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. Will Q take more than 8 hours to complete the job alone? I. P works faster than Q. II. P and Q can together finish the job in 5 hours. (J.M.E.T., 2005)
Verbal Reasoning
Data Sufficiency
Difficulty: Hard
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
In Data Sufficiency, we evaluate if the given statements provide enough information to definitively answer the question. The question asks if $q > 8$ hours. The relation between rates of work is $$ \text{Rate} = \frac{1}{\text{Time}} $$
### Step-by-Step Solution
From Statement I:
* P works faster than Q.
* This implies P takes less time than Q: $p < q$.
* Consequently, the rate of P is greater than the rate of Q: $\frac{1}{p} > \frac{1}{q}$.
* This alone is insufficient to find Q's specific time.
From Statement II:
* P and Q together finish the job in 5 hours.
* Thus, $\frac{1}{p} + \frac{1}{q} = \frac{1}{5}$.
* This alone is insufficient to find Q's specific time.
Combining I and II:
* We know $\frac{1}{p} > \frac{1}{q}$.
* Substituting this into the equation from II: $\frac{1}{q} + \frac{1}{q} < \frac{1}{p} + \frac{1}{q} = \frac{1}{5}$.
* Therefore, $\frac{2}{q} < \frac{1}{5} \Rightarrow q > 10$.
* Since $q > 10$, we can definitively answer "Yes, Q takes more than 8 hours".
### Exam Strategy & Shortcut
When rates are unequal, if both workers worked at the slower rate (Q's rate), they would take more than twice the time they take together. Since they take 5 hours together, two Q's would take $> 10$ hours. So Q alone takes $> 10$ hours. This immediately shows $q > 8$. Both statements are needed.
### Common Pitfall
A common mistake is trying to find the exact numerical value of $q$. Data sufficiency only requires knowing if there is enough information to answer the yes/no question logically.
### Final Answer
Therefore, the correct answer is **if the data in both Statements I and II together are necessary to answer the question.**