabbit has 2 more legs than a chicken, the number of rabbits is 6 divided by 2, which equals 3 rabbits. Step five: the number of chickens is 8 total heads minus 3 rabbits, which equals 5 chickens.
The second method is the algebraic method using a system of equations. Let's use the same example with 8 heads and 22 legs. Step one: define variables where C equals the number of chickens and R equals the number of rabbits. Step two: set up two equations. Equation one: C plus R equals 8 total heads. Equation two: 2C plus 4R equals 22 total legs. Step three: solve the system. From equation one, C equals 8 minus R. Substitute this into equation two: 2 times 8 minus R plus 4R equals 22. This simplifies to 16 minus 2R plus 4R equals 22, which becomes 16 plus 2R equals 22. Therefore 2R equals 6, so R equals 3 rabbits. And C equals 8 minus 3, which equals 5 chickens.
Now let's practice with a new problem. In a cage, there are chickens and rabbits with 12 heads and 32 legs total. How many of each animal are there? Let's solve this using the assumption method. Step one: assume all 12 animals are chickens. Step two: if all were chickens, we'd have 12 times 2 equals 24 legs. Step three: the difference is 32 minus 24 equals 8 extra legs. Step four: since each rabbit has 2 more legs than a chicken, we have 8 divided by 2 equals 4 rabbits. Step five: the number of chickens is 12 minus 4 equals 8 chickens. Let's verify: 8 plus 4 equals 12 heads, check. For legs: 8 times 2 plus 4 times 4 equals 16 plus 16 equals 32 legs, check. So the answer is 8 chickens and 4 rabbits.
Let's summarize the key points of the Chicken and Rabbit problem. We learned two main solution methods: the assumption method, which is easier and involves assuming all animals are one type then adjusting, and the algebraic method, which is more systematic using a system of equations. Both methods give the same correct answer. Always remember to verify your solution by checking that the total heads and total legs match the given numbers. This classic problem teaches us about systems of equations, logical reasoning, and effective problem-solving strategies. The general formulas are: C plus R equals total heads, and 2C plus 4R equals total legs, where C is chickens and R is rabbits. Mastering this problem helps build strong mathematical thinking skills that apply to many other areas.