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Sunday, October 14, 2012

Venn Diagram Problem

Category: Algebra

"Published in Newark, California, USA"

A school has 63 students who will take Physics, Chemistry and Biology. 33 students are taking Physics, 25 students are taking Chemistry, and 26 students are taking Biology. 10 students are taking Physics and Chemistry, 9 students are taking Biology and Chemistry, and 8 students are taking Physics and Biology. If a number of students are taking all three subjects is the same as a number of students are not taking either of the subjects, how many students are taking all the three subjects? How many students are taking only one of the three subjects?

Solution:

From the given problem above, it is a Venn Diagram Problem because it involves the intersection or mutual items of the sets. Consider the figure below:


Photo by Math Principles in Everyday Life

Let x be the number of students are taking Physics, Chemistry and Biology

By further analysis,

Number of students are taking Physics and Chemistry alone = 10 - x

Number of students are taking Physics and Biology alone = 8 - x

Number of students are taking Chemistry and Biology alone = 9 - x

Number of students are taking Physics alone = 33 - (10 - x) - x - (8 - x) = 33 - 10 + x - x - 8 + x = 15 + x

Number of students are taking Chemistry alone = 25 - (10 - x) - x - (9 - x) = 25 - 10 + x - x - 9 + x = 6 + x

Number of students are taking Biology alone = 26 - (8 - x) - x - (9 - x) = 26 - 8 + x - x - 9 + x = 9 + x

Next, we put all the items and parameters in the figure, we have


Photo by Math Principles in Everyday Life

Referring to the problem statement, "if a number of students are taking all three subjects is the same as a number of students are not taking either of the subjects


(15 + x) + (10 - x) + x + (8 - x) + (6 + x) + (9 - x) + (9 + x ) = 63 - x

(Let 63 - x be the number of students are taking Physics, Chemistry, and Biology. 63 is the total number of students in a school. x is the number of students are not taking either of the subjects.)

                             57 + x = 63 - x

                               x + x = 63 - 57

                                   2x = 6

                                     x = 3

Therefore, there are 3 students taking Physics, Chemistry, and Biology. 

Number of students are taking Physics alone = 15 + x = 15 + 3 = 18

Number of students are taking Chemistry alone = 6 + x = 6 + 3 = 9

Number of students are taking Biology alone = 9 + x = 9 + 3 = 12

Therefore, there are 18 + 9 + 12 = 39 students taking only one of the three subjects.