Question

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

• A. Data in statement I alone is sufficient.
• B. Data in statement II alone is sufficient.
• C. Data in both statements together is sufficient.
• D. Either of the Statements I and II is sufficient
• E. Data in both statements together is not sufficient.

Answer 1

Answer: (D)

Either of the Statements I and II is sufficient

First, set up an RTW chart:

Recall that the question is "Is $m<n$?"

(1): SUFFICIENT. Find out how much time it would take for the task to be performed with both Mary and Nancy working:

$(\frac{1}{m}+\frac{1}{n})t=1$

$\left(\frac{m+n}{mn}\right)t=1$

$t=\frac{mn}{m+n}$

Now, set up the inequality described in the statement (that is, twice this time is greater than m):

$2t>m$

$2\left(\frac{mn}{m+n}\right)>m$

$2mn>mn+m^2$$Youcancrossmultiplyby$m + n$because$m _ n$ is positive.

$mn>m^2$

$n>m$ You can divide by $m$ because $m$ is positive.

Alternatively, you can rearrange the original inequality thus:

$t>\frac{m}{2}$

If both Mary and Nancy worked at Mary's rate, then together, they would complete the task in $\frac{m}{2}$ hours. Since the actual time is longer, Nancy must work more slowly than Mary, and thus $n>m$.

(2): SUFFICIENT. You can reuse the computation of t, the time needed for the task to be jointly performed:

$2t<n$

$2\left(\frac{mn}{m+n}\right)<n$

$2mn<nm+n^2$ Again, you can cross multiply by $m+n$ because $m+n$ is positive.

$mn<n^2$

$m<n$ You can divide by $n$ because $n$ is positive.

Alternatively, you can rearrange the original inequality thus:

$t<\frac{n}{2}$

If both Mary and Nancy worked at Nancy's rate, then together, they would complete the task $\frac{n}{2}$ hours. Since the actual time is shorter, Mary must work faster than Nancy, and thus $m<n$.