Explain this math puzzle step by step with visuals and narration:
"There are 60 animals. Some are chickens (2 legs), some are dogs (4 legs). Together, they have 194 legs. How many of each?"
Use the **assumption method**:
1. First, assume all animals are chickens → calculate total legs.
2. Compare with actual legs → calculate the extra legs.
3. Each dog adds 2 legs → divide the extra legs by 2.
4. Find number of dogs and subtract from 60 to get chickens.
Include an alternate path:
- Assume all are dogs → overshoot → correct with chickens.
- Show how both paths give same result.
Then add:
- A bonus segment using algebra: x = dogs, 60–x = chickens, write and solve the equation.
- Optional challenge: "What if all 194 legs were dog legs?" — prompt user to reflect.
Close by saying:
"This is Strategy #01 of Hypothetical Thinking. You’ll see it again in average, mixture, and work problems."
Use calm, smart narration, step-by-step whiteboard writing, and highlight key steps visually.
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Let's solve this math puzzle step by step. We have 60 animals - some are chickens with 2 legs, and some are dogs with 4 legs. Together, they have 194 legs. We need to find out how many of each animal there are. We'll use the assumption method to solve this problem.
Let's try our first approach. We'll assume all 60 animals are chickens. Each chicken has 2 legs, so that's 60 times 2, which gives us 120 legs. But we know the total is 194 legs. That means we have 74 extra legs. Since replacing a chicken with a dog adds 2 extra legs, we need to convert 74 divided by 2, which is 37 animals from chickens to dogs.
Now we can finalize our answer. If we have 37 dogs, then the number of chickens must be 60 minus 37, which gives us 23 chickens. Let's verify: 23 chickens times 2 legs equals 46 legs. 37 dogs times 4 legs equals 148 legs. Adding 46 and 148 gives us 194 legs total, which matches our problem statement. So our answer is 23 chickens and 37 dogs.
Let's try an alternative approach. We could also assume all 60 animals are dogs. Each dog has 4 legs, so that would be 60 times 4, which gives us 240 legs. But that's 46 legs too many compared to our target of 194 legs. When we convert a dog to a chicken, we reduce the leg count by 2. So we need to convert 46 divided by 2, which is 23 animals from dogs to chickens. That means we have 23 chickens and 37 dogs, which matches our previous answer.
Finally, let's solve this using algebra. If we let x be the number of dogs, then 60 minus x is the number of chickens. We can write the equation: 4x plus 2 times 60 minus x equals 194. Simplifying: 4x plus 120 minus 2x equals 194. This simplifies to 2x plus 120 equals 194, so 2x equals 74, and x equals 37 dogs. Therefore, we have 23 chickens. As a bonus challenge, think about how many dogs we could have if all 194 legs were dog legs. This problem demonstrates Strategy Number 1 of Hypothetical Thinking, which you'll see again in average, mixture, and work problems.
Let's try our first approach. We'll assume all 60 animals are chickens. Each chicken has 2 legs, so that's 60 times 2, which gives us 120 legs. But we know the total is 194 legs. That means we have 74 extra legs. Since replacing a chicken with a dog adds 2 extra legs, we need to convert 74 divided by 2, which is 37 animals from chickens to dogs.
Now we can finalize our answer. If we have 37 dogs, then the number of chickens must be 60 minus 37, which gives us 23 chickens. Let's verify: 23 chickens times 2 legs equals 46 legs. 37 dogs times 4 legs equals 148 legs. Adding 46 and 148 gives us 194 legs total, which matches our problem statement. So our answer is 23 chickens and 37 dogs.
Let's try an alternative approach. We could also assume all 60 animals are dogs. Each dog has 4 legs, so that would be 60 times 4, which gives us 240 legs. But that's 46 legs too many compared to our target of 194 legs. When we convert a dog to a chicken, we reduce the leg count by 2. So we need to convert 46 divided by 2, which is 23 animals from dogs to chickens. That means we have 23 chickens and 37 dogs, which matches our previous answer.
Finally, let's solve this using algebra. If we let x be the number of dogs, then 60 minus x is the number of chickens. We can write the equation: 4x plus 2 times 60 minus x equals 194. Simplifying: 4x plus 120 minus 2x equals 194. This simplifies to 2x plus 120 equals 194, so 2x equals 74, and x equals 37 dogs. Therefore, we have 23 chickens. As a bonus challenge, think about how many dogs we could have if all 194 legs were dog legs. This problem demonstrates Strategy Number 1 of Hypothetical Thinking, which you'll see again in average, mixture, and work problems.