Directions: Each of the questions given below consists of a statement and/or a question followed by two statements labelled I and II. Read both the statements and Give answer (a) if the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question; Give answer (b) if the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question; Give answer (c) if the data either in Statement I or in Statement II alone are sufficient to answer the question; Give answer (d) if the data even in both Statements I and II together are not sufficient to answer the question; Give answer (e) if the data in both Statements I and II together are necessary to answer the question. How long will Machine Y, working alone, take to produce $x$ candles ? (M.B.A., 2002) I. Machine X produces $x$ candles in 5 minutes. II. Machine X and Machine Y working at the same time produce $x$ candles in 2 minutes.
Verbal Reasoning
Data Sufficiency
Difficulty: Medium
Choose an option
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Aif the data in Statement I alone are sufficient to answer the question, while the data in Statement II alone are not sufficient to answer the question;
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Bif the data in Statement II alone are sufficient to answer the question, while the data in Statement I alone are not sufficient to answer the question;
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Cif the data either in Statement I or in Statement II alone are sufficient to answer the question;
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Dif the data even in both Statements I and II together are not sufficient to answer the question;
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Eif the data in both Statements I and II together are necessary to answer the question.
Answer
Correct Answer: if the data in both Statements I and II together are necessary to answer the question.
Explanation
### Concept & Logic
This is a standard time and work problem where the combined rate equals the sum of individual rates:
$$ \frac{1}{X} + \frac{1}{Y} = \frac{1}{\text{Together}} $$
### Step-by-Step Solution
From Statement I:
* Machine X takes 5 minutes to produce $x$ candles.
* Time for X = 5 minutes.
* This is insufficient to find Y.
From Statement II:
* Machines X and Y together take 2 minutes.
* $\frac{1}{X} + \frac{1}{Y} = \frac{1}{2}$.
* This is insufficient to find Y alone.
Combining I and II:
* Substitute $X = 5$ into the combined rate equation:
* $\frac{1}{5} + \frac{1}{Y} = \frac{1}{2}$.
* $\frac{1}{Y} = \frac{1}{2} - \frac{1}{5} = \frac{3}{10}$.
* $Y = \frac{10}{3}$ minutes.
* Both statements are required to find the exact value.
### Exam Strategy & Shortcut
Recognize the classic "working together" formula structure. You need two of the three variables (Time X, Time Y, Time Together) to find the third. Statement I provides Time X, Statement II provides Time Together. Thus both are necessary.
### Common Pitfall
Overcomplicating the variable $x$. In this context, "$x$ candles" simply represents 1 whole, constant unit of work.
### Final Answer
Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question.**