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by j_shreyans » Sun Mar 29, 2015 5:32 am
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
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by GMATGuruNY » Sun Mar 29, 2015 6:17 am
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

119
We can plug in the answers, which represent the number of students in the orchestra only.
Given that the percentages are all multiples of 10, when the correct answer is added to 119 -- the total number of students in the band -- the resulting sum almost certainly will be a multiple of 10.
Thus, the most likely answer choice is B.

Answer choice B: 51 in the orchestra only.
Thus, the total number of students = 119+51 = 170.
Total in the band only = .5(170) = 85.
Students in only one group/Total = (51+85)/170 = 136/170 = 8/10 = 80%. Success!

The correct answer is B.

An algebraic solution:
Since 80% are in only one group, and 50% are in the band only, 30% are in the orchestra only.
Since 30% are in the orchestra only, 70% are in the band.
Thus, 119 is 70% of the total:
119 = .7x
x = 119/.7 = 170.
Thus, the number in the orchestra only = .3(170) = 51.

The correct answer is B.
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by [email protected] » Sun Mar 29, 2015 4:50 pm
Hi j_shreyans,

This prompt is one big 'logic' problem with a little bit of arithmetic thrown in.

We're given a number of facts to work with:
1) ALL students are in the band, the orchestra or BOTH.
2) 80% of students are ONLY in 1 group.
3) There are 119 students in the band.
4) 50% of the students are in the band ONLY.

We're asked how many students are in the orchestra ONLY.

From facts 2 and 4, we can break the students down into groups (by percent):

80% are in ONLY 1 group and 50% are in the band ONLY.

This means that 100% - 80% = 20% are in BOTH groups.
This also means that 80% - 50% = 30% are in the orchestra ONLY.

From fact 3, we know that 119 students are in the band (which includes the students in band ONLY and the students in BOTH)

Band only = 50%
Both = 20%

50% + 20% = 70% = 119 students

.7(Total) = 119
Total = 119/.7
Total = 170

Finally, we're asked for the number of students in the orchestra:

30% of 170 = 51

Final Answer: B

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