At a certain school of 200 students, the students can study French, Sp

Let total Spanish = x

Students without French without Spanish \(= \frac{x}{4}\)

Students with both French & Spanish \(= \frac{x}{4}\)

Students with only French \(= x - \frac{x}{4} = \frac{3x}{4}\)

Total French students \(= \frac{3x}{4} - 10\)

Only French students \(= \frac{3x}{4} - 10 - \frac{x}{4} = \frac{2x}{4} - 10\)

Total \(= \frac{3x}{4} - 10 + x = 200\)

x = 120

Only French students \(= \frac{x}{2} - 10 = 60 - 10 = 50\)

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MAGOOSH OFFICIAL SOLUTION:

Let x be the number of folks studying both, which means it is also the number of folks studying neither.

“One quarter of those who study Spanish also study French.” If the Spanish students studying French are x, then all Spanish students are 4x, and those who do not study French are 3x. Also, let y be the number of students who study French but not Spanish.

“The total number who study French is 10 fewer than those who study Spanish only.” In other words,
x + y = 3x – 10
10 = 2x – y

Also, notice that the total number of students is 200:
3x + x + y + x = 200
5x + y = 200

We have two equations with two unknowns. Add the equations (2x – y = 10) and (5x + y = 200), and we get
7x = 210
x = 30
y = 50

And the number who study French is x + y = 80.

Answer = (B)

STEP1) READ THE QUESTION AND CHOOSE BETWEEN VENN DIAGRAM AND DOUBLE MATRIX. I DID THIS QUESTION BOTH WAYS AND FOUND THAT VENN DIAGRAM IS EASIER. IF ANYONE HAS A WAY TO QUICKLY DISCERN THE EASIEST WAY TO SOLVE THIS QUESTION LET ME KNOW. I DON'T I WAS QUICK ENOUGH TO CHOOSE VENN DIAGRAM FOR THIS QUESTION BUT I WANT TO GET BETTER AT DISCERNING BETWEEN THE TWO (VENN DIAGRAM VS. DOUBLE MATRIX)

STEP2) IDENTIFY FOR WHICH VARIABLE YOU WILL BE SOLVING. THIS QUESTION ASKS FOR FRENCH WHICH I HAVE LABELED "C". THIS IS IMPORTANT BECAUSE AT THE VERY END OF THE SOLUTION I MUST DOUBLE CHECK THAT MY ANSWER IS FOR C AND NOT FOR OTHER VARIABLE.

STEP3) TRANSLATE INTO FORMULAS
FORMULA 1) A+B+C+D=200
FORMULA 2) B=D
FORMULA 3) 1/4(A+B)=B =>HERE SIMPLIFY TO A=3B. AT THIS POINT YOU SHOULD REALIZE THAT YOU HAVE D AND A IN TERMS OF B, YOU ONLY NEED C TO BE IN TERMS OF B IN ORDER TO GO BACK TO FIRST FORMULA AND SOLVE.
FORMULA 4) B+C=A-10 => REPLACE A WITH A=3B => B+C=3B-10 => C=2B-10

STEP4) REPLACE TO THE FORMULA 1) NOW THAT ALL THE FORMULAS ARE IN TERMS OF B.
A+B+C+D=200 => 3B+B+2B-10+B=200 => B=30

STEP5) SOLVE FOR C IN THE FORMULA, C=2B-10 => C=2(30)-10 => 60-10 => C=50

The easiest way to solve I believe is to use

200= PS + PF -both + Neither
since both and neither are equal, they cancel out

So spanish only= PS- PS/4= 3PS/4
French= 3PS/4 - 10

PS +3PS/4 - 10 = 200
7PS/4=210
PS=120

Only French is French - Both

3PS/4 -10 - PS/4= PS/2 -10

Plug in the value of PS

120/2 -10
60-10= 50

We can use the formula:

Total = French + Spanish - Both + Neither

We can let Spanish = x; then Both = x/4 (since it’s given that one quarter of those who study Spanish also study French). Since Both = Neither, then Neither = x/4. Since the total number who study French is 10 fewer than those who study Spanish only, we have French = x - x/4 - 10 = 3x/4 - 10. Since Total = 200, in terms of x, we have:

200 = 3x/4 - 10 + x - x/4 + x/4

200 = 3x/4 - 10 + x

800 = 3x - 40 + 4x

840 = 7x

120 = x

Since Spanish = 120, Both = 120/4 = 30 and French = 3(120)/4 - 10 = 80 and French Only = French - Both = 80 - 30 = 50.

Answer: B

Harsha
GMAT Tutor | gmatanchor.com