Working together, Rafael and Salvador can tabulate a certain set of da

Let 'r' and 's' are the no of hours taken by Rafael & Salvador, working alone, to complete the job respectively.

Given, \(\frac{1}{r}+\frac{1}{s}=\frac{1}{2}\)-------------(1)

Question stem:- r=?

St1:- Working alone, Rafael can tabulate the data in 3 hours less time than Salvador, working alone, can tabulate the data.
Or, r=s-3 Or, s=r+3
from(1), we have, \(\frac{1}{r}+\frac{1}{s}=\frac{1}{2}\)
Now substituting, \(\frac{1}{r}+\frac{1}{(r+3)}=\frac{1}{2}\)
Or, \(r^2-r-6=0\)
Or, (r+2)(r-3)=0
Or, r=3
Sufficient.

St2:- Working alone, Rafael can tabulate the data in 1/2 the time that Salvador, working alone, can tabulate the data.
Or, \(r=\frac{s}{2}\) Or, \(s=2r\)
from(1), we have, \(\frac{1}{r}+\frac{1}{s}=\frac{1}{2}\)
Now substituting, \(\frac{1}{r}+\frac{1}{2r}=\frac{1}{2}\)
Or, 2r=6
Or, r=3
Sufficient.

Ans. (D)
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Information given:

Working together, Rafael and Salvador can tabulate a certain set of data in 2 hours.

Question:

In how many hours can Rafael tabulate the data working alone?

This question may seem complicated, but the truth is that since we know how long it takes for the two of them to complete the job, if we have any information that we can use to accurately compare how fast one works to how fast the other works, then we can answer the question.

Statement 1: Working alone, Rafael can tabulate the data in 3 hours less time than Salvador, working alone, can tabulate the data.

We do not need to use any math to determine whether this statement is sufficient. We have only to notice that we know how long it takes for both of them to do the work together and that we have information that we can use to determine their relative speeds. With that information, we could work our way to the answer to the question, how many hours Rafael would take to do the job alone.

To see why, consider this:

Time for Rafael to do the job = R

Time for Salvador to do the job = R + 3

We are now working with one variable, and we have the time it takes for them to do the job together. So, we could work our way to Rafael's time.

Sufficient.

Statement 2: Working alone, Rafael can tabulate the data in 1/2 the time that Salvador, working alone, can tabulate the data.

This tells us that Rafael works twice as fast as Salvador works. Since we know how long they take to do the job together, we can clearly work our way to Rafael's speed and then his time to do the job himself.

To see why, consider this:

Rafael's speed = 1/r

Salvador's speed = 1/2r

We are now working with one variable, and we have the time it takes for them to do the job together. So, we could work our way to Rafael's time, but we won't because this is a DS question.

Sufficient.

The correct answer is D.
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How did you transform \(\frac{1}{(r+3)}+\frac{1}{r}=\frac{1}{2}\) into \(r^2-r-6=0\) ? PKN

Hi redwood,
You must know the addition and subtraction of fractional expressions.

Here, \(\frac{1}{(r+3)}+\frac{1}{r}=\frac{1}{2}\)

Or, \(\frac{[1*r+1*(r+3)]}{(r+3)r}=\frac{1}{2}\)
Or, \(\frac{(2r+3)}{(r^2+3r)}=\frac{1}{2}\)
Cross-multiplying, We have
\(2(2r+3)=r^2+3r\)
Or, \(4r+6=r^2+3r\)
Or, \(r^2-r-6=0\)

Hope it helps.
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PKN Thanks a lot, got it now!

Thanks. OG had a very confusing explanation. This is quite clear, and straightforward to follow.

Hi Marty,

I've solved this with similar logic and didn't solve the equation. Although, I knew that this equation will come out to be a quadratic which might have both imaginary roots or 2 positive roots.
So, my question is about the assumption that Gmat will give a true scenario in statements which will for sure have one positive root only that will be an answer. Thus, we don't need to check if the equation has only 1 positive root.
Please, let me know if this assumption is wrong.

Regards,
Pankaj

Hi Pahkaj.

If a problem Solving question could be solved via the use of a quadratic, the solution will have only one root that works. Of course, Problem Solving questions have to work that way. Otherwise, they would not make sense, as a problem solving question has to have only one correct answer.

However, in answering a Data Sufficiency question, you can't assume that there will be only one root that works. It could be that there are multiple roots that work and that a statement or both statements are, therefore, insufficient.
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Hello guys ,
lets say time needed for Salvador=x and time needed for Rafael=x-3 so
1/x+1/x-3=1/2 solving this I get x^2-7x+6=0 so x could be 1 or 6 and we reject 1 because 1-3=negative number.
If I had chosen to set Rafael's time x and Salvador's x+3 i would end up with a quadratic that would give me one answer choice (with the first approach I almost lost the question because i thought that I had 2 valid options for A) . What I want to ask is the following, is there a way to choose beforehand the option that will require the least additional steps at least in this type of questions?

MartyTargetTestPrep How can we tell that we'll have just have 1 positive value and not 2 positive values from statement 1 without forming the quadratic and doing the steps shown below?

1/R + 1/(R+3) = 1/2
(R+3 + R)/(R(R+3)) = 1/2
4R + 6 = R^2 + 3R
R^2 - R - 6 = 0
No need to solve further since C is negative, implying we'll have 2 roots: one + and one -. Since only + value is allowed, sufficient.

Logic dictates that, if we have the length of time it takes for them to complete the task working together and also information that we can use to determine their relative rates, only one result is possible. There is no way in which the information provided could generate two rates.

Think about it. If we have the time it takes for them to do the job together, and we know that one is faster than the other, and we know how much faster, we at that point know that only one rate set of rates will work. There's just no way that there would be two different results. How would there be more than one? The relative rates have been defined and the total rate has been defined. There's only one way in which all are going to work with each other.
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Question: To fill an order, a manufacturer had to produce 1000 tools per day for n days. what is the value of n?

Lets take a look at the statements:

(1) Because of production problems, the manufacturer produced only 600 tools per day during the first 5 days

600 * 5 = 3000 tools

We know the manufacturer needed to produce 5,000 tools; thus, the manufacturer was 2,000 tools short over the span of the first 5 days. However, we can't determine the value of n. Insufficient.

(2) Because of production problems, the manufacturer had to produce 1,500 tools per day on each of the last 4 days in order to meet the schedule

1,500 * 4 = 6,000

We know the manufacturer needed to produce 4,000 tools over the span of 4 days; thus, the manufacturer produced 2,000 more tools to compensate for earlier days. However, we still can't determine the value of n. Insufficient.

(1&2) From the two statements combined, we know the manufacturer made 9,000 tools. However, we still don't know the number of days the manufacturer worked. There could have been days where the manufacturer made 1,000 tools a day. We can only conclude that the manufacturer worked at least 9 days. Insufficient.

Answer is E.
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They both do in 1 hour =1/2 job

(1) Rafael does in \(x\) hours, then Salvador does \(x+ 3;\) So, \(\frac{1}{x}+\frac{1}{x+3}=\frac{1}{2};\) Sufficient.

(2) Rafael x, Salvador 2x, So, \(\frac{1}{x}+\frac{1}{2x}=\frac{1}{2}\) Sufficient.

The answer is \(D\)
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Hi MartyTargetTestPrep - For S1 - You mention, just because there is one variable, you know this statement is sufficient (without pen to paper and no math calculation)

If the Rafael time's is t and Salvador's time is t+3 for example

How are you so sure - there is only ONE value of T and not multiple values of T ?

T after all does not have an integer constraint and can be any positive value over zero.

Think about it.

If Rafael's rate is 1/T and Salvadore's rate is 1/(T + 3), and they are both postiive and must add up to 1/2, how would there be different postiive numbers that would work?

We also know at this point that Rafael's rate is greater since 1/T > 1/(T+3).

Without calculating the rates, we can see that there is specific relationship between them. Only one set up numbers will add up to 1/2 and fit those other constraints.
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Hi MartyTargetTestPrep

The equation that I got from the question stem is (R+S) * 2 = 1, in which I assumed R and S to be the rates of Rafael and Salvador.

Can you please help me understand how people are getting 1/R + 1/S = 1/2? I think I have a conceptual gap here.

That equation makes sense, since the question says, "Working together, Rafael and Salvador can tabulate a certain set of data in 2 hours."


The R and S in that equation are not their rates. They are the numbers of hours they take to complete the job alone.

So, in that equation, 1/R is Rafael's rate alone, and 1/S is Salvador's rate alone, which add up to 1/2, since together they take two hours to tabulate one set.
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ThatDudeKnows avigutman I have the same doubt. MartyTargetTestPrep

gmexamtaker1 and Elite097 no, I can't think of a way to know ahead of time whether it's better to go with x & (x+3) or (x-3) & x.
Having said that, in my humble opinion we shouldn't put pen to paper at all for this question. In a 2-minute-per-question kind of test, this question should be solved conceptually. We know how long the job takes them together, and with statement (1) on its own, if we were to draw a number line, we could plot the amounts of time each would take on his own as two tick marks with 3 hours in between. The question is, essentially, can those two tick marks be moved up or down the number line without changing the amount of time they take to do the job together (since that's already given)? The answer is no: if we push the tick marks to the right, the job is going to take them longer if they were to work together, and vice versa if we push the tick marks to the left. Therefore, knowing the delta in their individual times is sufficient to answer the question.
I would evaluate statement (2) conceptually as well. If you find yourself building a quadratic equation on the GMAT, it's likely that you're doing something inappropriate.