Question

Buckets X and Y contained only water and bucket Y was 1/2 full. If all of the water in bucket X was then poured into bucket Y, what fraction of the capacity of Y was then filled with water?
(1) Before the water from X was poured, X was 1/3 full.
(2) X and Y have the same capacity.

• 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:

For Data Sufficiency questions involving unknown quantities, it's often helpful to set up an algebraic expression representing the value you need to find. Then, analyze each statement to see if it provides the missing variables or relationships to solve for that expression. Here, the key was realizing we needed the ratio \(\frac{W_X}{C_Y}\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a Data Sufficiency problem involving fractions and ratios. We need to determine if we have enough information to calculate the final water level in bucket Y as a fraction of its total capacity.
Step 2: Analyzing the Question Stem:
Let \(C_X\) be the capacity of bucket X and \(C_Y\) be the capacity of bucket Y.
Let \(W_X\) be the initial amount of water in bucket X and \(W_Y\) be the initial amount of water in bucket Y.
From the stem, we know:
\[ W_Y = \frac{1}{2} C_Y \] After pouring, the total water in bucket Y is \(W_Y + W_X\).
The question asks for the value of the fraction: \(\frac{W_Y + W_X}{C_Y}\).
Substituting what we know:
\[ \frac{\frac{1}{2}C_Y + W_X}{C_Y} = \frac{1}{2} + \frac{W_X}{C_Y} \] To answer the question, we must be able to find a specific numerical value for the ratio \(\frac{W_X}{C_Y}\).
Step 3: Detailed Explanation:
Evaluating Statement (1) Alone:
"Before the water from X was poured, X was 1/3 full."
This tells us: \(W_X = \frac{1}{3} C_X\).
Substituting this into the ratio we need to find:
\[ \frac{W_X}{C_Y} = \frac{\frac{1}{3}C_X}{C_Y} = \frac{1}{3} \left( \frac{C_X}{C_Y} \right) \] We cannot determine the value of this expression because we do not know the relationship between the capacities of X and Y (the ratio \(\frac{C_X}{C_Y}\)).
Therefore, Statement (1) alone is not sufficient.
Evaluating Statement (2) Alone:
"X and Y have the same capacity."
This tells us: \(C_X = C_Y\).
The ratio we need to find is \(\frac{W_X}{C_Y}\). The stem does not provide any information about the amount of water in bucket X, \(W_X\). It could be any value from 0 to \(C_X\).
Therefore, Statement (2) alone is not sufficient.
Evaluating Statements (1) and (2) Together:
From Statement (1): \(W_X = \frac{1}{3} C_X\).
From Statement (2): \(C_X = C_Y\).
We can combine these to find the ratio we need.
\[ \frac{W_X}{C_Y} = \frac{\frac{1}{3} C_X}{C_Y} \] Since \(C_X = C_Y\), we can substitute \(C_Y\) for \(C_X\):
\[ \frac{W_X}{C_Y} = \frac{\frac{1}{3} C_Y}{C_Y} = \frac{1}{3} \] Now we can calculate the final fraction in bucket Y:
\[ \text{Final Fraction} = \frac{1}{2} + \frac{W_X}{C_Y} = \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \] Since we can find a unique value, the statements together are sufficient.
Step 4: Final Answer:
Neither statement is sufficient on its own, but combining them provides enough information to solve the problem. This corresponds to option (C).