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?
A. 30
B. 51
C. 60
D. 85
E. 11
We can let d, c, b, and n be the number of students in the band, orchestra, both band and orchestra, and school, respectively. We can create the equations:
d + c - b = n,
(d - b) + (c - b) = 0.8n,
d = 119,
and
d - b = 0.5n
Substituting the fourth equation into the second equation, we have:
0.5n + (c - b) = 0.8n
c - b = 0.3n
Substituting the above in the first equation, we have:
d + 0.3n = n
d = 0.7n
Finally, substituting the above in the third equation, we have:
0.7n = 119
n = 119/0.7 = 1190/7 = 170
Since orchestra only is c - b, or 0.3n, the number of students in the orchestra only is 0.3 x 170 = 51.
Alternate Solution:
If 80% of the students belong to only one group, then 100 - 80 = 20% of the students belong to both groups, i.e. both orchestra and band. Since 50% of the students are in the band only, 50 + 20 = 70% of the students in the band (including the students who are both in the orchestra and the band). We are given that the number of students in the band is 119; thus there are in total 119/0.7 = 170 students in the school. The number of students who are only in the orchestra is 100 - 70 = 30% of all students; thus there are 170 * 0.3 = 51 students who are only in the orchestra.
Answer: B