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

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.

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

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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Collections:-
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Mixture problems in a file with best solutions: http://gmatclub.com/forum/mixture-problems-with-best-and-easy-solutions-all-together-124644.html

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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The Kudo, please

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"

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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Press Kudos, if I have helped.
Thanks!