Sets / Counting Problem

[bfman]

Here is the problem.

70 students are enrolled in Math, English, or German. 40 students are in Math, 35 are in English, 30 are in German. 15 students are enrolled in all 3 of the courses. How many of the students are enrolled in exactly two of the courses: Math, English, and German?

The book has very bad explanation and possibly wrong answer for this problem. Using my own logic I get a different answer than the book.

Please let me know what you guys get as answer for this problem.

thank you.

[joostinshu]

The answer is 5.

You have 105 "spots" in English, Math, German combined and with overlap.

The problem states 15 students take all 3 classes which means those 15 students account for 45 "spots" leaving 60 "spots".

We know there are only 70 real students total which we subtract 15 (that take 2 classes) from to get 55. We have 55 people and 60 "spots" remaining. It becomes clear that 5 students must take 2 classes.

[ssmiles08]

I got 5 as well.

You might want to set up a ven diagriam with 3 circles and plug in the following values:


all 3 classes: 15

M+E = x
M+G = y
E+G = z

Only M = 25 -(x+y)
Only E = 20-(x+z)
Only G = 15 -(y+z)

The question is asking for x+y+z

(25-x-y) + (20-x-z) + (15-y-z) + (x+y+z) +15 = 70

75-x-y-z = 70

x+y+z = 5

[bfman]

Thank you both of you for helpful explanations!
The book's answer is also 5, but very bad explanation. Thanks!