The chicken and rabbit problem is a classic mathematical puzzle. We have a cage with chickens and rabbits. We know the total number of heads is 35, and the total number of legs is 94. We need to find how many chickens and how many rabbits are in the cage.
The first method is the algebraic approach using a system of equations. We define variables: let c be the number of chickens and r be the number of rabbits. Then we set up two equations. For heads, since each animal has one head, we get c plus r equals 35. For legs, chickens have 2 legs and rabbits have 4 legs, so we get 2c plus 4r equals 94.
Now let's solve the system of equations step by step. We have 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: 2 times 35 minus r plus 4r equals 94. This simplifies to 70 minus 2r plus 4r equals 94, which becomes 70 plus 2r equals 94. Solving for r: 2r equals 24, so r equals 12. Therefore c equals 35 minus 12, which is 23. The answer is 23 chickens and 12 rabbits.
The second method is the assumption method. Let's assume all 35 animals are chickens. If all were chickens, the total legs would be 35 times 2, which equals 70 legs. But we actually have 94 legs, so the difference is 94 minus 70, which is 24 extra legs. Since each rabbit has 2 more legs than a chicken, the number of rabbits is 24 divided by 2, which equals 12 rabbits. Therefore, the number of chickens is 35 minus 12, which equals 23 chickens.
Let's verify our answer. We found 23 chickens and 12 rabbits. Checking heads: 23 plus 12 equals 35, which is correct. Checking legs: 23 times 2 plus 12 times 4 equals 46 plus 48, which equals 94 legs, also correct. Both the algebraic method and assumption method give us the same result. The chicken and rabbit problem is an excellent example that teaches us systematic thinking and shows how multiple approaches can solve the same problem.