Multiple choice

This data sufficiency problem consists of a question and two statements labelled (1) and (2), which contain certain data. Using these data and your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), decide whether the data given are sufficient for answering the question and then indicate one of the following answer choices: Mary and Nancy can each perform a certain task in $m$ and $n$ hours, respectively. Is $m < n$? (1) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is greater than m. (2) Twice the time it would take both Mary and Nancy to perform the task together, each working at their respective constant rates, is less than n

  1. Data in statement I alone is sufficient.

  2. Data in statement II alone is sufficient.

  3. Data in both statements together is sufficient.

  4. Either of the Statements I and II is sufficient

  5. Data in both statements together is not sufficient.
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let T = mn/(m+n) be the time taken together. (1) 2T > m implies 2n/(m+n) > 1, so 2n > m+n, n > m. (2) 2T < n implies 2m/(m+n) < 1, so 2m < m+n, m < n. Both statements independently prove m < n.

AI explanation

The combined rate formula gives the time taken by both as mn/(m+n). Statement (1) establishes that 2mn/(m+n) > m, which simplifies to 2n > m+n and thus n > m. Statement (2) establishes that 2mn/(m+n) < n, which simplifies to 2m < m+n and thus m < n. Because each statement independently proves that m is less than n, either statement alone is sufficient to answer the question.

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