Let’s assume the work done to be 1 unit.

Mary can do this work in m hours; therefore, Mary works at the rate of (\(\frac{1}{m}\)) units per hour.
Nadir can do this work in n hours; therefore, Nadir works at the rate of (\(\frac{1}{n}\)) units per hour.

If they work together, their combined rate = \(\frac{1}{m}\) + \(\frac{1}{n}\) = \(\frac{(m+n) }{ mn}\).
When work is constant, time and rates are reciprocal of each other. Therefore, the time taken by them when working together = \(\frac{mn }{ (m+n)}\).

From statement I alone, \(\frac{mn }{ (m+n)}\) > \(\frac{m}{2}\).

Note that m and n are positive values since they represent time. Therefore, we can cancel off m from both sides of the inequality. When we do this, we get,
\(\frac{n }{ (m+n)}\) > ½ or 2n>m+n or n>m.

Is m<n? Yes. Statement I alone is sufficient. Answer options B, C and E can be eliminated. Possible answer options are A or D.

From statement II alone, \(\frac{mn }{ (m+n)}\) < \(\frac{n}{2}\). Following a similar approach as we did in the previous statement, we can simplify the inequality as shown below:

\(\frac{m }{ (m+n)}\) < ½ or 2m < m + n or m<n.

Is m<n? Yes. Statement II alone is sufficient. Answer option A can be eliminated.

The correct answer option is D.

Hope that helps!
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Time of Mary = m
Rate of Mary = 1/m

Time of Nadir = n
Rate of Nadir = 1/n

Question asks is Time of Mary (m) < Time of Nadir (n)
**For time (m) to be less than time (n), the rate Mary works at needs to be greater than the rate Nadir works at.

So, need to find whether (1/m) > (1/n)!

(1) 1/m +1/n > 2/m (rate is the reciprocal of time)
1/n > 2/m - 1/m
1/n > 1/m
This statement can answer the question => is 1/m > 1/n?
Sufficient.

(2) 1/m + 1/n < 2/n
1/m < 2/n - 1/n
1/m < 1/n
This statement can answer the question => is 1/m > 1/n?
Sufficient.

Answer: D

I have a doubt since the question is asking for m<n?? So why not consider the condition in which m>n also??

Mary and Nadir are both working constant rates. Lets take a look at the statements:

(1) The time it would take Mary and Nadir to perform the task together, each working at their respective constant rates, is greater than m/2

If Mary and Nadir work together at their respective rates, the only way for their combined rate to be greater than m/2 is if n > m. If n = m, the combined rate will be exactly m/2. If n < m, the combine rate will be less than m/2. Sufficient.

(2) The time it would take Mary and Nadir to perform the task together, each working at their respective constant rates, is less than n/2

If Mary and Nadir work together at their respective rates, the only way for their combined rate to be less than n/2 is if n > m. If n = m, the combined rate will be exactly n/2. If n < m, the combine rate will be greater than n/2. Sufficient.
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Hi everyone,

I solved the question with a different approach than the ones above. I chose to pick numbers that satisfy the constraint. I picked M=2 AND N=4
rate of m= 1/2
rate of n= 1/4
rate of m+n= 3/4 therefore time m+n = 4/3 or 1h20

1) The time it would take to do the job together, 1h20 is greater than m/2 which is 1 ==> sufficient
2) The time it would take to do the job together 1h20 is less than n/2 which is 2 ==> sufficient

Each of the statements are sufficient : Answer D

Does this approach always work?
Thank you!

I understood this problem as below.
Suppose m=n then working together they would have completed their task in half of the time they used to take work alone.

But st.1 says than Marry has taken longer time which indicates that Mary is taking longer than Nadir to complete the same task alone. m>n.

St. 2 says n takes less than n/2 hours while working with mary in simultaneously. It indicates that Nadir will take shorter time to complete the work alone. m>n

D is answer.

Unfortunately, I did this problem by picking numbers, which took me much longer.

Given the problem and statements are asking about the combined work for both Mary and Nadir, I decided to use the Combined Work Formula, which is \(\frac{mn}{(m+n)}\)

S1) I decided to pick numbers originally I chose m=3 and n=5, but then I realized in order for statement 1 to be true I needed m=4 and n=5. This was to test for m<n so an always or sometimes Yes.

I then flipped those numbers to m=5 and n=4, to test for an always or sometimes No. However, when I tried to test this the numbers didn't comply with statement 1 and so the answer became an Always Yes, which is Sufficient.

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

\(\frac{mn}{(m+n)}=\frac{4*5}{(4+5)}=\frac{20}{9}=2\frac{2}{9}>\frac{4}{2}\)
Yes

\(\frac{mn}{(m+n)}=\frac{5*4}{(5+4)}=\frac{20}{9}=2\frac{2}{9}<\frac{5}{2}\)
Since \(2\frac{2}{9}<2\frac{1}{2}\), the statement is not True and these numbers can't be chosen

S2) I used the same numbers m=4 and n=5 to test for an always/sometimes Yes and I used m=5 and n=4 for an always/sometimes No. Once I put the numbers into the combined work formula only m=4 and n=5 complied with Statment 2. I also tried m=5 and n=6, which also turned out to work with Statement 2 and gave me an Always Yes, which makes the statement Sufficient

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

\(\frac{mn}{(m+n)}=\frac{4*5}{(4+5)}=\frac{20}{9}=2\frac{2}{9}<\frac{5}{2}\)
Yes

\(\frac{mn}{(m+n)}=\frac{4*5}{(4+5)}=\frac{20}{9}=2\frac{2}{9}>\frac{4}{2}\)
Since \(2\frac{2}{9}>2\), the statement is not True and these numbers can't be chosen

Both statements also work when m=5 and n=6, but doesn't work when m=6 and n=5

The answer IMO is Either or D

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