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All of the students of Music High School are in the band

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All of the students of Music High School are in the band [#permalink] New post 27 Nov 2010, 06:12
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All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only, how many students are in the orchestra only?

A. 30
B. 51
C. 60
D. 85
E. 11
[Reveal] Spoiler: OA

Last edited by Bunuel on 11 Nov 2012, 04:57, edited 1 time in total.
Renamed the topic and edited the question.
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Re: music high school [#permalink] New post 27 Nov 2010, 07:28
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anilnandyala wrote:
All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only, how many students are in the orchestra only?

30

51

60

85

11


{total} = {band only} + {orchestra only} + {both};

As {band only} + {orchestra only} = 80% and {band only} = 50% then {orchestra only} = 30%.

So, {band}= 100% - {orchestra only} = 70% --> 70% of {total} = 119 --> {total} = 170 --> {orchestra only} = 30% of {total} = 51.

Answer: B.
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Re: music high school [#permalink] New post 27 Nov 2010, 12:55
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Look at the diagram below:
If total 80% students are in one group, and 50% are in band only then 30% must be in orchestra only. Remaining 20% must be in both.
Total 70% students are in band and this number is equal to 119.
Attachment:
Ques2.jpg
Ques2.jpg [ 13.21 KiB | Viewed 5259 times ]

70% of total students = 119 so total students = 170
No of students in orchestra only = 30% of 170 = 51
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Re: Sets [#permalink] New post 14 Mar 2011, 00:48
AnkitK wrote:
All of the students of music high school are in the band ,the orchestra or both.80 percent of students are in one group.There are 119 students in the bands.If the 50 percent of the students are in the band only,how many students are in the orchestra?

A.30
B.51
C.60
D.85
E.119


If 50% of the students are in the band only; then the group that contains 80% of the students must be the band group.

However, if I try to find 0.8x = 119; x is coming as a decimal, which is not possible.

Wonder, what am I doing wrong!!!
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Re: Sets [#permalink] New post 14 Mar 2011, 00:54
Dear Fluke ,even i am facing the same problem and not able to find the solution.lets see if anybody else has better technique.
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Re: Sets [#permalink] New post 14 Mar 2011, 01:18
I also tried it like this, but getting a fraction :

Let a be the total # of students only in orchestra, b be the total # of students only in Band and Let x be the common # of students in Band and Orchestra

b +x = 119

b = 1/2(a + b +x)

2b = a + b + 119 - b => 2b = 119 + a

x = 30/100(a + b + x)

=> 119 - b = 3/10( a + 119)

119 - (119 + a)/2 = 3/10(a + 119)

119/2 - a/2 = 3a/10 + 119 * 3/10

119 ( 1/2 - 3/10) = a(1/2 + 3/10)


119 * 2/10 = a ( 8)/10)

=> a = 119 * 2/8
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Re: music high school [#permalink] New post 14 Mar 2011, 03:28
My solution:

Let's define variables as follows:

w = # students in orchestra only
x = # students in band only
y = # students in both band & orchestra
z = # students who are neither in band nor orchestra
t = total # students

From question we know the following

All of the students of Music High School are in the band, the orchestra, or both
=> (i) z = 0

80 percent of the students are in only one group
=> (ii) 0.8*t = w + x

There are 119 students in the band
=> (iii) x + y = 119

If 50 percent of the students are in the band only
=> (iv) x = 0.5*t

From (i) and (iv) follows (v) w = 0.3*t

Given that (iv) x = 0.5*t and (v) w=0.3*t => y = 0.2*t

Since (iii) x + y = 119 => t = 170

Hence, w = 0.3*170 = 51
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Re: music high school [#permalink] New post 14 Mar 2011, 23:21
Ok, here is my retake, I couldn't figure this line earlier :

Let x be the total # of students.

B + C - B&C = x

Given that -> B - B&C = 0.5x

=> C = 0.5x


NOw, given that 80 percent of the students are in only one group => B - B&c + C - B&C = 0.8x
=> B&C = 0.2x and C - B&C = 0.3x

Also given that B = 119, so B + B&C = 0.7x = 119 => x = 170 and 0.3x = 51, so the answer is B.
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Re: music high school [#permalink] New post 15 Mar 2011, 04:51
VeritasPrepKarishma wrote:
Look at the diagram below:
If total 80% students are in one group, and 50% are in band only then 30% must be in orchestra only. Remaining 20% must be in both.
Total 70% students are in band and this number is equal to 119.
Attachment:
Ques2.jpg

70% of total students = 119 so total students = 170
No of students in orchestra only = 30% of 170 = 51




How can i reason that "total 80% students are in one group" 30 percent for orchestra and 50%band. Why not 50 percent for band and 30% for both?
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Re: music high school [#permalink] New post 15 Mar 2011, 08:01
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Baten80 wrote:
VeritasPrepKarishma wrote:
Look at the diagram below:
If total 80% students are in one group, and 50% are in band only then 30% must be in orchestra only. Remaining 20% must be in both.
Total 70% students are in band and this number is equal to 119.
Attachment:
Ques2.jpg

70% of total students = 119 so total students = 170
No of students in orchestra only = 30% of 170 = 51




How can i reason that "total 80% students are in one group" 30 percent for orchestra and 50%band. Why not 50 percent for band and 30% for both?


Look at the question again:

All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only, how many students are in the orchestra only?

It says 80% students are in 1 group only (either orchestra only or band only but not both). The leftover 20% must be in both the groups, orchestra as well as band. (since every student is in at least one group)
Say, out of 100 students, 80 are in only one group. 50% students are in band only. So we now know how the 80% of 'one group only' is split: 50% in band only and 30% in orchestra only.
Look at the diagram in my previous post for more clarity.
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Re: All of the students of Music High School are in the band [#permalink] New post 01 Apr 2013, 15:27
As it is an overlapping set problem, could someone show us how to solve this pb with the Double Matrix method ?

I find it hard to model the pb using this mthd

Thx in advance
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Re: All of the students of Music High School are in the band [#permalink] New post 19 May 2013, 05:26
The official solution is below.


This is an overlapping sets problem concerning two groups (students in either band or orchestra) and the overlap between them (students in both band and orchestra).

If the problem gave information about the students only in terms of percents, then a smart number to use for the total number of students would be 100. However, this problem gives an actual number of students (“there are 119 students in the band”) in addition to the percentages given. Therefore, we cannot assume that the total number of students is 100.

Instead, first do the problem in terms of percents. There are three types of students: those in band, those in orchestra, and those in both. 80% of the students are in only one group. Thus, 20% of the students are in both groups. 50% of the students are in the band only. We can use those two figures to determine the percentage of students left over: 100% - 20% - 50% = 30% of the students are in the orchestra only.

Great - so 30% of the students are in the orchestra only. But although 30 is an answer choice, watch out! The question doesn't ask for the percentage of students in the orchestra only, it asks for the number of students in the orchestra only. We must figure out how many students are in Music High School altogether.

The question tells us that 119 students are in the band. We know that 70% of the students are in the band: 50% in band only, plus 20% in both band and orchestra. If we let x be the total number of students, then 119 students are 70% of x, or 119 = .7x. Therefore, x = 119 / .7 = 170 students total.

The number of students in the orchestra only is 30% of 170, or .3 × 170 = 51.

The correct answer is B.


I'm in confusion in the below sentence. Can someone please elaborate?

"We know that 70% of the students are in the band: 50% in band only, plus 20% in both band and orchestra"
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Re: All of the students of Music High School are in the band [#permalink] New post 19 May 2013, 05:49
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connexion wrote:
All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students
are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only,
how many students are in the orchestra only?

A)30
B)51
C)60
D)85
E)119


Attachment:
Untitled4.jpg
Untitled4.jpg [ 16.74 KiB | Viewed 3493 times ]


Let the total number of students be x

119 = only band + both

0.8x = only band + only orchestra

which also implies 0.2x students are in both(x-0.8x)

0.5x = only band

which implies only orchestra = 0.3x


So number of students in band = 0.5x + 0.2x = 0.7x = 119 which gives x as 170

So the number of students in orchestra only is 0.3*170 = 51

Answer B
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Re: All of the students of Music High School are in the band [#permalink] New post 19 May 2013, 07:23
connexion wrote:
All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students
are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only,
how many students are in the orchestra only?

A)30
B)51
C)60
D)85
E)119


80% Students are only in one group. => 20% are in both group
50% students are in band only =>

i. 50% (Band Only)+ 20%(both Band and Orchetsra) = 70% are in the band = 119 Students(given)
ii. 80% (Either only in Band or Only in Orchestra) - 50%(only in Band) = 30% are in only Orchestra

If \(70% = 119 => 30% = 119*(30/70) = 51\)(Ans = B)
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Re: All of the students of Music High School are in the band [#permalink] New post 25 Jan 2014, 02:26
Initially it was difficult for me to understand the question. But solved in 3 mins.

Let t be total number of students.

80% is in only one group => 0.8t
50% is in only band => 0.5t so 0.8t-0.5t = 0.3t is in Only Orchestra.

Now,
Orchestra Only + both + Band Only = t
0.3t + x + 0.5t = t

It is given that 119 are in band means 0.5t+x = 119
So,
0.3t+119 = t
t = 119/0.7 = 170

Orchestra Only = 0.3t = 0.3*170 = 51 ---B is the answer.
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Re: All of the students of Music High School are in the band [#permalink] New post 01 Jun 2015, 22:23
B
St. in both band + both = 50+(100-80) = 70% of total = 119
Total st. = 119/0.7
St. in orchestra only = (80-50) % of total students = 0.3 * 119/0.7 = 51
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Re: All of the students of Music High School are in the band [#permalink] New post 01 Jun 2015, 23:09
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Presenting the matrix approach to the solution

Let's assume the total number of students to be x.

We are given that 50% of the students are in band only. Number of students in band only =0.5x.

Also, we are given that there are 80% students in one group only = 0.8x

Students in one group only = students in band only + students in orchestra only

0.8x = 0.5x +students in orchestra only i.e. students in orchestra only = 0.3x

Image


As there are no students who are in neither band nor orchestra, number of students who are not in orchestra = 0.5x. Therefore, number of students who are in orchestra = x - 0.5x = 0.5x.

Students in orchestra = students in orchestra only + students in both band and orchestra

0.5x = 0.3x + students in both band and orchestra
students in both band and orchestra = 0.2x

We are told that there are 119 students in band.

Students in band = students in band only + students in both band and orchestra

119 = 0.5x + 0.2x i.e. x = 170.

We are asked to find the number of people in orchestra only = 0.3x = 0.3 * 170 = 51

Hope this helps :)

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Re: All of the students of Music High School are in the band   [#permalink] 01 Jun 2015, 23:09
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