Question

If a certain vase contains only roses and tulips, how many tulips are there in the vase?
(1) The number of roses in the vase is 4 times the number of tulips in the vase.
(2) There is a total of 20 flowers in the vase.

• A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
• B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
• C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
• D. EACH statement ALONE is sufficient.
• E. Statements (1) and (2) TOGETHER are NOT sufficient.

Hint:

Word problems in Data Sufficiency often translate into a system of equations. To solve for \(n\) unique variables, you generally need \(n\) independent equations. Each statement here provides one equation, so two statements are needed for two variables.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency problem. We need to determine if the given statements provide enough information to find a unique number of tulips. Let \(R\) be the number of roses and \(T\) be the number of tulips. The question asks for the value of \(T\).
Step 2: Detailed Explanation:
Analyze Statement (1):
This statement gives us the equation \(R = 4T\). This is a single equation with two unknown variables (\(R\) and \(T\)). We cannot solve for a unique value of \(T\). For example, if \(T=2\), \(R=8\); if \(T=3\), \(R=12\). Thus, statement (1) alone is not sufficient.
Analyze Statement (2):
This statement gives us the equation \(R + T = 20\). Again, this is a single equation with two unknowns. We cannot determine a unique value for \(T\). For example, if \(T=5\), \(R=15\); if \(T=10\), \(R=10\). Thus, statement (2) alone is not sufficient.
Analyze Statements (1) and (2) Together:
Combining both statements, we have a system of two linear equations:
1) \(R = 4T\)
2) \(R + T = 20\)
We can substitute the expression for \(R\) from the first equation into the second equation:
\[ (4T) + T = 20 \]
\[ 5T = 20 \]
\[ T = 4 \]
Since we can find a unique value for \(T\), the statements together are sufficient.
Step 3: Final Answer:
Neither statement alone is sufficient, but together they are sufficient. This corresponds to option (C).