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

BTGmoderatorDC wrote:At a certain school of 200 students, the students can study French, Spanish, both or neither. Just as many study both as study neither. One quarter of those who study Spanish also study French. The total number who study French is 10 fewer than those who study Spanish only. How many students study French only?

A. 30
B. 50
C. 70
D. 90
E. 120

OA B

Source: Magoosh

Say,

the number of students who study only French = f;
the number of students who study only Spanish = s;
the number of students who study both French & Spanish = b;

Thus,

the number of students who study French = f + b;
the number of students who study Spanish = s + b;

the number of students who study neither French nor Spanish = n = b (given)

=> f + s + b + n = 200 => f + s + 2b = 200 ---(1)

Given 'One-quarter of those who study Spanish also study French,' we have (s + b)/4 = b => s = 3b ---(2)

Given 'The total number who study French is 10 fewer than those who study Spanish only,' we have f + b = s – 10 => s = f + b + 10 ---(3)

From (2) and (3), we have f + b + 10 = 3b => f = 2b – 10 ---(4)

From (1) and (4), we have

f + s + 2b = 200
=> 2b – 10 + 3b + 2b = 200
b = 30

Thus, f = 2b – 10 = 2*30 – 10 = 50

The correct answer: B

Hope this helps!

-Jay
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BTGmoderatorDC wrote:At a certain school of 200 students, the students can study French, Spanish, both or neither. Just as many study both as study neither. One quarter of those who study Spanish also study French. The total number who study French is 10 fewer than those who study Spanish only. How many students study French only?

A. 30
B. 50
C. 70
D. 90
E. 120

OA B

Source: Magoosh

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