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19 Jan 2011, 00:03
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This is the paraphrased question from gmatprep v1:
Each employee of Company C is either from Division X or Division Y. Each division has full-time and part-time employees. Is the ratio of full-time to part-time employees greater for Division X than it is for Company C?
I. The ratio of full-time to part time employees for Division Y is less than that of Company C.
II. More than half of full-time employees are in Division X and more than half of part-time employees are from Division Y.
I got the correct answer by picking numbers. I was wondering however if there was a more "mathematical" (dealing with variables) approach to justify the answer.
Thank you!
Each employee of Company C is either from Division X or Division Y. Each division has full-time and part-time employees. Is the ratio of full-time to part-time employees greater for Division X than it is for Company C?
I. The ratio of full-time to part time employees for Division Y is less than that of Company C.
II. More than half of full-time employees are in Division X and more than half of part-time employees are from Division Y.
I got the correct answer by picking numbers. I was wondering however if there was a more "mathematical" (dealing with variables) approach to justify the answer.
Thank you!
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This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Data Sufficiency (DS) Forum
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20 Jan 2011, 11:56
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I wouldn't advise doing this with variables; I'd advise doing it with logic. The Ratio in Company C is the weighted average of the ratios of Division X and Division Y. Therefore, its ratio must be between the X and Y ratios. That is, if Division X has a ratio bigger than Division Y, the ratio of C overall must be between those two (not necessarily half-way between, since the average is weighted, but definitely between them).
Thus, Statement (1) tells us that Y has the "smaller" ratio of ft:pt. This means that X must have the "bigger" ratio, which is all that they're asking us. Sufficient.
Statement (2), meanwhile, tells us that X has the bigger chunk of full-time employees and that Y has the bigger chunk of part-time employees. Thus, again, we know that X must have the "bigger" of the two ft:pt ratios, and thus must again be "bigger" than the weighted average of C overall. Sufficient.
Mathematically, it's awfully ugly, because it's all about inequalities and no actual equations. So, I do not recommend that anyone do this problem with algebra. Nevertheless, to satisfy everyone's curiosity (including, I freely admit, my own), I'll give it a try:
Let a=part-time employees in Division X; b=part-time employees in Division Y; c=full-time employees in Division X; d=full-time employees in Division Y.
The ratios, therefore, are c/a and d/b for the individual divisions, and, in Company C overall, it would be:
(c + d) / (a + b) note: this is the one truly useful line of this entire mathematical equation, because it explains how to properly calculate a weighted ratio.
The question is, "Is c/a > c+d / a+b ?"
Let's play around with this a little. Remember, when cross-multiplying over an inequality, two things must be kept in mind: 1) ONLY do this if you know if the variables are + or - (here we know everything's positive because we're talking about numbers of people), and 2) Keep the products on the same side as the NUMERATORS when cross-multiplying.
Is c/a > c+d / a+b ?
Is c(a+b) > a(c+d)?
Is ac + bc > ac + ad?
Is bc > ad?
That's a much simpler question. Let's see how the Statements work:
(1) says that d/b < c+d / a+b. Following the same mathematics as above, we get:
d/b < c+d / a+b
d(a+b) < b(c+d)
ad + bd < bc + bd
ad < bc
This is exactly what we're asked about, so, it's Sufficient!
Statement (2) talks about "more than half," but the best way to phrase it so that it will match what we're already working with is to put it in terms of inequalities. Really, what it means is that there are more full-time employees in Division X than in Division Y (because that's what "more than half" implies), and vice-versa for part-time employees. Thus, using our variables above:
c > d and b > a
As a result, given that all of our numbers are positive, it's easy to see that, in fact, bc > ad. Again, this is what we're asked about, so, Sufficient.
***
After doing it, I admit it wasn't that hard, except for the big deduction on how to properly write out and include Statement (2). Nevertheless, it's VERY work-heavy, very open to major errors, and logically seeing through the problem (or plugging in numbers, as karenhipol did) will serve you far better on far more questions on Test Day.
Thus, Statement (1) tells us that Y has the "smaller" ratio of ft:pt. This means that X must have the "bigger" ratio, which is all that they're asking us. Sufficient.
Statement (2), meanwhile, tells us that X has the bigger chunk of full-time employees and that Y has the bigger chunk of part-time employees. Thus, again, we know that X must have the "bigger" of the two ft:pt ratios, and thus must again be "bigger" than the weighted average of C overall. Sufficient.
Mathematically, it's awfully ugly, because it's all about inequalities and no actual equations. So, I do not recommend that anyone do this problem with algebra. Nevertheless, to satisfy everyone's curiosity (including, I freely admit, my own), I'll give it a try:
Let a=part-time employees in Division X; b=part-time employees in Division Y; c=full-time employees in Division X; d=full-time employees in Division Y.
The ratios, therefore, are c/a and d/b for the individual divisions, and, in Company C overall, it would be:
(c + d) / (a + b) note: this is the one truly useful line of this entire mathematical equation, because it explains how to properly calculate a weighted ratio.
The question is, "Is c/a > c+d / a+b ?"
Let's play around with this a little. Remember, when cross-multiplying over an inequality, two things must be kept in mind: 1) ONLY do this if you know if the variables are + or - (here we know everything's positive because we're talking about numbers of people), and 2) Keep the products on the same side as the NUMERATORS when cross-multiplying.
Is c/a > c+d / a+b ?
Is c(a+b) > a(c+d)?
Is ac + bc > ac + ad?
Is bc > ad?
That's a much simpler question. Let's see how the Statements work:
(1) says that d/b < c+d / a+b. Following the same mathematics as above, we get:
d/b < c+d / a+b
d(a+b) < b(c+d)
ad + bd < bc + bd
ad < bc
This is exactly what we're asked about, so, it's Sufficient!
Statement (2) talks about "more than half," but the best way to phrase it so that it will match what we're already working with is to put it in terms of inequalities. Really, what it means is that there are more full-time employees in Division X than in Division Y (because that's what "more than half" implies), and vice-versa for part-time employees. Thus, using our variables above:
c > d and b > a
As a result, given that all of our numbers are positive, it's easy to see that, in fact, bc > ad. Again, this is what we're asked about, so, Sufficient.
***
After doing it, I admit it wasn't that hard, except for the big deduction on how to properly write out and include Statement (2). Nevertheless, it's VERY work-heavy, very open to major errors, and logically seeing through the problem (or plugging in numbers, as karenhipol did) will serve you far better on far more questions on Test Day.
20 Jan 2011, 12:44
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Thanks Adam!
I am happy that you were able to satisfy your curiosity and mine, and of course everyone else.
This problem shows that one should really know all the possible approaches. Due to constraints of time though, one should know when one should just go for the "good enough" solution - but still logical and sound, instead of using "solid math" -equations, etc, which will definitely take more time.
Karen
I am happy that you were able to satisfy your curiosity and mine, and of course everyone else.
This problem shows that one should really know all the possible approaches. Due to constraints of time though, one should know when one should just go for the "good enough" solution - but still logical and sound, instead of using "solid math" -equations, etc, which will definitely take more time.
Karen
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Data Sufficiency (DS) Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
