A department manager distributed a number of pens, pencils

Question

A department manager distributed a number of pens, pencils, and pads among the staff in the department, with each staff member receiving x pens, y pencils, and z pads. How many staff members were in the department?

(1) The numbers of pens, pencils, and pads that each staff member received were in the ratio 2:3:4, respectively.
(2) The manager distributed a total of 18 pens, 27 pencils, and 36 pads.

ANSWER I DID: 
BOTH statements TOGETHER are sufficient, but neither statement alone is sufficient.

As per OG12: 
(1) Each of 10 staff members could have received 2 pens, 3 pencils, and 4 pads, or each of 20 staff members could have received 2 pens, 3 pencils, and 4 pads; NOT sufficient.  Agree ]
(2) There could have been 1 staff member who received 18 pens, 27 pencils, and 36 pads, or 3 staff members each of whom received 6 pens, 9 pencils, and 12 pads; NOT sufficient. Agree ]

Assuming both (1) and (2), use the fact that 18:27:36 is equivalent to both 6:9:12 and 2:3:4 to obtain diff erent possibilities for the number of staff . Each of 3 staff members could have received 6 pens, 9 pencils, and 12 pads, or each of 9 staff members could have received 2 pens, 3 pencils, and 4 pads.

Now Here I defer . As per Point 1, its already shown that the ratio is ratio 2:3:4. So, we can clearly choose the staff number is 9.

Can anybody please help me to understand where I am making the mistake in this question?

KarishmaB · Accepted Answer

Ok, look the question tells us the following:
Staff Member 1 - x pens, y pencils, z pads
Staff Member 2 - x pens, y pencils, z pads
.
.
.
Staff Member n - x pens, y pencils, z pads
Total no of pens - nx, total no of pencils - ny and total no of pads - nz
Question: What is n?

Stmnt 1: x:y:z = 2:3:4. So values of x, y and z can be 2, 3 and 4 or 4, 6 and 8 or 6, 9 and 12 or any other values in the ratio 2:3:4. They needn't necessarily be 2, 3 and 4. Just the ratio required is 2:3:4.
Of course n can be anything here. Not sufficient.

Stmnt 2: nx = 18, ny = 27 and nz = 36.
Note here that nx:ny:nz = 18:27:36 = 2:3:4 (They had 9 as a common factor)
Since n is a common factor on left side, x:y:z = 2:3:4 (Ratios are best expressed in the lowest form.)

This is a case of what we call "We already knew that." Information given in stmnt 1 is already part of stmnt 2 so it is not possible that stmnt 2 alone is not sufficient but together stmnt 1 and 2 are.

Now to your question:
Why can't we say that the number of staff members must be 9?
Because ratio of 2:3:4 is same as ratio of 6:9:12 which is same as 18:27:36 (When you multiply each number of a ratio by the same number, the ratio remains unchanged).
If 18, 27 and 36 pens, pencils and pads are distributed in the ratio 2:3:4, I could give them all to one person (18:27:36 is the same ratio as 2:3:4), to 3 people (giving them 6 pens, 9 pencils and 12 pads each. 6:9:12 is the same ratio as 2:3:4) or to 9 people (giving them 2 pens, 3 pencils and 4 pads). Hence I don't know how many staff members are there.

Ivan91 · Answer

I got it only by using common sense. Its not tricky if you think about it .
(1) The numbers of pens, pencils, and pads that each staff member received were in the ratio 2:3:4, respectively.
This shows you how many more pads  one person  gets compared to the number of pencils or pens he gets.

(2)The manager distributed a total of 18 pens, 27 pencils, and 36 pads. 
This can give you an idea how many gadgets are there in total to distribute, but we cannot know how many people will take them.

There can be just one person taking all the 18 pens, 27 pencils, 36 pads (in the proportion of 2:3:4 as stated in (1) )
There can be three people, each taking 6 pens, 9 pencils, 12 pads (2:3:4)
There can be 9 people, each taking 2 pens, 3 pencils, 4 pads (2:3:4)

bubuja · Answer

6:9:12 relate to each other just like 2:3:4 and you get 3 member of staff. That is why you cannot find out exactly how many members tehre are 9 or 3

Patrizio · Answer

Answer to this question is E:

Start considering (2) ---> if the manager distributes a total of 18 pens, 27 pencils and 36 pads with each person receiving x pens, y pencils and z pads, means that the number of the staff has to be a factor of both 18,27 and 36. That value could be 3 or 9 --> Not sufficient

Now move to (1) ---> Clearly not sufficient as we do not have any information about the total number of persons or pens,pencils and pads distributed. We have just three ratios.

Considering Together ---> We know form (1) that the number of person should be 3 or 9. If people are 3 we have to distribute 18,27 and 36 --> 6pens, 9pencils, 12 pads for each person. If people are 9 we distribute 2 pens, 3 pencils, 4 pads. In both cases the ratio is the same 2:3:4. Hence not Sufficient.

Hope it's clear.

By the way AMAZING FORUM!

Sachin9 · Answer

Karishma,

Thanks for your reply. I have the following doubt..

How do I learn to recognize that there can be multiple instances of sets who have the same ratio..

2:3:4 is the ratio of distribution.

18, 27 and 36 are the actuals distributed..

9 was the first no that came to my mind..

I then multiplied 2 to the ratio to get 4:6:8 and the actuals cannot be distributed in this ratio and I picked the answer to be sufficient...

had I multiplied the ratio by 3, I would have got 6:9:12 and I would have known that the answer is insufficient..

Please help me understand and learn thought process..

KarishmaB · Answer

When you saw 18, 27 and 36, what came to your mind was that the number of people could have been 9 which would mean that he gave 2 pens, 3 pencils and 4 pads. You know that 9 is divisible by 3. That should make you realize that the number of people could have been 3 too which would mean that the manager distributed 6 pens, 9 pencils and 12 pads. 9 does not have 2 as a factor so it will not work.

sanjaykvbsingh · Answer

I think answer is B. the explanation from the people considering the answer as E is missing the point that option b also tells the total. Hence don't just conclude that b provide the same information as a ( ratio), why are you not considering the total provided in b and the fact that each employee receives the pen, pencils,.. In same ratio.

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Bunuel · Answer

The correct answer is E.

There can be two cases:
There can be 9 managers each receiving 2 pens, 3 pencils and 4 pads.
There can be 3 managers each receiving 6 pens, 9 pencils and 12 pads.

Does this make sense?

honchos · Answer

Karishma I have a doubt here:

NX:NY:NZ = 18:27:36

N  = 9

Karishma I agree that N is unknown, but on Right hand side when we reduced to the lowest ration, 9 is the only viable possibility. Hence N=9, May be I am missing something will need your insight.

KarishmaB · Answer

If N is 9, 
X = 2, Y = 3 and Z = 4
There were 9 staff members and each got 2 pens, 3 pencils and 4 pads.
Ratio of pens:pencils:pads = 2:3:4

But if N is 3,
X = 6, Y = 9 and Z = 12
There were 3 staff members and each got 6 pens, 9 pencils and 12 pads.
Ratio of pens:pencils:pads = 2:3:4

N could even take the unlikely value of 1
X = 18, Y = 27 and Z = 36
There was only 1 staff member and he/she got 18 pens, 27 pencils and 36 pads.
Ratio of pens:pencils:pads = 2:3:4

cfpenteado · Answer

Could we also consider the case of 1 employee getting all the 18 pens, 27 pencils and 36 pads ?

quantified · Answer

Hi,

Do you know where I can find more DS questions of Ratio type. Exactly or close to this? Thanks for your help.

Bunuel · Answer

3. Fractions, Decimals, Ratios and Proportions  
     Theory   
 GMAT Shortcut: Adding to the Numerator and Denominator 
 Rounding Rules on the GMAT: Slip to the Side and Look for a Five! 
 How to Identify Terminating Decimals on the GMAT 
 Rounding Up Some Official GMAT Questions! 
 Terminating Decimals in Data Sufficiency on the GMAT 
 GMAT Quantitative: Ratio and Proportions 
 Ratios in GMAT Data Sufficiency 
 Why You Should Convert Fractions to Decimals on the GMAT 
 Use This Strategy for Fractions and Save Time on GMAT Test Day 
 Variations On The GMAT - All In One Topic 
     Questions   
 Terminating and Recurring Decimals 
 Units digits, exponents, remainders problems 
 DS Questions 
 PS Questions

Check below for more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope it helps.

TheUltimateWinner · Answer

VeritasKarishma 
Thanks for the nice explanation with kudos.
In the highlighted part, the question ask about member (S) . So, can we limit the possibility of  one person ? I mean: Is it possible to distribute to  one member  as the question stem used the plural sign (memberS)?
Thanks__

KarishmaB · Answer

Yes, number of members can be 1. We will use plural because we don't know the actual number. The only constraint is that since he does distribute it to members, number of members cannot be 0.

It's like - How many people are in the room?
A valid answer is 1 person.

BrentGMATPrepNow · Answer

Given : A department manager distributed a number of pens, pencils, and pads among the staff in the department, with each staff member receiving x pens, y pencils, and z pads.

Target question :    How many staff members were in the department?

Statement 1 : The numbers of pens, pencils, and pads that each staff member received were in the ratio 2:3:4, respectively.  
2 + 3 + 4 = 9
So, the TOTAL number of pens, pencils, and pads each worker receives is a multiple of 9.
There's no way we can use this information to determine the number of workers
Statement 1 is NOT SUFFICIENT

Statement 2 : The manager distributed a total of 18 pens, 27 pencils, and 36 pads. 
 Important: Notice that 18 pens, 27 pencils, and 36 pads is in the same ratio as noted in statement 1. So, statement 2 doesn't seem to be adding a whole lot of new information. So I'm already thinking that statement 2 it's not sufficient. Let's see if we can find two conflicting cases that satisfy statement 2  
 Case a : It could be the case that there's 1 worker, and that worker receives 18 pens, 27 pencils, and 36 pads. In this case, the answer to the target question is   the department has 1 staff member 
 Case b : It could be the case that there are 3 workers, and each worker receives 6 pens, 9 pencils, and 12 pads. In this case, the answer to the target question is   the department has 3 staff members 
Since we can’t answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
 IMPORTANT: Notice that the  same counter-examples  I used to show that statement 2 is not sufficient also satisfy the conditions stated in statement 1. So, the same counter-examples will satisfy the two statements COMBINED.  
In other words, 
 Case a : It could be the case that there's 1 worker, and that worker receives 18 pens, 27 pencils, and 36 pads. In this case, the answer to the target question is   the department has 1 staff member 
 Case b : It could be the case that there are 3 workers, and each worker receives 6 pens, 9 pencils, and 12 pads. In this case, the answer to the target question is   the department has 3 staff members 
Since we can’t answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

Kavicogsci · Answer

Absolutely cool Q!
Explaining a mistake I did so that others could be alerted

So i said as each person gets x pens, y pencils and z pads let total people be N so,
Nx be total pens
Ny be total pencils
Nz be total pads

A. Only gives us RATIOS that each person gets it in ratio of 2:3:4
The only inference to be drawn here is that the number of pads are more than pencils and number of pencils are more than pads. DO NOT assume x=2, y=3 and z=4.

B. Total qty is given
Nx = 18
Ny = 27
Nz = 36

Now,
See common factors = 1,3,9

1*18
1*27
1*36
x=18, y=27, z=36
x:y:z = 18:27:26 = 2:3:4
N=1

3*6
3*9
3*12
x=6, y=9, z=12
x:y:z = 6:9:12 = 2:3:4
N=3

9*2
9*3
9*4
x=2, y=3, z=4
x:y:z = 2:3:4 
N=9

Not sufficient

Combine A& B not sufficient

When I was combining I assumed x=2, y=3 and z=4 which was wrong because x,y,z were the actual number that was divided and we cannot get there just basis ratios cause infinite combinations of 3 numbers can end up with the same ratio.

Usernamevisible · Answer

Let n be the number of staff members, and let each person receive x pens, y pencils, and z pads. Then the totals are nx, ny, and nz.

Statement (1): x : y : z = 2 : 3 : 4. This gives only a ratio, not actual quantities. Many values of n are possible, so the statement is insufficient.

Statement (2): The totals are 18, 27, and 36. The number of staff members could be any common divisor of these numbers. For example, n = 9 gives (2, 3, 4) per person, while n = 3 gives (6, 9, 12) per person. Insufficient.

Together: Since 18 : 27 : 36 = 2 : 3 : 4, Statement (1) adds no new restriction. We get nk = 9, which has multiple integer solutions such as (n, k) = (9, 1), (3, 3), and (1, 9). Therefore, n is not uniquely determined.

Answer: E. Both statements together are insufficient.

A department manager distributed a number of pens, pencils

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A department manager distributed a number of pens, pencils

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