Welcome to the rabbit and chicken problem! This is a classic word problem that teaches us how to solve systems of equations. In this problem, we have a farm with chickens and rabbits. We know the total number of heads and the total number of legs, and we need to find how many of each animal there are. Let's start with an example: 10 heads and 28 legs total.
The first method is guess and check. We systematically try different combinations of chickens and rabbits that add up to 10 heads. Let's start with 5 chickens and 5 rabbits. This gives us 10 heads, which is correct, but 30 legs, which is too many. Since we have too many legs, we need fewer rabbits. Let's try 6 chickens and 4 rabbits. This gives us 10 heads and exactly 28 legs. Perfect! So we have 6 chickens and 4 rabbits.
The second method uses logical deduction. First, we assume all 10 animals are chickens. This would give us 20 legs total. But we actually have 28 legs, so we have 8 extra legs. Each time we replace a chicken with a rabbit, we add 2 extra legs, since rabbits have 4 legs and chickens have 2. So we need 8 divided by 2, which equals 4 rabbits. That means we have 4 rabbits and 6 chickens. This method is more systematic and works well for larger numbers.
The third method uses algebraic equations, which is the most systematic approach. We define variables: C for chickens and R for rabbits. From the problem, we get two equations. First, C plus R equals 10, representing total heads. Second, 2C plus 4R equals 28, representing total legs. We solve by substitution: from the first equation, C equals 10 minus R. Substituting into the second equation gives us 2 times 10 minus R plus 4R equals 28. Simplifying: 20 plus 2R equals 28, so R equals 4 and C equals 6.
Let's summarize what we've learned. We explored three different methods to solve the rabbit and chicken problem, and all three gave us the same answer: 6 chickens and 4 rabbits. Let's verify this is correct. We have 6 plus 4 equals 10 heads, which matches our given information. For legs, we have 6 times 2 plus 4 times 4, which equals 12 plus 16, equals 28 legs. Perfect! Each method has its strengths. Guess and check works well for small numbers. Logical deduction builds reasoning skills. Algebraic equations work for complex problems. Choose the method that works best for you and the situation!