If 200 cats kill 200 mice in 200 days at a constant rate, then, under the same conditions, how many days will 8 cats take to kill 8 mice?

Introduction / Context:
This is a classic work and time style puzzle framed in terms of cats and mice. It tests proportional reasoning and the ability to interpret rates correctly. The data indicate how many cats are needed to kill a certain number of mice in a given time. The question then changes the numbers of cats and mice but expects us to assume constant individual rates of work. Understanding that each cat works at the same rate is crucial to answering correctly.

Given Data / Assumptions:
- Two hundred cats kill two hundred mice in two hundred days.
- Each cat works at the same constant rate of killing mice per day.
- We are asked how long eight cats will take to kill eight mice under the same conditions.
- The rate of work does not change over time, and there are no other influences.

Concept / Approach:
We interpret the statement 200 cats kill 200 mice in 200 days in terms of unit rates. Since the numbers of cats and mice are equal, we can simplify the situation to find how many mice one cat can kill in a certain period. The principle of direct proportion applies: if every cat kills mice independently at the same speed, then scaling down both cats and mice by the same factor leaves the required time unchanged. This is similar to many work problems where identical workers perform identical tasks.

Step-by-Step Solution:
Step 1: The statement 200 cats kill 200 mice in 200 days suggests that the total work is killing 200 mice. Step 2: Because 200 cats together kill 200 mice in 200 days, we can see that each cat effectively kills one mouse in 200 days, assuming work is evenly distributed. Step 3: Express the rate of one cat as 1 mouse per 200 days. Step 4: Now consider eight cats. Together, their combined rate is 8 times the rate of one cat, so they kill 8 mice in 200 days as well. Step 5: Equivalently, we can think of the second situation as scaling the first by a factor of 8/200, reducing both cats and mice in the same proportion. When cats and mice are reduced by the same factor, the time required remains the same. Step 6: Therefore, eight cats will also take 200 days to kill eight mice under the same conditions.

Verification / Alternative check:
Compute the rate explicitly. Each cat kills 1 mouse in 200 days, which is 1/200 mouse per day. Eight cats therefore kill 8 * 1/200 = 8/200 = 1/25 mouse per day. To kill 8 mice at this combined rate, the time needed is total mice divided by rate, which is 8 divided by 1/25 = 8 * 25 = 200 days. This alternative calculation confirms our earlier reasoning and shows that the time required does not change when both cats and mice are scaled equally.

Why Other Options Are Wrong:
Option 8 days: This would be correct if the total amount of work were scaled differently, but here each cat still needs 200 days on average to kill a single mouse.
Option 800 days: This incorrectly assumes that reducing the number of cats slows the process far more than proportional, which conflicts with the constant rate assumption.
Option 1 day: This would require an extremely high killing rate that does not match the given data.
Option Cannot be determined: The information is sufficient to find the rate and compute the time, so this option is incorrect.

Common Pitfalls:
Many learners mistakenly assume that if both the number of cats and the number of mice are reduced, the time must also reduce in the same ratio. However, here the key is that the ratio of cats to mice remains 1 to 1, so the time does not change. Another pitfall is forgetting to compute the rate of one cat and instead guessing based on intuition. Writing down the rate explicitly helps avoid such confusion.

Final Answer:
Eight cats will take 200 days to kill eight mice.