Every student at the Performing Arts Academy must take at

Question

Every student at the Performing Arts Academy must take at least one of the two drama courses offered, Classical Theater or Improvisation. If 15% of the students who take Classical Theater also take Improvisation, how many students take both Classical Theater and Improvisation?

(1) Ten percent of the students who take Improvisation also take Classical Theater.
(2) The Performing Arts Academy has a total of 450 students.

(1) Ten percent of the students who take Improvisation also take Classical Theater.
(2) The Performing Arts Academy has a total of 450 students.

OE
 (1): 10% of the students enrolled in Improvisation take both classes, while 90% of the students take only Improvisation. But with no actual numbers given, we cannot determine a single value for the number of students who take both classes. There may be 100 students or 1,000 students
y + 0.85x = x + 0.9y → 0.1y = 0.15x
Insufficient
(2): Without knowing how many do not take Classical Theater, we cannot calculate the value we need.
Insufficient
Combined:
x + 0.90y = 450 and 0.15x = 0.10y
we can solve for the values of both x and y
Sufficient

Hi, this question is wordy and does not look straight forward.
I want to know if we have simple solution for this question, please.

Bunuel · Accepted Answer

Every student at the Performing Arts Academy must take at least one of the two drama courses offered, Classical Theater or Improvisation. If 15% of the students who take Classical Theater also take Improvisation, how many students take both Classical Theater and Improvisation?

(1) Ten percent of the students who take Improvisation also take Classical Theater.
(2) The Performing Arts Academy has a total of 450 students.

Given:
 {Total} = {Classical Theater} + {Improvisation} - {Both}  (notice that we are told that every student must take  at least one  of the two courses, which implies that  {neither}=0 ).
 0.15{Classical Theater} = {Both}  -->  {Classical Theater} = {Both}/0.15 .

The question asks to find the value of  {Both} .

(1) Ten percent of the students who take Improvisation also take Classical Theater -->  0.1{Improvisation} = {Both}  -->  {Improvisation} = {Both}/0.1 . Not sufficient.

(2) The Performing Arts Academy has a total of 450 students -->  {Total} = 450 . Not sufficient.

(1)+(2) From above:  450 = {Both}/0.15 + {Both}/0.1 - {Both}  --> we can solve for  {Both} . Sufficient.

Answer: C.

Hope it's clear.

BrentGMATPrepNow · Answer

Given : Every student at the Performing Arts Academy must take at least one of the two drama courses offered, Classical Theater or Improvisation. 15% of the students who take Classical Theater also take Improvisation  
Let's use the  Double Matrix Method . 
This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions). 
Here, we have a population of students, and the two characteristics are: 
- takes Classical Theater or doesn't take Classical Theater
- takes Improvisation or doesn't take Improvisation

Since " Every student at the Performing Arts Academy must take at least one of the two drama courses offered ", we know that ZERO students take neither course. 
Also, if we let x = the number of students taking Classical Theater, then 0.15x = the number of students taking Classical Theater AND Improvisation
We can set up our matrix as follows: 
 https://i.imgur.com/jJKfmWo.png

Target question :    How many students take both Classical Theater and Improvisation?  
In other words, we want to find the value in the top-left box.

Statement 1 : Ten percent of the students who take Improvisation also take Classical Theater.  
If we let y = the number of students taking Improvisation , then 0.1y = the number of students taking Classical Theater AND Improvisation
We get: 
 https://i.imgur.com/qHuoBXE.png 
So, we have two ways to represents the value in the top-left box. 
HOWEVER, since we don't know the value of x or y (or the total number of students), we cannot answer the  target question  with certainty
Statement 1 is NOT SUFFICIENT

Statement 2 : The Performing Arts Academy has a total of 450 students. 
Add this to our original diagram to get: 
 https://i.imgur.com/NLShRbA.png 
We can see that we do not have enough information to answer the  target question  with certainty
Statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
We have: 
 https://i.imgur.com/HlBe1WV.png

Okay, if there are 450 students, and x of them take Classical Theater, then  450-x  students do NOT take Classical Theater
Likewise, if there are 450 students, and y of them Improvisation , then  450-y  students do NOT take Improvisation 
 https://i.imgur.com/9zzHgtm.png

Now that we know the sums of each row and column, we can add the following info to our diagram:
 https://i.imgur.com/WIc11Bp.png

Now, if we focus on the top-left box . . . 
 https://i.imgur.com/0O87wa5.png 
. . . we can see that we have 2 different ways to represent the same value. 
So, we can conclude that   0.15x = 0.1y

Next, if we focus on the left column . . . 
 https://i.imgur.com/6Pxe4Mz.png 
. . . we can see that the two boxes must add to y. 
So, we can write: 0.15x + (450 - x) = y
Simplify to get:   450 - 0.85x = y

At this point, we should recognize that we have a system of 2 linear equations with 2 variables:
  0.15x = 0.1y  
  450 - 0.85x = y  
As such, we COULD solve this system for x and y, which means we COULD answer the  target question .
ASIDE: Although we COULD solve the system of equations, we would never waste valuable time on test day doing so. We need only determine that we COULD answer the target question.

Since we COULD answer the  target question  with certainty, the combined statements are SUFFICIENT

Answer: C

This question type is  VERY COMMON  on the GMAT, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video:
 https://www.youtube.com/watch?v=jK-tiBrrf04

One practice question:
 https://www.youtube.com/watch?v=DsbPUSH_Wu4

stne · Answer

Something may be wrong with the question and also there is a  typo :

final equation should be Both/.15 +both/.1  -  both = 450

even if we were to solve Both/.15 +both/.1  -  both= 450 we would get Both = 1350/47 not an integer.

Number of students should be an integer.

amz14 · Answer

Maybe i misread the question but when the question states that 15% of the students who take classical theater also take Improvisation , i do not think that means that both = 0.15C. That means that Both = 0.15C + x where x would be 0.10I so both here would equal 0.15C + 010I . 15% of Classical Theater students do not equal all students who take both but rather a portion of those who take both which we do not know how much.

0.10I + 0.15C = Both

and C + I - (0.10I + 0.15C) = 450

I solved it to be E.

ramalo · Answer

Whats wrong with the question? I'm not getting an integer figure myself

Someone help!!!

Bunuel · Answer

I think it should be 470 instead of 450.

bumpbot · Answer

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Every student at the Performing Arts Academy must take at

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