hazelnut01 wrote: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
This is an EITHER/OR group problem.
Every child EITHER takes Spanish OR does not.
Every child EITHER takes French OR does not.
For an EITHER/OR group problem, we can use a GROUP GRID (also known as a double-matrix) to organize the data.
Let S = Spanish, NS = not Spanish, F = French, NF = not French.
In the grids below, the entries in any given row or column must add up to the TOTAL of that row or column.
Let the total number of students = 10x.
The following grid is yielded:
10% of students do not take a foreign language class.
Half of all students are in a French class.
Students who take neither language = 10% of 10x) = x.
Students who take French = (half of 10x) = 5x.
Entering these values into the grid and completing the rest of the middle row and the rightmost column, we get:
70% of students take exactly one foreign language class.
70% of 10x = 7x.
In the grid just above, only Spanish = 4x.
Thus, only French = (total taking exactly one language) - (only Spanish) = 7x - 4x = 3x.
Entering this value into the grid and completing the remainder of the grid, we get:
50 students take classes in both languages.
In the final grid, the number of students taking both French and Spanish = 2x.
Thus:
2x = 50
x = 25.
How many students are in a Spanish class?
In the final grid, the total number of students taking Spanish = 6x.
Thus:
6x = 6*25 = 150.
The correct answer is
B.
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