A certain high school offers two foreign languages, Spanish and French

Question

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

(A) 100 
(B) 150
(C) 200
(D) 240
(E) 250

CrackverbalGMAT · Accepted Answer

Hi Ziyuenlau,

For overlapping sets, Venn – Diagram is the best way to solve the question:

From the Venn-diagram we can see that,

a+b+50+neither = total …….(1)

Given that, a+b that is exactly one foreign language is 70% and neither is 10%

Then,

70%+ 50+10% = 100%

So 50 is equal to 20 percent of total

20% * total = 50

Then total = 250

We know that,

According to the Venn- diagram, we have to find “b+50”,

Since french is half of the total,

we can write,

a+50 = 125

So a = 75,

Substitute in the equation 1,

75+b+50+25 = 250

b+50 = 150

So the answer is B.

Hope this is clear.

BrentGMATPrepNow · Answer

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: 
 https://i.imgur.com/3Xna3GQ.png

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:
 https://i.imgur.com/30L1GDQ.png

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:
 https://i.imgur.com/grJ5DVr.png

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: 
 https://i.imgur.com/EevbD8s.png

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:
 https://i.imgur.com/VoGOUkC.png

Since the two left-hand boxex must add to 0.5x, the bottom-left box must contain 0.3x students
 https://i.imgur.com/CNCa0xe.png

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: 
 https://i.imgur.com/ioo0jaw.png

When we add the boxes in the top row, we see that 0.6x students are in Spanish. 
 https://i.imgur.com/lANeK6k.png

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:
 https://www.youtube.com/watch?v=jK-tiBrrf04

Kurtosis · Answer

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

Answer: B

BillyZ · Answer

https://uploads.tapatalk-cdn.com/20170127/a79aa67a4f34f60194d27170a89b759c.jpg

Why my solution is not workable?

Sent from my iPhone using  GMAT Club Forum mobile app

Kurtosis · Answer

Hi  ziyuenlau ,

Your solution is not correct as you have incorrectly assumed the number of students opting for spanish as "0.5x".

Refer the corrected solution in the figure.

chetan2u · Answer

Hi
 if you use logic, we will not require Venn diagram..

10% study nothing and 70% study exactly ONE..
SO 100-10-70=20% study both and this is given as 50..
20% of Total =50... Total= 50*100/20=250..

Only Spanish = total-french+both=90-50+20=60%
60% of Total= 60/100 * 250=150..
B

BillyZ · Answer

Dear  chetan2u , How do you able to get the total equal to 90?

Bunuel · Answer

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

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

GMATinsight · Answer

Make a grid and solve as mentione dbelow... Follow the sequence of steps as per color coding

Answer: option B

GovindAgrawal · Answer

we are given 70% students take only one language which means 20% take both (100-70-10), 10 for none. further we are given 50 students take both classes therefore if 20% make up 50 then 250 will make up 100% (50/20 * 100). so we get total students 250. now apply formula:  ] , we get: 250- 25= 125 + spanish - 50 which is equal to 150. so ans is B.

JeffTargetTestPrep · Answer

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

1) Total = French Only + Spanish Only + Both + Neither

2) 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 = 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 number of students, we will use the aforementioned second formula:

250 = 125 + Spanish - 50 + 25

250 = 100 + Spanish

Spanish = 150

Answer: B

rohit8865 · Answer

ziyuen

chetan2u is talking of 90% which he takes in account after deducting 10% (which is neither case).

ScottTargetTestPrep · Answer

We can create the following equation:

Total number of students = # who take Spanish + # who take French - # who take both + # who take neither

We can let n = the total number of students

Since 10% of students do not take a foreign language class, 0.1n = # number who take neither class

We are also given that 50 students take both classes. Thus:

n = S + F - 50 + 0.1n

50 + 0.9n = S + F

The number of students who take only Spanish is S - 50. Similarly, the number of students who take only French is F - 50. Thus, the number of students who take only one class is S - 50 + F - 50 = S + F - 100. We are given that this makes up 70% of the class; thus S + F - 100 = 0.7n, or equivalently, S + F = 0.7n + 100. Recall that S + F = 50 + 0.9n; thus, we have:

50 + 0.9n = 0.7n + 100

0.2n = 50

n = 250
 
Thus, the total number of students is 250. We know that half of all students take French; thus, F = 125. Since S + F = 0.7n + 100, we have S + 125 = 0.7(250) + 100 = 275; thus S = 150.

Answer: B

Nunuboy1994 · Answer

If .10 of students take neither and .70 students are in exactly one set then P(A ∩ B)= .20

50= x(.20)

50/.20 =250

250/2 = 125

250(.10) = 25 students neither

Now apply set theory formula

250 = x + 125 -25

x= 150

B

pushpitkc · Answer

Let the total students be x

Given data:
 \frac{x}{10}  is the number of students studying neither language
Number of students, learning both languages are 50
Number of students, studying exactly one language is  \frac{7x}{10}

We know that 
 Total - Neither = Only one + Both

x -  \frac{x}{10}  =  \frac{7x}{10}  + 50
 \frac{9x}{10}  -  \frac{7x}{10}  = 50
x=250

If x=250, students studying Only one is  \frac{7x}{10}  = 175

Since students studying French are half of the total(125) and the students studying both are 50,
students studying Only French = 75

Students studying Only Spanish = Only one - Only French = 175 - 75 = 100
Hence, students studying Spanish = Only Spanish + Both = 100 + 50 = 150(Option B)

SamBoyle · Answer

Venn Diagram is the way to go.
A= Only Spanish
B= Only French
C= Both
D= Neither

A+B+C+D=1
D=0.1 (stated)
Therefore A+B+C=0.9
A+B =0.7 (70% take only one language class)

A+B+C=0.9
A+B =0.7
Therefore C =0.2

C= 50 so therefore there are 250 students 50/0.2 (as C = 0.2 if C is 20% of the pie and is 50 then 100% of the pie is 250)

B+C=0.5 (stated, half the students take French)
Therefore B+C = 125 (250*0.5)
C=50 therefore B = 75 (125-50)

A+B =0.7
Therefore A+B =175 (250*0.7)
B=75 therefore A=100, B =75, C =50 and D=25

Now the question asks how many in Spanish??

Here is a little trick at the end. Only Spanish is 100 =A However C also takes Spanish so it is 100+50 for a total of 150.

How many take French? It is 75+50 = 125?

How many take Spanish or French it is 75+50+100 or 225.

So ensure you don't get tricked at the end. Had they said, how many only take Spanish then the answer is 100

fskilnik · Answer

https://i.ibb.co/sbhmDrV/05-Dec18-4r.gif

? = 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.

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A certain high school offers two foreign languages, Spanish and French

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A certain high school offers two foreign languages, Spanish and French

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