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Every student at the Performing Arts Academy must take at le

by ktrout2020 » Thu Oct 31, 2019 6:19 am
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.

Source: Kaplan
Answer: C
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by Brent@GMATPrepNow » Sat Nov 02, 2019 5:08 am
ktrout2020 wrote: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.

Source: Kaplan
Answer: C
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:
Image

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:
Image
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:
Image
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:
Image


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
Image

Now that we know the sums of each row and column, we can add the following info to our diagram:
Image

Now, if we focus on the top-left box . . .
Image
. . . 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 . . .
Image
. . . 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.gmatprepnow.com/module/gmat- ... ems?id=919

Once you're familiar with this technique, you can attempt these additional practice questions:

Easy Problem Solving questions
- https://www.beatthegmat.com/finance-majo ... 67425.html

Medium Problem Solving questions
- https://www.gmatprepnow.com/module/gmat- ... /video/920
- https://www.beatthegmat.com/posted-speed ... 72374.html
- https://www.beatthegmat.com/motel-t271938.html
- https://www.beatthegmat.com/of-the-appli ... 70255.html
- https://www.beatthegmat.com/opening-nigh ... 64869.html
- https://www.beatthegmat.com/at-least-100 ... 74669.html
- https://www.beatthegmat.com/prblem-solving-t279424.html

Difficult Problem Solving questions
- https://www.gmatprepnow.com/module/gmat- ... /video/946
- https://www.beatthegmat.com/ratio-problem-t268339.html
- https://www.beatthegmat.com/overlapping- ... 65223.html
- https://www.beatthegmat.com/fractions-t264254.html
- https://www.beatthegmat.com/overlapping- ... 64092.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-2

Easy Data Sufficiency questions
- https://www.gmatprepnow.com/module/gmat- ... /video/943
- https://www.beatthegmat.com/for-what-per ... 70596.html
- https://www.beatthegmat.com/ds-quest-t187706.html

Medium Data Sufficiency questions
- https://www.beatthegmat.com/sets-matrix-ds-t271914.html
- https://www.beatthegmat.com/each-of-peop ... 71375.html
- https://www.beatthegmat.com/a-manufacturer-t270331.html
- https://www.beatthegmat.com/in-costume-f ... 69355.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-1

Difficult Data Sufficiency questions
- https://youtu.be/dsCeqF9Kbk8
- https://www.beatthegmat.com/double-set-m ... 71423.html
- https://youtu.be/dOZ9KM1m5Hs
- https://www.beatthegmat.com/sets-t269449.html
- https://www.beatthegmat.com/mba/2011/05/ ... question-3

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by deloitte247 » Sun Nov 03, 2019 1:27 pm
Let the number of people taking classical Theatre = x
Number of people taking both Classical Theater and Improvisation = 15% of x = 15x / 100
Let the number of people taking Improvisation = y

Statement 1: Ten percent of the student who takes Improvisation also take Classical Theater
Number of people taking Improvisation and Classical Theater = 10% of y = 10y / 100
The exact value of y is unknown, hence, statement 1 is INSUFFICIENT.

Statement 2: The performing Arts Academy has a total of 450 students.
x + y = 450
There are 2 unknown variables in the above equation, so, the total number of people taking Classical Theater and Improvisation cannot be estimated.
Hence, statement 2 is INSUFFICIENT.

Combining both statements together:
Statement 1: 10y / 100
Statement 2: x+y = 450
Number of students taking Classical Theater alone = 450 - y
Number of students taking Improvisation alone = 450 - x
Number of students taking Improvisation and Classical Theater = Number of students taking Classical Theater and Improvisation
Therefore; 15x / 100 = 10y / 100 ==> 0.15x - 0.1y = 0 --- (i)
y = (450 - x) + 0.15x ==> 450 - 0.85x --- (ii)
x + y = 450
x = 450 - y
y = (450 - x) + 0.15x
y = (450 - (450-y)) + 0.15 (450 - y)
y = y + 67.5 - 0.15y
0.15y = 67.5
y = 67.5 / 0.15 = 450
Since 0.15x = 0.1y
0.15x = 0.1 * 450
x = 45/0.15 = 300

Total number of students in both classes are ;
15% of x = 0.15 * 300 = 45

Conclusively, both statements together are SUFFICIENT but neither statement alone is sufficient.
The answer is, therefore, option C.
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