set theory

[uptowngirl92]

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?

[asamaverick]

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.

[mehravikas]

Why it could not be 80.

Raspberry + Str = 0
Raspberry + apple = 0

Raspberry = 80

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.

[Stuart@KaplanGMAT]

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.

[uptowngirl92]

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"...

[Stuart@KaplanGMAT]

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

[mehravikas]

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

[Stuart@KaplanGMAT]

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)

[mehravikas]

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

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)

[Stuart@KaplanGMAT]

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

[papgust]

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

[uptowngirl92]

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!!

[Stuart@KaplanGMAT]

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.