A company produces a certain toy in only 2 sizes, small or large, and in only 2 colors, red or...

Understanding the Question

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:

• Number of large green toys ÷ Total number of green toys
• Which equals: \(\mathrm{(L/2) ÷ (S/2 + L/2) = L/(S + L)}\)

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.

Analyzing Statement 1

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:

• Small green toys = 400
• Small red toys = 400
• Total small toys S = 800

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:

• Scenario 1: If there are 200 large toys total
  • \(\mathrm{L/(S + L) = 200/1000 = 1/5}\)
  • So \(\mathrm{1/5}\) of green toys would be large
• Scenario 2: If there are 800 large toys total
  • \(\mathrm{L/(S + L) = 800/1600 = 1/2}\)
  • So \(\mathrm{1/2}\) of green toys would be large

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.

Analyzing Statement 2

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:

• Fraction of all toys that are large = \(\mathrm{1/3}\)
• Fraction of green toys that are large = \(\mathrm{1/3}\)

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.

The Answer: B

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."