set theory

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uptowngirl92: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 19, 2009 9:08 pm; Location: Kolkata,India; Thanked: 7 times; GMAT Score:670

set theory

by uptowngirl92 » Tue Oct 06, 2009 12:12 am

Among 200 people 56% like strawberry jam, 44% like apple jam, and 40% like raspberry jam. If 30% of people like both strawberry and apple jam, what is the largest possible number of people who like raspberry jam but do not like either strawberry or apple jam?

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Source: Beat The GMAT — Problem Solving | Tue Oct 06, 2009 12:12 am

asamaverick: Master | Next Rank: 500 Posts; Posts: 106; Joined: Mon Sep 28, 2009 9:29 am; Location: Boston, MA; Thanked: 26 times; Followed by:2 members; GMAT Score:700

Re: set theory

by asamaverick » Tue Oct 06, 2009 7:18 am

Strawberry 56%
Apply 44%
Strawberry & Apple 30%

Maximizing Raspberry eaters would mean assuming that the above bunch does not include any Raspberry eaters. With that assumption we can boil down to
Strawberry only 26%
Apply Only 14%
Str & App 30%
Total 70%
Remaining is 30%, which is 60 people. So IMO answer is 60.

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mehravikas: Legendary Member; Posts: 1161; Joined: Mon May 12, 2008 2:52 am; Location: Sydney; Thanked: 23 times; Followed by:1 members

Re: set theory

by mehravikas » Tue Oct 06, 2009 12:49 pm

Why it could not be 80.

Raspberry + Str = 0
Raspberry + apple = 0

Raspberry = 80

asamaverick wrote:Strawberry 56%
Apply 44%
Strawberry & Apple 30%

Maximizing Raspberry eaters would mean assuming that the above bunch does not include any Raspberry eaters. With that assumption we can boil down to
Strawberry only 26%
Apply Only 14%
Str & App 30%
Total 70%
Remaining is 30%, which is 60 people. So IMO answer is 60.

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

Re: set theory

by Stuart@KaplanGMAT » Tue Oct 06, 2009 1:50 pm

mehravikas wrote:Why it could not be 80.

Raspberry + Str = 0
Raspberry + apple = 0

Raspberry = 80

It can't be 80 because you'd have more than 100% of 200 people.

Yes, 40% of the 200 like Raspberry (I'm assuming you got 80 by simply taking 40% of 200). However, there are people who like other jams as well.

asamaverick explained the math very well, but let's look at it again.

In general, when we're asked to maximize one thing in a GMAT question, we want to minimize everything else.

In this question, to maximize the number of people who like just raspberry, we need to minimize the number of people who like strawberry and/or apple PLUS raspberry.

Here's what we know about Strawberry/Apple:

112 people like Strawberry
88 people like Apple.
60 people like both of them.

Since only 60 people like both of them, this means that:

52 people like only strawberry;
28 people like only apple; and
60 people like both.

That's already 140 people. We only started with 200, so the maximum possible number of people who could dislike both apple and strawberry is 60.

Stuart Kovinsky | Kaplan GMAT Faculty | Toronto

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uptowngirl92: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 19, 2009 9:08 pm; Location: Kolkata,India; Thanked: 7 times; GMAT Score:670

by uptowngirl92 » Tue Oct 06, 2009 10:24 pm

Stuart,
I have drawn a venn diagram.Please see where my error lies:
We need to find the shaded area= 80-x-y-z =??

Applying formula:
200=112+88-60+80-y-z+x
80-y-z+x=60
Hmm..Should'nt that be "-x"...

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Tue Oct 06, 2009 11:27 pm

uptowngirl92 wrote:Stuart,
I have drawn a venn diagram.Please see where my error lies:
We need to find the shaded area= 80-x-y-z =??

Applying formula:
200=112+88-60+80-y-z+x
80-y-z+x=60
Hmm..Should'nt that be "-x"...

Hi!

For 3 groups, the generic formula is:

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) - 2(# in 1/2/3)

Applying it to your diagram:

200 = 112 + 88 + 80 - 60 - y - z - 2x

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mehravikas: Legendary Member; Posts: 1161; Joined: Mon May 12, 2008 2:52 am; Location: Sydney; Thanked: 23 times; Followed by:1 members

by mehravikas » Wed Oct 07, 2009 12:49 am

Hi Stuart,

Just checking...the formula should be -

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) - 2(# in 1/2/3)

or

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) + (# in 1/2/3)

Thanks,
Vikas Mehra

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Wed Oct 07, 2009 1:08 am

mehravikas wrote:Hi Stuart,

Just checking...the formula should be -

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) - 2(# in 1/2/3)

or

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) + (# in 1/2/3)

Thanks,
Vikas Mehra

It's the way I originally posted it, I promise!

There's a different version of the formula in which you add:

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in at least 1/2) - (# in at least 1/3) - (# in at least 2/3) + (# in 1/2/3)

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mehravikas: Legendary Member; Posts: 1161; Joined: Mon May 12, 2008 2:52 am; Location: Sydney; Thanked: 23 times; Followed by:1 members

by mehravikas » Wed Oct 07, 2009 1:23 am

Can you please give an example where the second formula can be used?

Stuart Kovinsky wrote:
mehravikas wrote:Hi Stuart,

Just checking...the formula should be -

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) - 2(# in 1/2/3)

or

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) + (# in 1/2/3)

Thanks,
Vikas Mehra
It's the way I originally posted it, I promise!

There's a different version of the formula in which you add:

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in at least 1/2) - (# in at least 1/3) - (# in at least 2/3) + (# in 1/2/3)

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Wed Oct 07, 2009 10:12 am

mehravikas wrote:Can you please give an example where the second formula can be used?

We can always use either formula, it really just depends on the information we're given.

At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry and English. 53 students study math, 88 study chemistry and 58 study english. If 6 students take all 3 classes, how many take exactly 2 classes?

In this case, we'd use the first formula, since we want the number who take exactly 2 classes:

150 = 53 + 88 + 58 - (doubles) - 2(triples)

150 = 199 - (doubles) - 2(6)

150 = 187 - doubles

doubles = 37

Let's just change the question a tiny bit:

At a certain school, each of the 150 students takes between 1 and 3 classes. The 3 classes available are Math, Chemistry and English. 53 students study math, 88 study chemistry and 58 study english. If 6 students take all 3 classes, how many take at least 2 classes?

In this case, we'd use the second formula, since we want the number who take at least 2 classes:

150 = 53 + 88 + 58 - (at least 2 of the 3) + (all 3)

150 = 199 - (at least 2 of 3) + 6

150 = 193 - (at least 2 of 3)

At least 2 of 3 = 43

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papgust: Community Manager; Posts: 1537; Joined: Mon Aug 10, 2009 6:10 pm; Thanked: 653 times; Followed by:252 members

by papgust » Wed Oct 07, 2009 9:53 pm

Thanks Stuart!! That really helped me a lot! I was never aware of these formulae.

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uptowngirl92: Master | Next Rank: 500 Posts; Posts: 447; Joined: Sun Apr 19, 2009 9:08 pm; Location: Kolkata,India; Thanked: 7 times; GMAT Score:670

by uptowngirl92 » Thu Oct 15, 2009 3:07 am

I am absolutly confused!!
I have always always used this formula:

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) + (# in 1/2/3)

I dugout my school books and rechecked!!

I googled it:
https://gmat-maths.blocked/2008/05/ ... mulas.html
First formula..HELP!!

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Stuart@KaplanGMAT: GMAT Instructor; Posts: 3225; Joined: Tue Jan 08, 2008 2:40 pm; Location: Toronto; Thanked: 1710 times; Followed by:614 members; GMAT Score:800

by Stuart@KaplanGMAT » Thu Oct 15, 2009 9:03 am

uptowngirl92 wrote:I am absolutly confused!!
I have always always used this formula:

True # of items = (total # in group 1) + (total # in group 2) + (total # in group 3) - (# in only 1/2) - (# in only 1/3) - (# in only 2/3) + (# in 1/2/3)

I dugout my school books and rechecked!!

I googled it:
https://gmat-maths.blocked/2008/05/ ... mulas.html
First formula..HELP!!

I'm not really sure what your question is - both formulas work, they each use slightly different information.