Each piece of candy in a certain jar is either brown, red, or green. If the jar has a total...

Understanding the Question

The question asks us to find the ratio of green pieces to brown pieces of candy. Let's think about what we need to determine sufficiency.

Given Information

• Total pieces of candy: 312
• Each piece is either brown, red, or green
• We need: the ratio of green to brown pieces

What We Need to Determine

For sufficiency, we need to determine a unique ratio of green:brown. This is crucial - we don't need the actual numbers of green and brown candies, just their relationship to each other.

Key Insight

Since we're looking for a ratio rather than specific values, a statement that tells us the relationship between green and brown pieces might be sufficient on its own, even without knowing exact quantities.

Analyzing Statement 1

Statement 1 tells us: "There are 200 percent more green pieces of candy than brown pieces of candy."

What This Means

Let's translate this percentage relationship step by step:

• "\(200\%\) more" means we ADD \(200\%\) of the original amount
• If brown pieces = \(100\%\), then green pieces = \(100\% + 200\% = 300\%\)
• In other words: \(\mathrm{Green\ pieces} = 3 \times \mathrm{brown\ pieces}\)

The Ratio Revelation

This directly gives us the ratio! No matter how many brown pieces there are:

• If brown = 10, then green = 30, so green:brown = \(30:10 = 3:1\)
• If brown = 50, then green = 150, so green:brown = \(150:50 = 3:1\)
• If brown = any number, green = 3 × that number, so the ratio is always \(3:1\)

[STOP - Statement 1 is SUFFICIENT!]

Statement 1 is sufficient to determine the ratio of green to brown pieces.

This eliminates choices B, C, and E. We're left with A or D.

Analyzing Statement 2

Important: Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: "There are 216 red pieces of candy in the jar."

What We Know and Don't Know

With 216 red pieces out of 312 total pieces:

• Brown + Green = \(312 - 216 = 96\) pieces

This tells us the sum of brown and green pieces, but not how they're split between the two colors.

Testing Different Scenarios

Let's check if different splits give us the same ratio:

Different valid splits produce different ratios. We cannot determine a unique ratio from Statement 2 alone.

Statement 2 is NOT sufficient.

This eliminates choice D.

The Answer: A

Since Statement 1 alone is sufficient but Statement 2 alone is not sufficient, the answer is A.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."

Why This Makes Sense

• Statement 1 gives us a direct percentage relationship between green and brown, which immediately translates to a ratio
• Statement 2 gives us only the remainder after accounting for red pieces, leaving multiple possibilities for the green:brown split