Raghavachari, Viswanathan and Venkatraman together finished

I found this exercise too complex to be a real GMAT problem.

For simplicity , let's call them Rag, Vis & Ven.

We know Rag and Vis working together spend 20 days for 4/5 of the job --> this means 25 days for the entire job, Rag and Vis working together, at that pace. (1)

And we have the formula used for work problems: the entire job time T, with the three guys working together, will be (2):

\(\frac{1}{T}=\frac{1}{Rag}+\frac{1}{Vis}+\frac{1}{Ven}\)

where: Rag is the time Rag needs to complete the work alone, Vis is the time Vis needs to complete the work alone, and Ven is the time Ven needs to complete the work alone.

Now, because of (1), we know that (3):

\(\frac{1}{25}=\frac{1}{Rag}+\frac{1}{Vis}\)

Therefore, we can substitute (3) on (2), to get (4):

\(\frac{1}{T}=\frac{1}{25}+\frac{1}{Ven}\)

Now, it said that Ven is faster that the other guys with long names.

This was the difficult part for me: if Rag and Vis can do the entire work in 25 days, reordering (3) this means that (5):

\(25=\frac{Rag*Vis}{Rag+Vis}\)

There are at least 5 options for this to be 25 (see attached graph):
Rag=50 and Vis=50
Rag=30 and Vis=150
Rag=26 and Vis=650
Rag=150 and Vis=30
Rag=650 and Vis=26.

But we have to analyze to choices they give us. We get (4) and solve for Ven depending on T (6):

\(Ven=\frac{25*T}{25-T}\)

If we plug T=16 (option A), we get that Ven=44 days approx.
If we plug T=18 (option B), we get that Ven=64 days approx.
If we plug T=20 (option C), we get that Ven=100 days approx. and so on

With the choices we are given, and knowing that Ven has to be the fast guy, we can only opt for the answer A, assuming that Rag and Vis take each 50 days to complete the task by their own.

Answer A

Solution:

Let us call the three men as R, VI and VE


1) Time taken by R and VI to complete 4/5 of the task is 20 days

2) VE is the fastest of the three workers

3) From (1), Time that would be taken by R and VI to complete the whole task is 25 days

4) We may assume that R and VI take 50 days each to complete the task

5) From (2) and (4) we have VE taking less than 50 days or \(1/VE > 1/50\)

6) From (4) and (5), we have , \(1/Total time = 1/R + 1/VI + 1/VE > 1/50 + 1/50+ 1/50\)

or\(1/ Total Time\) > 3/50 or Total time taken < 16.67 days.

The only choice that fits the above value is choice A.

Note: The number of days taken to complete the task cannot be greater than the above because VE being the fastest worker the maximum number of days that VE could possibly take is 49. Lesser values would lessen the total time taken.
_________________

Not a typical GMAT question IMHO !

Assume Task = Writing 50 books

In 20 days, Ra and Vi finish writing 40 books

Assuming they both work at equal speed (since we want to minimize Ve's rate), each of them writes at 1 book per day.

If Ve worked at the same rate as Ra and Vi, the rate would be 3 books per day and the task would be completed in 50/3 = 16.6 days.

Since Ve works the fastest, the work should be completed in less than 16.6 days.

A is the only option available.

Dude why complicate things and use those weird names.
KISS (Keep it simple Sravna)

Thanks
Cheers

let 1/25=rate of R+Vi
1/x=rate of Ve
d=number of days taken by all three to complete task
d(1/25+1/x)=1
x=25d/(25-d)
plugging in 16 as d, x=400/9
1/x=9/400
1/25=16/400
1/25+1/x=25/400=1/16
16 days(1/16 combined daily rate)=1 completed task
16 works as possible number of days R, Vi, and V worked together to complete task

I got confused of the wording..
does R and V need 20 days to complete 4/5 of the job or 1/5 of the job?

This one took me by surprise. Anyway, here goes:

Prelims:
\(R\): rate of Raghavachari
\(V_i\): rate of Viswanathan
\(V_e\): rate of Venkatraman

Task = rate * time (days)
\(T=r * t\)


From the narrative: \((R+V_i)*20days=\frac{4}{5}T \implies (R+V_i)=\frac{1}{25} Task/day\)


If either \(R\) or \(V_i\) does nothing, maximum \(V_e\) is \(\frac{1}{25}\)

If \(R = V_i\) (i.e. half the rate of either one), minimum \(V_e\) is \(\frac{1}{50}\)

In other words, \(\frac{1}{50}<V_e<\frac{1}{25}\)


The time it takes to complete the task:
\(t=\frac{1}{(R+V_i) + V_e}\)


Considering \(V_e\) boundaries:
\(\frac{1}{\frac{1}{25}+\frac{1}{25}} <t<\frac{1}{\frac{1}{25}+\frac{1}{50}}\)

\(\implies 12.5days <t< 16.7days\)


The only answer meeting this range is A.

I can't get my head around this one. As it received a Kudos from OP, it had to make sense. I would like to understand the successful solution, so I learn.

How does one arrives at the reciprocal relationship above?

Thanks for any help.

Hi every one,
there have been very long solutions for this Q..
But a very simple and totally valid solution would be--

All three do 4/5 of work -- this doesn't have any value apart from helping in the preceding task..
Two of them can do same amount of work in 20 days..
so 4/5th work is done in 20 days..
complete work is done by the two in 20*5/4=25 days..

Now the third is fastest, but lets take his speed the least possible..
for that the two others should be of same speed and the fastest third slightly more or nearly same..
so if two of them did the work in 25 days, each at same speed would do it in 50 days..
let the fastest do it in slightly less than 50 or say 50 itself..
so all three will do the work in 1/50 + 1/50 +1/50= 3/50 or 50/3 days = 16 2/3 days..
so the total time in no way can be more than 16 2/3..
only A is left
_________________

A - raghavachari, B - viswanathan, C - venkatraman
Given: A + B can finish 4/5 task in 20 days => so A + B can finish the whole task in 25 days
if A work at same speed as B, then A alone can finish the task in 50 days.

worst case, if C were to work at speed of A and B, then total time taken to finish by three altogether = 50/3 = 16.67 days
but given C is definitely better than A and B, then total time taken to finish must be less than 16.67, only option (A) stands.

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________