Discussion
In a school, a total of 180 students are enrolled in the foreign languages program. The foreign languages offered are : French, German and Chinese and the number of people studying each of these subjects are 80, 90 and 100 respectively. The number of students who study more than one of the three subjects is 50% more than the number of students who study all the three subjects. How many students study all three foreign languages ?
Answer: C
36 students study all three languages.
Let x study all three and y exactly two. Those studying more than one number 1.5x, so y + x = 1.5x and y = 0.5x. Let a study exactly one language.
\[ \begin{aligned} a + y + x &= 180 \\ a + 2y + 3x &= 80 + 90 + 100 = 270 \\ y + 2x &= 90 \\ 0.5x + 2x &= 90 \Rightarrow x = 36 \end{aligned} \]
Check: 18 study exactly two, 126 exactly one, and 126 + 2(18) + 3(36) = 270.
18 is the number studying exactly two, a likely confusion; 24 and 42 fail the 270 total.
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