M02-07

Question

Andy, George, and Sally are a team of consultants working on Project Alpha. They have an eight-hour deadline to complete the project. The team members work at constant rates throughout the eight-hour period. If the team of three has to start work immediately and no one else can work on this project, will Project Alpha be completed by the deadline?

(1) Sally can finish the project alone in  4k+7  hours, where  k  is a positive integer ranging from 1 to 5, inclusive.
 
(2) Working alone, George will take  2k+1  hours, and Andy will take  3+2k  hours to complete the project, where  k  is a positive integer ranging from 1 to 5, inclusive.

Bunuel · Accepted Answer

Official Solution:

Andy, George, and Sally are a team of consultants working on Project Alpha. They have an eight-hour deadline to complete the project. The team members work at constant rates throughout the eight-hour period. If the team of three has to start work immediately and no one else can work on this project, will Project Alpha be completed by the deadline? 
 
(1) Sally can finish the project alone in  4k+7  hours, where  k  is a positive integer ranging from 1 to 5, inclusive.
 
We know Sally can finish the project in  4k+7  hours, but we know nothing about George's or Andy's time. If  k = 5 , Sally can finish the work by herself in 27 hours. In this worst-case scenario, Sally alone cannot complete the project within the deadline, which means we need more information about George and Andy to determine if the team as a whole can complete the project within the deadline. Suppose Andy and George also need a considerable amount of time to finish the job by themselves, say 100 hours each. In that case, the team won't be able to complete the project by the deadline, as their combined rate of  \frac{1}{27} + \frac{1}{100} +\frac{1}{100}  job/hour would be less than the required rate of  \frac{1}{8}  job/hour. On the other hand, if Andy and George need a small amount of time to finish the job by themselves, say 1 hour each, then the team will be able to complete the project by the deadline. This is because their combined rate of  \frac{1}{27} + \frac{1}{1} +\frac{1}{1}  job/hour would be greater than the required rate of  \frac{1}{8}  job/hour. Not sufficient.
 
(2) Working alone, George will take  2k+1  hours, and Andy will take  3+2k  hours to complete the project, where  k  is a positive integer ranging from 1 to 5, inclusive.
 
Let's check whether George and Andy can finish the project by themselves for the greatest possible value of  k . We do this because if they can finish the project by themselves for the greatest possible value of  k , it would mean that they don't even need Sally's input to complete the project. For the maximum value of  k = 5 , George needs  2k+1=11  hours and Andy needs  3+2k=13  hours to finish the project alone. Their combined rate is  \frac{1}{11} + \frac{1}{13}  job/hour. Since this is more than the required rate of  \frac{1}{8}  job/hour ( \frac{1}{11} + \frac{1}{13} > (\frac{1}{16} + \frac{1}{16}=\frac{1}{8})  ), George and Andy can complete the job by themselves irrespective of Sally's rate. Sufficient.

Answer: B

FightToSurvive · Answer

Deeuk,

Firstly consider the below cases:

1. say, k =5 (Maximum)
Then also, on further simplification, 1/11 + 1/13 > 1/8

2. say, k =1 (Minimum)
Then also, on further simplification, 1/3 + 1/5 > 1/8

Both cases, its sufficient to decipher that both person sufficient to do alpha within the schedule of 8 hours. Hence B is the answer to the said question.

Yes, coming to your question, yes if we could find different solution to both cases, then we would have to consider the first option, then in that case, C would have been the answer.

Hope its clear.

mpcroaming · Answer

A really easy way to look at this is as follows:

for section (2) put 1 into the equation ( the minimum work value) for George and andy and you get G = 3 and A = 5. This add up to 8 so together they can complete it in 8 hours which is already enough to meet the deadline. therefore it doesn't even matter what Sallys rate is. Not sure if this is the correct rational but it worked for me.

chetan2u · Answer

hi,

your method is wrong on two counts..
 George will take 2k+1 hours and Andy will take 3+2k hours,
1) firstly by substituting k as 1, you are taking the minimum time they will take.
so G alone can finish in 3 hrs, and he doesn't require anyone to finish before 8 hrs....
SO, you look at the max time they take to be certain that even at the lowest speed they can finish the work in <8 hrs..

2) second fault is if G can do it in 3 hrs and A can do in 5 hrs...
both combined will not do it in 3+5 hrs, but less than 3 hrs..
if some one alone was doing in 3 hrs, and he gets help, he will do it in lesser time..

Hope the concepts have become slightly clearer..

RenanBragion · Answer

I think this is a  high-quality  question and I agree with explanation.

jjt0407 · Answer

I'm confused by the question. It says all 3 of them have to work together. How is statement 2 SUFF, if it only has two of the workers?

Bunuel · Answer

From (2) it follows that George and Andy can alone finish the work in less than required 8 hours, so no matter what the rate of Sally is we can answer the question: yes, the project will be completed by the deadline.

agusadboi · Answer

I arrived at the correct answer but I used a slightly different approach, not sure if it's appropriate so I welcome any feedback

ST1: The least amount of time that Sally will need to finish the project is in 11h (1/11) , which is over 8 hours. Insufficient

ST2 :The Max Value (Slowest Rates) for Andy and George is when k = 5. Their respective rates are (1/11) and (1/13)

They have 8 hours to finish the project. Let's test if 8 hours will be sufficient.

8x (1/11 )+ 8x (1/13) = 192/143 > 1 . This means that they A & G alone can complete the entire project within the allocated 8 hours, even at their slowest rates.

Sufficient - Answer B.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

pcdr · Answer

Can someone please explain why 1/11 + 1/13 being greater than 1/8 means that they can together finish in less than 8 hours? Thank you!

Posted from my mobile device

Bunuel · Answer

The question asks whether the project can be completed in 8 hours, or in other words, whether the combined rate of the team is greater than 1/8. Since the combined rate of just two team members is already greater than 1/8, they can complete the project in under 8 hours by themselves, regardless of Sally's rate.

Hope it helps.

Gemmie · Answer

S: time needed for Sally to finish the project alone
A: time needed for Andy ...
G: time needed for George

Question: Whether \(\frac{1}{S} + \frac{1}{A} + \frac{1}{G} >= \frac{1}{8}\)

Statement 1: S = 4k+7, k can be from 1 to 5

(i) S = 4k+7, k can be from 1 to 5
=> 11 <= S <= 27
=> \(\frac{1}{27} <= \frac{1}{S} <= \frac{1}{11}\)

Even \(\frac{1}{11} < \frac{1}{8}\)

(ii) We know nothing about A and G

=> Statement 1 is insufficient

Statement 2: G = 2k + 1, A = 3 + 2k, k can be from 1 to 5

3 <= G <= 11
\(\frac{1}{11} <= \frac{1}{G} <= \frac{1}{3}\)

5 <= A < = 13
\(\frac{1}{13} <= \frac{1}{A} <= \frac{1}{5}\)

\(\frac{1}{G} + \frac{1}{A} >= \frac{1}{11} + \frac{1}{13} = 0.09 + 0.07 = 0.16 > 0.125 = \frac{1}{8}\)

=> Statement 2 is sufficient

==> Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

dnguyen1203 · Answer

I think this is a  high-quality  question and I agree with explanation.

gummallasunil · Answer

I like the solution - it’s helpful.

ki870409 · Answer

I like the solution - it’s helpful.

idlch · Answer

I like the solution - it’s helpful.

Sunny00731 · Answer

I don’t quite agree with the solution. Since K can take up any value from 1 to 5, so we cant say with certainty that it is sufficient. it will depend on k value. 
so mostly the answer should be E

Bunuel · Answer

You are wrong. In Statement (2), we check the worst case, meaning the largest possible value of k, because if George and Andy can complete the project even when k = 5 (the slowest case), then they will definitely be able to complete it for smaller values of k (faster cases). Since at k = 5 they are fast enough to meet the 8-hour deadline, the answer is fully determined. Statement (2) alone is sufficient. The correct answer is B, not E.

suman94 · Answer

I like the solution - it’s helpful.

M02-07

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