BTGmoderatorRO 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

OA is B

OA says B but my answer was option A.Pls an Expert is needed here. Thanks

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.

Answer: B

BTGmoderatorRO 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

OA is B

OA says B but my answer was option A.Pls an Expert is needed here. Thanks

If 80% are in only one group, 20% are in both groups. If 50% are in the band only, then, counting the 20% who are in both groups, 70% of the students are in the band in total. So 30% are only in the orchestra. From here, you can either notice that the ratio of people in the band to people only in the orchestra is 70 to 30, or 7 to 3, so if we have 119 = (17)(7) people in the band, we must have (17)(3) = 51 people in the orchestra only. Or you could notice 119 is 70% of the total number of people, so the total number of people must be 170, and since 30% of all people are only in the orchestra, the answer is 30% of 170, which is 51.