If Pei ordered a total of 63 bottles of cola, root beer, and ginger ale for a party, how many...

Understanding the Question

We need to find the exact number of cola bottles that Pei ordered.

Given Information

• Total bottles ordered = 63
• Three types: cola (C), root beer (R), and ginger ale (G)
• \(\mathrm{C + R + G = 63}\)

What We Need to Determine

For this value question to be sufficient, we need to be able to calculate a unique numerical value for C (the number of cola bottles).

Key Insight

Since we have 3 unknowns (C, R, G) and only 1 equation from the setup, we need 2 more independent equations to solve for a unique value. Each statement will provide at most one additional constraint, so let's see if either statement alone gives us enough information.

Analyzing Statement 1

Statement 1 tells us: The number of root beer bottles was 80% of the number of ginger ale bottles.

This gives us the relationship: \(\mathrm{R = 0.8G}\)

Now we have:

• \(\mathrm{C + R + G = 63}\)
• \(\mathrm{R = 0.8G}\)

Substituting the second equation into the first:
\(\mathrm{C + 0.8G + G = 63}\)
\(\mathrm{C + 1.8G = 63}\)

We still have one equation with two unknowns (C and G). We cannot determine unique values for either variable.

For example:

• If G = 10, then C = 45
• If G = 20, then C = 27
• Different values of G give different values of C

Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2 tells us: The number of cola bottles was 75% of the total number of root beer and ginger ale bottles combined.

This gives us: \(\mathrm{C = 0.75(R + G)}\)

Since we know \(\mathrm{C + R + G = 63}\), we can express \(\mathrm{R + G = 63 - C}\)

Substituting this into our Statement 2 equation:
\(\mathrm{C = 0.75(63 - C)}\)
\(\mathrm{C = 47.25 - 0.75C}\)
\(\mathrm{C + 0.75C = 47.25}\)
\(\mathrm{1.75C = 47.25}\)
\(\mathrm{C = 27}\)

We get a unique value: C = 27 bottles of cola.

[STOP - Sufficient!]

Statement 2 is sufficient.

This eliminates choices C and E.

The Answer: B

Statement 2 alone is sufficient because it provides the exact relationship needed to solve for C uniquely, while Statement 1 alone leaves us with two unknowns and insufficient constraints.

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