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A company produces a certain toy in only 2 sizes, small or large, and in only 2 colors, red or green. If, for each size, there are equal numbers of red and green toys in a certain production lot, what fraction of the total number of green toys is large?
The company produces toys in only 2 sizes (small and large) and only 2 colors (red and green). We're told that for each size, there are equal numbers of red and green toys. We need to find: what fraction of the total number of green toys is large?
Let's translate this carefully. If we have S small toys total, then \(\mathrm{S/2}\) are red and \(\mathrm{S/2}\) are green (because of the equal distribution). Similarly, if we have L large toys total, then \(\mathrm{L/2}\) are red and \(\mathrm{L/2}\) are green.
The question asks for:
Key insight: Due to the equal red/green distribution within each size, the fraction of green toys that are large is exactly the same as the fraction of ALL toys that are large. This elegant symmetry will be crucial to our analysis.
For sufficiency: We need to determine the exact value of \(\mathrm{L/(S + L)}\), or equivalently, we need to know the ratio of large toys to total toys.
Statement 1 tells us that 400 of the small toys are green.
Since there are equal numbers of red and green toys for each size:
But here's what we don't know: anything about the large toys. We have no information about L, so we cannot determine the ratio \(\mathrm{L/(S + L)}\).
Let's consider two scenarios to demonstrate why this isn't sufficient:
Different values of L give us different answers to our question. Statement 1 is NOT sufficient.
[STOP - Not Sufficient!] We eliminate choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us that \(\mathrm{2/3}\) of the toys produced are small.
This means \(\mathrm{S/(S + L) = 2/3}\), which directly tells us that \(\mathrm{L/(S + L) = 1 - 2/3 = 1/3}\).
Remember our key insight from the beginning: due to the equal red/green distribution within each size, the fraction of green toys that are large equals the fraction of all toys that are large.
Therefore:
We can answer the question definitively: \(\mathrm{1/3}\) of the green toys are large. Statement 2 is sufficient.
[STOP - Sufficient!] We eliminate choices C and E.
Statement 2 alone provides the exact ratio we need through the elegant symmetry of the problem, while Statement 1 alone leaves us uncertain about the number of large toys.
Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."