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Inequality & Word Problem: Gift Wrapping

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Inequality & Word Problem: Gift Wrapping

by beeparoo » Thu Jul 24, 2008 7:27 pm
At a certain department store present-wrapping counter, each clerk will wrap no fewer than 20 and no more than 30 presents per hour. If seventy people are waiting in line, will all their presents be wrapped after one hour?

1) Each person in line has at least one present to be wrapped by one of the six clerks at the counter.

2) If each person in line had one more present to be wrapped, nine clerks would be required to guarantee that every gift would be wrapped in one hour.

--

Had some difficulty agreeing with the explanation to this problem, so I'm curious to see how others treat it. Will happily post OA shortly.
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Re: Inequality & Word Problem: Gift Wrapping

by Ian Stewart » Fri Jul 25, 2008 4:47 am
beeparoo wrote:At a certain department store present-wrapping counter, each clerk will wrap no fewer than 20 and no more than 30 presents per hour. If seventy people are waiting in line, will all their presents be wrapped after one hour?

1) Each person in line has at least one present to be wrapped by one of the six clerks at the counter.

2) If each person in line had one more present to be wrapped, nine clerks would be required to guarantee that every gift would be wrapped in one hour.
(1) is not sufficient: there might be exactly 70 gifts to wrap, or there might be 7,000,000 gifts to wrap.

(2) is not sufficient: there might be no clerks at all at the counter, or there might be 1,000,000 clerks.

(1)+(2) together: Well, what does (2) tell us? If we need to guarantee that every gift could be wrapped, we'd need to assume the clerks were as slow as possible. That is, we need to assume they wrap 20 gifts per hour, not 30 gifts per hour. If there were 160 gifts or less, we could be sure that 8 clerks could do the job. If there were more than 180 gifts, and all the clerks are slow, then we might need 10 clerks to do the job. So, from (2), we know that if each person in line had one more gift, there would be between 161 and 180 gifts in total. Since there are 70 people in line, and (2) assumes each person has one more gift than they actually have, there are actually between 91 and 110 gifts in total (subtract 70). We have six clerks, from Statement (1), and we know that six clerks can wrap at least 120 gifts in an hour. C.

I'm interested in the source of the question- it doesn't strike me as a real GMAT question, because of the phrase about 'guaranteeing' the work gets done in Statement 2.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by beeparoo » Fri Jul 25, 2008 8:33 am
Hi Ian!

Bless your little mathematical heart, you are such an institution on this site.

OK, the OA is C afterall <- For those of you dying to know the answer. : P
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Re: Inequality & Word Problem: Gift Wrapping

by beeparoo » Fri Jul 25, 2008 9:05 am
Ian Stewart wrote:(1) is not sufficient: there might be exactly 70 gifts to wrap, or there might be 7,000,000 gifts to wrap.

(2) is not sufficient: there might be no clerks at all at the counter, or there might be 1,000,000 clerks.

(1)+(2) together: Well, what does (2) tell us? If we need to guarantee that every gift could be wrapped, we'd need to assume the clerks were as slow as possible. That is, we need to assume they wrap 20 gifts per hour, not 30 gifts per hour. If there were 160 gifts or less, we could be sure that 8 clerks could do the job. If there were more than 180 gifts, and all the clerks are slow, then we might need 10 clerks to do the job. So, from (2), we know that if each person in line had one more gift, there would be between 161 and 180 gifts in total. Since there are 70 people in line, and (2) assumes each person has one more gift than they actually have, there are actually between 91 and 110 gifts in total (subtract 70).

Hey, um, how did you get your values for total gifts per hr? My range calculation is between 110 to 200 gifts per hour (the required rate to guarantee, say, 'customer satisfaction'.

Min rate: 20 presents per clerk per hr [or p/c/hr]
Max rate: 30 p/c/hr

20 p/c/hr </= (Total + 70 presents)/(9 clerks•hr) </= 30 p/c/hr
110 p/hr </= Total presents/hr </= 200 p/hr

[Where '</=' means less than or equal to]

We have six clerks, from Statement (1), and we know that six clerks can wrap at least 120 gifts in an hour. C.

I'm interested in the source of the question- it doesn't strike me as a real GMAT question, because of the phrase about 'guaranteeing' the work gets done in Statement 2.
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Re: Inequality & Word Problem: Gift Wrapping

by Ian Stewart » Fri Jul 25, 2008 9:58 am
beeparoo wrote: Hey, um, how did you get your values for total gifts per hr? My range calculation is between 110 to 200 gifts per hour (the required rate to guarantee, say, 'customer satisfaction'.

Min rate: 20 presents per clerk per hr [or p/c/hr]
Max rate: 30 p/c/hr

20 p/c/hr </= (Total + 70 presents)/(9 clerks•hr) </= 30 p/c/hr
110 p/hr </= Total presents/hr </= 200 p/hr

[Where '</=' means less than or equal to]
Yes, this is where things get a bit confusing. Statement 2) tells us about what situation we'd have if there were 70 more gifts (one more per person), so we'll subtract 70 in the end. Now, imagine you're running a big department store, and you know that your worst employees wrap 20 gifts an hour, and your best employees wrap 30 gifts an hour. Say someone asks you to absolutely guarantee that 161 gifts will be wrapped in one hour. What assumptions should you make? Well, if you need to be absolutely sure the work will get done, you should assume that your worst employees are working- we need to be absolutey sure the job gets finished. So we should assume that everyone at the counter wraps 20 gifts per hour. If you need to wrap 161 gifts, and you only have 8 employees, there's a chance they'll only wrap 160 gifts, and you will not succeed in wrapping every gift. With nine employees, even if they're as bad as they can be, you could be certain to wrap up to 180 gifts in an hour. Because the statement says that nine employees are required, I assumed that 8 employees would not, at least in the worst case, be enough- and that there would need to be between 161 and 180 gifts in total. Then, subtracting 70, we get the 91 to 110 range.

I don't much like the question, though- it reads like it was originally an algebra problem that someone has tried to convert into a word problem, and in the translation, it has become awkward and subject to misinterpretation. That's why I was curious about where the question came from. It doesn't seem like a real GMAT question, and if it isn't, I wouldn't worry about it too much!
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by anandr84 » Tue Nov 03, 2009 4:41 am
its a Kaplan CAT question....nice explanation Ian. Thanks
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