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Out of 75 students, 17 students enrolled in a Physics class, 28 studen

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Out of 75 students, 17 students enrolled in a Physics class, 28 studen  [#permalink]

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New post 14 Mar 2018, 00:25
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[GMAT math practice question]

Out of 75 students, 17 students enrolled in a Physics class, 28 students enrolled in a Chemistry class, and 39 students enrolled in a Biology class. 5 students enrolled in both the Physics and Chemistry classes, 7 students enrolled in both the Chemistry and Biology classes, and 6 students enrolled in both the Biology and Physics classes. If 4 students enrolled in Physics, Chemistry, and Biology, how many students did not enroll in any of the three science classes?

A. 2
B. 3
C. 4
D. 5
E. 6

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Re: Out of 75 students, 17 students enrolled in a Physics class, 28 studen  [#permalink]

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New post 14 Mar 2018, 01:06
It is a three category Venn diagram problem.

Start from the inside ( with no. Of students participating in all three being 4) and build as you go out.

It all falls down to 70 students inside the circles and 5 not participating in any subjects.

Certain subsets are : only physics - 10.
Only chemistry - 20.
Only biology - 30.

Only P & B - 2.
Only P & C - 1.
Only B&C - 3.
All three - 4.

Hence none =5.

Hence D.

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Re: Out of 75 students, 17 students enrolled in a Physics class, 28 studen  [#permalink]

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New post 15 Mar 2018, 15:52
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This can be solved by directly applying the formula for overlapping sets

\(Total=A+B+C−(sum of 2−group overlaps)+(all three)+Neither\)

Let students who enrolled in none of the courses be x

\(75 = 17 + 28 + 39 - (5 + 7 + 6) + 4 + x\)
\(75 = 70 + x\)
\(x = 5\)


Answer: D
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Re: Out of 75 students, 17 students enrolled in a Physics class, 28 studen  [#permalink]

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New post 16 Mar 2018, 00:54
=>

Attachment:
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The total number of students is given by
\(a + b + c + d + e + f + g + h = 75.\)
Of these,
\(a + d + f + g = 17\) are enrolled in Physics,
\(b + d + e + g = 28\) are enrolled in Chemistry, and
\(c + e + f + g = 39\) are enrolled in Biology.
Adding these three equations gives \(( a + b + c ) + 2( d + e + f ) + 3g = 84.\)
We also know that
\(d + g = 5\) study both Physics and Chemistry,
\(e + g = 7\) study both Chemistry and Biology, and
\(f + g = 6\) study both Physics and Biology.
Adding these three questions yields \(( d + e + f ) + 3g = 18.\)
Since \(4\) students are enrolled in all three classes, we have \(g = 4,\)

Plugging this value for \(g\) into the above equation yields
\((d + e + f) + 3g = 18\)
\((d + e + f) + 12 = 18\)
\(d + e + f = 6.\)

So,
\(( a + b + c ) + 2( d + e + f ) + 3g = 84\) yields
\((a + b + c) + 2(6) + 3(4) = 84\)
\((a + b + c) + 12 + 12 = 84\)
\(a + b + c = 60.\)

Finally, using the equation, \(a + b + c + d + e + f + g + h = 75\), we see that
\((a + b + c) + (d + e + f) + g + h = 75\)
\(60 + 6 + 4 + h = 75\)
\(h = 5.\)

Therefore, the answer is D.
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Re: Out of 75 students, 17 students enrolled in a Physics class, 28 studen  [#permalink]

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New post 07 Mar 2019, 20:08
Hi chetan2u Bunuel VeritasKarishma Gladiator59 generis,

I was wondering could one of you experts please help me solve this question via the overlap formula? If I am not mistaken, the formula for three overlap set is Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither.

Based on this we get: 75=17+28+39-18-8+x or x=17. Could one of you please help me and tell me where I am making a mistake? Would greatly appreciate it!
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Re: Out of 75 students, 17 students enrolled in a Physics class, 28 studen  [#permalink]

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New post 07 Mar 2019, 20:30
JS1290 wrote:
Hi chetan2u Bunuel VeritasKarishma Gladiator59 generis,

I was wondering could one of you experts please help me solve this question via the overlap formula? If I am not mistaken, the formula for three overlap set is Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither.

Based on this we get: 75=17+28+39-18-8+x or x=17. Could one of you please help me and tell me where I am making a mistake? Would greatly appreciate it!



Hi,
you are going wrong in formula..
It is Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three)+all three + Neither.... 75 = 17+28+39-18+4+x or x=5..

Logic is when you add Group 1, 2 and 3... all three portion gets added 3 times, and when you subtract 'two together', you are subtracting 'all three' thrice, so finally your solution is without the group ' all three', that is why we have to add 'all three' once..
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Re: Out of 75 students, 17 students enrolled in a Physics class, 28 studen  [#permalink]

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New post 07 Mar 2019, 21:07
JS1290 wrote:
Hi chetan2u Bunuel VeritasKarishma Gladiator59 generis,

I was wondering could one of you experts please help me solve this question via the overlap formula? If I am not mistaken, the formula for three overlap set is Total = Group1 + Group2 + Group3 - (sum of 2-group overlaps) - 2*(all three) + Neither.

Based on this we get: 75=17+28+39-18-8+x or x=17. Could one of you please help me and tell me where I am making a mistake? Would greatly appreciate it!



Check out this post: https://www.veritasprep.com/blog/2015/1 ... ping-sets/
It discusses the various formulas you can use to solve overlapping sets questions.
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Re: Out of 75 students, 17 students enrolled in a Physics class, 28 studen   [#permalink] 07 Mar 2019, 21:07
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