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A certain high school offers two foreign languages, Spanish

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A certain high school offers two foreign languages, Spanish

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Source: Veritas Prep

A certain high school offers two foreign languages, Spanish and French. 10% of students do not take a foreign language class, and 70% of students take exactly one foreign language class. If half of all students are in a French class and 50 students take classes in both languages, how many students are in a Spanish class?

A. 100
B. 150
C. 200
D. 240
E. 250

The OA is B.

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BTGmoderatorLU wrote:
Source: Veritas Prep

A certain high school offers two foreign languages, Spanish and French. 10% of students do not take a foreign language class, and 70% of students take exactly one foreign language class. If half of all students are in a French class and 50 students take classes in both languages, how many students are in a Spanish class?

A. 100
B. 150
C. 200
D. 240
E. 250
Excellent opportunity for the Venn diagram (aka overlapping sets)!



$$? = S = a + 50$$
$$\left\{ \matrix{
a + b = 7T\,\,\,\left( {{\rm{given}}} \right) \hfill \cr
a + \underbrace {b + 50}_{5T} = 9T \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,\,\left\{ \matrix{
\,2T = 50 \hfill \cr
\,a = 4T = 100 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,? = 150$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or https://GMATH.com.br (Portuguese version)
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BTGmoderatorLU wrote:
A certain high school offers two foreign languages, Spanish and French. 10% of students do not take a foreign language class, and 70% of students take exactly one foreign language class. If half of all students are in a French class and 50 students take classes in both languages, how many students are in a Spanish class?

A. 100
B. 150
C. 200
D. 240
E. 250
Another approach is to 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:
- taking Spanish or not taking Spanish
- taking French or not taking French

Let x = the TOTAL number of students.
We get the following diagram:


10% of students do not take a foreign language class
In other words, 10% of x (aka 0.1x) are taking NEITHER language.
Add this to our diagram:


70% of students take exactly one foreign language class.
The highlighted boxes below represent students who are taking exactly one foreign language class.
We know that these two boxes add to 0.7x:


Since all 4 boxes must add to x students, we can conclude that there are 0.2x students in the unaccounted for box in the top-left corner:


Half of all students are in a French class
In other words, 50% of x (aka 0.5x) are taking French.
So, the two left-hand boxes must add to 0.5x
Add this to our diagram:


Since the two left-hand boxex must add to 0.5x, the bottom-left box must contain 0.3x students


Also, since all 4 boxes must add to x students, we can conclude that there are 0.4x students in the remaining box in the top-right corner:


When we add the boxes in the top row, we see that 0.6x students are in Spanish.


50 students take classes in both languages
Diagram tells us that 0.2x students take classes in both languages
So, we can write: 0.2x = 50, which means x = 250

How many students are in a Spanish class?
There are 0.6x students in Spanish.
x = 250, so the number of students in Spanish = 0.6(250) = 150

Answer: B

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: http://www.gmatprepnow.com/module/gmat-word-problems?id=919

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

Easy Problem Solving questions
- http://www.beatthegmat.com/finance-majors-non-finance-majors-overlapping-set-question-t167425.html

Medium Problem Solving questions
- http://www.gmatprepnow.com/module/gmat-word-problems/video/920
- http://www.beatthegmat.com/posted-speed-limit-t272374.html
- http://www.beatthegmat.com/motel-t271938.html
- http://www.beatthegmat.com/of-the-applicants-passes-a-certain-test-15-t270255.html
- http://www.beatthegmat.com/opening-night-and-late-patrons-t264869.html
- http://www.beatthegmat.com/at-least-100-students-at-a-certain-high-school-study-japanes-t274669.html
- http://www.beatthegmat.com/prblem-solving-t279424.html

Difficult Problem Solving questions
- http://www.gmatprepnow.com/module/gmat-word-problems/video/946
- http://www.beatthegmat.com/ratio-problem-t268339.html
- http://www.beatthegmat.com/overlapping-sets-question-t265223.html
- http://www.beatthegmat.com/fractions-t264254.html
- http://www.beatthegmat.com/overlapping-sets-t264092.html
- http://www.beatthegmat.com/mba/2011/05/09/random-double-matrix-question-2

Easy Data Sufficiency questions
- http://www.gmatprepnow.com/module/gmat-word-problems/video/943
- http://www.beatthegmat.com/for-what-percent-of-those-tested-for-a-certain-t270596.html
- http://www.beatthegmat.com/ds-quest-t187706.html

Medium Data Sufficiency questions
- http://www.beatthegmat.com/sets-matrix-ds-t271914.html
- http://www.beatthegmat.com/each-of-people-voted-once-in-an-election-x-t271375.html
- http://www.beatthegmat.com/a-manufacturer-t270331.html
- http://www.beatthegmat.com/in-costume-for-halloween-t269355.html
- http://www.beatthegmat.com/mba/2011/05/05/random-double-matrix-question-1

Difficult Data Sufficiency questions
- https://youtu.be/dsCeqF9Kbk8
- http://www.beatthegmat.com/double-set-matrix-question-t271423.html
- https://youtu.be/dOZ9KM1m5Hs
- http://www.beatthegmat.com/sets-t269449.html
- http://www.beatthegmat.com/mba/2011/05/16/random-double-matrix-question-3

Cheers,
Brent

_________________
Brent Hanneson – Creator of GMATPrepNow.com
Use our video course along with Beat The GMAT's free 60-Day Study Guide

Sign up for our free Question of the Day emails
And check out all of our free resources

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Neither French nor Spanish = 10%
Only French or only Spanish = 70%
Both Spanish and French = 100 - 10 - 70 = 20%

Given: Both Spanish and French = 50 = 20% --> 100% = 250 = Total number of students

French = 0.5 * 250 = 125 --> Only French = 125 - 50 = 75
Neither = 0.1 * 250 = 25
Spanish = 250 - 25 - 75 = 150

Hence, the correct answer is B.

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