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Question

Difficulty: MediumData Sufficiency

A large water reservoir is equipped with two inlet pipes, Pipe PP and Pipe QQ, each filling the reservoir at its own constant rate. How many hours will it take to fill the empty reservoir if both pipes operate simultaneously from the start?

Statement (I): Pipe PP alone can fill the empty reservoir in 15 hours15\text{ hours}.
Statement (II): Pipe QQ fills the reservoir at a rate that is 50%50\% higher than the filling rate of Pipe PP.

Which of the following options correctly evaluates the sufficiency of the statements?

  1. A
    Statement (I) alone is sufficient to answer the question, while Statement (II) alone is not sufficient.
  2. B
    Statement (II) alone is sufficient to answer the question, while Statement (I) alone is not sufficient.
  3. C
    Either Statement (I) alone or Statement (II) alone is sufficient to answer the question.
  4. D
    Neither Statement (I) nor Statement (II) is sufficient, and even together they are not sufficient to answer the question.
  5. Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.Answer

Answer

Both Statement (I) and Statement (II) together are sufficient to answer the question, but neither statement alone is sufficient.
Evaluating Statement (I) alone gives only the individual performance of Pipe P, which is insufficient to determine the combined time. Statement (II) alone provides only a relative ratio between the rates of Pipe P and Pipe Q without any concrete time metric, making it insufficient on its own. When both statements are combined, Statement (I) provides the base rate for Pipe P and Statement (II) allows calculation of Pipe Q's rate, leading to a unique answer of 6 hours for the combined operation.

Step-by-Step Solution

1
Evaluate Statement (I) alone
Pipe PP's rate is 115\frac{1}{15} of the reservoir per hour. However, no information is given regarding Pipe QQ's rate.
Statement (I) alone is insufficient to calculate the combined time.
2
Evaluate Statement (II) alone
Pipe QQ's rate is 1.51.5 times Pipe PP's rate, meaning Rate(QQ) =1.5×= 1.5 \times Rate(PP).
Statement (II) alone gives only a relative ratio of work rates, with no numerical time value given, so it is insufficient.
3
Evaluate Statements (I) and (II) together
From Statement (I), Rate(PP) =115= \frac{1}{15} reservoir/hour. From Statement (II), Rate(QQ) =1.5×115=110= 1.5 \times \frac{1}{15} = \frac{1}{10} reservoir/hour. Combined rate =115+110=16= \frac{1}{15} + \frac{1}{10} = \frac{1}{6} reservoir/hour. Thus, total combined time =6 hours= 6\text{ hours}.
Combining both statements yields a unique and definitive answer.

Key Concept

Data Sufficiency in Work and Time / Rate Problems
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Topics

Data Sufficiency

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