Question

Each of the 256 solid-colored marbles in a box is either blue, green, or purple. What is the ratio of the number of blue marbles to the number of purple marbles in the box?
(1) The number of green marbles in the box is 4 times the number of blue marbles in the box.
(2) There are 192 green marbles in the box.

Hint:

In Data Sufficiency problems involving multiple variables, you generally need as many independent linear equations as you have variables to find unique values. To find a ratio, you sometimes need one fewer equation. Here, we needed to find values for B and P, which required eliminating G and then using information from both statements.

Answer 1

Step 1: Understanding the Concept:
This is a Data Sufficiency question. We need to determine if the given information is sufficient to find a unique value for the ratio of blue to purple marbles.
Let B, G, and P be the number of blue, green, and purple marbles, respectively.
From the question stem, we have the equation: \(B + G + P = 256\).
The question asks for the ratio \(B:P\), which is the value of the fraction \(B/P\).
Step 2: Detailed Explanation:
Evaluating Statement (1) Alone:
"The number of green marbles in the box is 4 times the number of blue marbles in the box."
This gives us the equation: \(G = 4B\).
Substitute this into our original equation:
\[ B + (4B) + P = 256 \] \[ 5B + P = 256 \] This is a single equation with two unknown variables, B and P. We cannot solve for a unique ratio. For instance, if \(B=40\), then \(P=56\), and the ratio is \(40:56 = 5:7\). If \(B=50\), then \(P=6\), and the ratio is \(50:6 = 25:3\). Since the ratio is not unique, this statement is not sufficient.
Evaluating Statement (2) Alone:
"There are 192 green marbles in the box."
This gives us: \(G = 192\).
Substitute this into our original equation:
\[ B + 192 + P = 256 \] \[ B + P = 256 - 192 = 64 \] Again, we have a single equation with two unknown variables. The ratio \(B:P\) is not unique. For example, if \(B=32\), then \(P=32\), and the ratio is \(1:1\). If \(B=16\), then \(P=48\), and the ratio is \(1:3\). This statement is not sufficient.
Evaluating Statements (1) and (2) Together:
From statement (1), we have \(G = 4B\).
From statement (2), we have \(G = 192\).
We can combine these to find B:
\[ 4B = 192 \] \[ B = \frac{192}{4} = 48 \] Now we have a value for B. We can use the equation from our analysis of statement (2), \(B + P = 64\), to find P:
\[ 48 + P = 64 \] \[ P = 64 - 48 = 16 \] Since we have unique values for B (48) and P (16), we can find the exact ratio: \(B:P = 48:16 = 3:1\).
Together, the statements are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but both statements together are sufficient to answer the question. This corresponds to option (C).