Welcome to the classic Chickens and Rabbits problem! This is an ancient Chinese mathematical puzzle. In this problem, we have chickens and rabbits in a cage. Each chicken has one head and two legs, while each rabbit has one head and four legs. We can count the total number of heads and legs, but we need to figure out how many chickens and how many rabbits there are.
Now let's set up a specific example to solve. Suppose we have 35 animals in total, and we count 94 legs altogether. We need to find how many chickens and how many rabbits there are. Let's use variables: let x represent the number of chickens, and y represent the number of rabbits. This gives us a system of two equations: x plus y equals 35, representing the total number of animals, and 2x plus 4y equals 94, representing the total number of legs.
Let's solve this system using the substitution method. From the first equation x plus y equals 35, we can express x as 35 minus y. Now we substitute this into the second equation: 2 times the quantity 35 minus y, plus 4y equals 94. Expanding this gives us 70 minus 2y plus 4y equals 94. Combining like terms: 70 plus 2y equals 94. Subtracting 70 from both sides: 2y equals 24. Dividing by 2: y equals 12. So we have 12 rabbits. Substituting back: x equals 35 minus 12, which equals 23 chickens.
Now let's solve the same problem using the assumption method, which is a more intuitive approach. First, we assume all 35 animals are chickens. If they were all chickens, we would have 35 times 2 equals 70 legs. But we actually have 94 legs, so there are 94 minus 70 equals 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. This gives us the same answer as the substitution method.