Problem Solving

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

OAB

j_shreyans 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

This is an EITHER/OR group problem.
Every student EITHER studies French OR does not.
Every student EITHER studies Spanish OR does not.
To organize the data in an either/or group problem, use a GROUP GRID (also known as a double-matrix).

Let F = French, NF = not French, S = Spanish, and NS = not Spanish.
Since the total number of students = 200, the following grid is yielded:

In a group grid, the entries in any given row or column must sum to the TOTAL of that row or column.

Just as many study both as study neither.
Let both = neither = x.
The following grid is yielded:


One quarter of those who study Spanish also study French.
Since the number who study both languages -- x -- is equal to 1/4 of the total number who study Spanish, the total number who study Spanish = 4x.
The following grid is yielded:


The total number who study French is 10 fewer than those who study Spanish only.
Since Spanish only = 3x, the total number who study French = 3x-10.
The following grid is yielded:


Adding together the values in the bottom row, we get:
(3x-10) + 4x = 200
7x = 210
x = 30.
The following grid is yielded:

In the resulting grid, French only = 50.

The correct answer is B.