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Of the 200 students at College T majoring in one or more of

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Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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Walkabout wrote:
Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150


{Total} = {Chemistry } + {Biology} - {Both} + {Neither}

200 = 130 + 150 - {Both} + {Neither} --> {Neither} = {Both} - 80.

Since given that {Neither}>=30, then {Both} - 80>=30 --> {Both}>=110.

Now, as the number of students majoring in both chemistry and biology cannot possibly be more than the number of students majoring in either of sciences, then 110<={Both}<=130.

Answer: D.
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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Let X be the number majoring in Both Chemistry and Biology .

Then

130-X+X+150-X+GT30 = 200

X= GT110 ( Here GT=Greater than )

X>= 110

And as one of the set consists of only 130 , the maximum value cannot exceed 130.

110<=X<=130
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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Bunuel wrote:
Walkabout wrote:
Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150


{Total} = {Chemistry } + {Biology} - {Both} + {Neither}

200 = 130 + 150 - {Both} + {Neither} --> {Neither} = {Both} - 80.

Since given that {Neither}>=30, then {Both} - 80>=30 --> {Both}>=110.

Now, as the number of students majoring in both chemistry and biology cannot possibly be more than the number of students majoring in either of sciences, then 110<={Both}<=130.

Answer: D.


Bunuel, I have a question. I'm a bit confused about the wording:

to me the current stem (....at least 30 of the students are not majoring in either chemistry or biology.....) sounds like "at least 30 students are majoring in only one subject, chem or bio".
However, the only way to solve the problem is to apply that at least 30 students relate to the section "neither".

Additionally, at the beginning of the stem it is provided that "Of the 200 students at College T majoring in one or more of the sciences.....". To me it sound like each of the students is majoring in at least one subject, bio or che.

Am I wrong?
Apologies if the question is stupid - I'm not a native speaker :)
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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New post 18 Sep 2014, 10:19
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Maksym wrote:
Bunuel wrote:
Walkabout wrote:
Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150


{Total} = {Chemistry } + {Biology} - {Both} + {Neither}

200 = 130 + 150 - {Both} + {Neither} --> {Neither} = {Both} - 80.

Since given that {Neither}>=30, then {Both} - 80>=30 --> {Both}>=110.

Now, as the number of students majoring in both chemistry and biology cannot possibly be more than the number of students majoring in either of sciences, then 110<={Both}<=130.

Answer: D.


Bunuel, I have a question. I'm a bit confused about the wording:

to me the current stem (....at least 30 of the students are not majoring in either chemistry or biology.....) sounds like "at least 30 students are majoring in only one subject, chem or bio".
However, the only way to solve the problem is to apply that at least 30 students relate to the section "neither".

Additionally, at the beginning of the stem it is provided that "Of the 200 students at College T majoring in one or more of the sciences.....". To me it sound like each of the students is majoring in at least one subject, bio or che.

Am I wrong?
Apologies if the question is stupid - I'm not a native speaker :)


1. At least 30 of the students are not majoring in either chemistry or biology, means that there are at least 30 of the students who are majoring in something else, maybe in math (so neither chemistry nor biology).

2. Of the 200 students at College T majoring in one or more of the sciences, means that all of them are majoring in at least one of the sciences BUT this does not mean that they are majoring only in chemistry or in biology. There might be some other sciences, so there might be some students who are not majoring in either chemistry or biology.

Hope it's clear.
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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New post 18 Sep 2014, 19:49
I realized that students in both subjects are >=110, so stuck between D and E. Thanks Bunuel for explanation
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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Refer diagram below:

Attachment:
sqa.png
sqa.png [ 4.78 KiB | Viewed 11392 times ]


130 + 150-x + 30 = 200

x = 110

Least of Chemistry(130) & Biology(150) is Chemistry = 130

Range = 110 to 130

Answer = D
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Of the 200 students at College T majoring in one or more of [#permalink]

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New post 20 Jan 2015, 11:12
I created a table like this:

__________B_________NB_________ALL
C.........110most........20most............130
NC.........40least........30least..................
ALL............150..............50...............200

C= Chemistry
NC= not Chemistry
B= Biology
NB= not Biology

We start by adding the values we know:200, 30 the least, 130, 150.
200-150= 50, so we add 50 under NB for ALL.
50-30= 20. Since 30 was the least possible, 20 is the most possible.
130-20=110. Since 20 was the most, 110 is the most. Otherwise, if BC is 111, C-ALL would be 131.

So, the first row gives us the answer: 110 to 130. ANS D
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Best answer I have seen indeed!
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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Walkabout wrote:
Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150


This is how I would do this question: Total 200 students, Chemistry (130), Biology(150), Neither (30)

So let's ignore 30 of the 200 and we only have to worry about the 170 students who do take Chemistry or Biology. So our relevant total is 170.

Out of these 170, 130 take Chemistry. Let's say that the rest 40 take Biology but then, the leftover 110 of Biology must overlap with Chemistry. So the overlap must be at least 110.
The other side of the situation is that all 130 of Chemistry take Biology too so the overlap in that case will be 130.

This gives us that the range of students majoring in both could be anywhere from 110 to 130.

Answer (D)
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We can use here a double matrix (see MGMAT - Word Problems)
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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New post 06 Nov 2015, 13:20
Walkabout wrote:
Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150


[from;to] inclusive or [min;max] denotes a range

1. [30;50]
2. [50-50;50-30]=[0;20]
3. [130-20;130-0]=[110;130]
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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New post 28 Apr 2016, 18:47
So I did this one with a grid set up
Given:
..............Bio...................Not
.
.
Chem......BC..................CN...........................130
.
.
Not.........BN..............NN 30<=x<=50............70=(200-130)
.
.
.............150..................50= (200-150)..........T = 200

^^This will hold true throughout the problem

Since we know that the minimum in NN has to be 30, we also know the max has to be 50
With NN at minimum 30, we know that BN must equal 40, CN must equal 20, and BC is minimum 110
With NN at maximum 50, we know that BN must equal 20, CN must equal 0, and BC is maximum 130
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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Walkabout wrote:
Of the 200 students at College T majoring in one or more of the sciences, 130 are majoring in chemistry and 150 are majoring in biology. If at least 30 of the students are not majoring in either chemistry or biology, then the number of students majoring in both chemistry and biology could be any number from

(A) 20 to 50
(B) 40 to 70
(C) 50 to 130
(D) 110 to 130
(E) 110 to 150


Solution:

This is an overlapping set question that is testing our ability to solve for maximum and minimum values. We know this because the answer choices each provide a range of values. The best way to solve this problem would be to set up a table with two categories: chemistry and biology. More specifically, our table will be labeled as follows:

1) Majored in chemistry (Chemistry)

2) Did not major in chemistry (No Chemistry)

3) Majored in biology (Biology)

4) Did not major in biology (No Biology)

We are given:

Total = 200

Chemistry = 130

Biology = 150

No Chemistry and No Biology = at least 30

We can fill all this into our table. Remember, all the columns and rows sum together.

Image

Now that we have our chart initially filled in, we can determine the greatest and least number of students who majored in biology and chemistry.

Because we are given that at least 30 students majored in neither biology or chemistry, let’s assume that this number is exactly 30 students. Using the value 30 as the "No Chemistry-No Bio" entry, we can solve for the remaining entries in the table by subtraction.

Image

We see that there are 110 students who majored in both biology and chemistry. Even though at this point we don’t know whether this is the maximum or the minimum number of students who could major in both sciences, we can guess it’s a minimum by looking at the answer choices given. Furthermore, if that is the case, the answer must be either choice D or E.

Now, if 110 is indeed the minimum, how can we find the maximum? Recall that we used 30 as the number of students who majored in neither science, and 30 is the smallest number we can use for the “No Chemistry-No Bio” entry. So, now let’s find the largest number we can use for that entry. Recall that there are a total of 50 students who did not major in Bio, so the “No Chemistry-No Bio” entry can be at most 50. That is, the largest number we can use for that entry is 50. Now let’s fill in the rest of the table.

Image

We see that there are now 130 students who majored in both biology and chemistry, which is also the maximum number of students who could major in both sciences.

The answer is D.
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Of the 200 students at College T majoring in one or more of [#permalink]

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My 2 cents with credi to Magoosh
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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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New post 05 Nov 2016, 09:55
My question is why is the maximum value of those in neither Bio or Chem 50? Why not 70?

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Re: Of the 200 students at College T majoring in one or more of [#permalink]

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thorinoakenshield wrote:

Wow, thank you! I was really struggling with that one and thanks to your post it's so clear and easy now.
Re: Of the 200 students at College T majoring in one or more of   [#permalink] 06 Nov 2016, 09:30
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