A certain high school offers two foreign languages, Spanish and French. 10% of students do not take a foreign language

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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


OA B

Source: Veritas Prep

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BTGmoderatorDC wrote:
Mon Mar 29, 2021 5:37 pm
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


OA B

Source: Veritas Prep
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:
- taking Spanish or not taking Spanish
- taking French or not taking French

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

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:
Image

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:
Image

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:
Image

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

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:
Image

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

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.
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BTGmoderatorDC wrote:
Mon Mar 29, 2021 5:37 pm
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


OA B

Solution:

To solve this problem, there are two useful formulas we can use:

Total = French Only + Spanish Only + Both + Neither
Total = French + Spanish - Both + Neither

In terms of percentage of students, we will use the first formula. We are given that the “Neither” group is 10%. Even though we don’t know “French Only” and “Spanish Only” individually, we know the total of these two groups is 70%; thus, we have:

100% = 70% + Both + 10%

Both = 20%

We are also given that 50 students take classes in both languages. If we let t = the total number of students, we have:

0.2t = 50
t = 250

Since half of all students take French, and 10% take neither, we have 125 students who take French and 25 who take neither. Therefore, in terms of numbers of students, we will use the aforementioned second formula:

250 = 125 + Spanish - 50 + 25

250 = 100 + Spanish

Spanish = 150

Answer: B

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