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

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.

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"

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

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




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

Regards
Harsh

not the best use of double-set matrix but a sure-thing. There are more time-efficient ways but good enough as a framework. Can be slow if not fast at long division.

Hi All,

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:

GMAT assassins aren't born, they're made,
Rich

let the total number of students be x.
By definition
- Band only will be \(\frac{5x}{10}\) i.e. [50%]
- Orchestra only will be \(\frac{3x}{10}\) i.e. [80% minus 50%]
- Both will be \(\frac{2x}{10}\) i.e. [20%]

Therefore, \(\frac{5x}{10} + \frac{3x}{10} + \frac{2x}{10} = x\)

But \(\frac{5x}{10} + \frac{2x}{10} = 119\)

Therefore, \(\frac{3x}{10} + 119 = x\)

Solving we get x = 170 AND (Orchestra only ) i.e. 3x/10 = 51

Correct answer is B

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

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