Answer to Question #164990 in Algebra for Joe

Question

In a school, all of the students study physics or chemistry. Sixty percent of the students who study physics also study chemistry, but only one third of the chemistry students study physics. If there are 110 students in this school, how many of them study both of these subjects?

Sorry, I’m not 100% sure if this is considered algebra

Answer 1

$\displaystyle \textsf{Let the number of students studying Physics}\\ \textsf{only be}\,\,A, \,\,\textsf{Chemistry be}\,\,C_{mixed}\\ \textsf{and both Physics and Chemistry be}\,\, B.\\ \textsf{also denote the number of students}\\ \textsf{studying Physics or Chemistry be}\,\, A_{mixed} \\ \textsf{and Chemistry be}\,\, B_{mixed} \\ \frac{40 A_{mixed}}{100} = A_{mixed} - B \\ \frac{2 C_{mixed}}{3} = C_{mixed} - B \\ C_{mixed} = 3B \\ A_{mixed} = \frac{5}{3}B \\ \begin{aligned} \textsf{Total number of students}\,\, &= A_{mixed} - B + C_{mixed} - B + B \\&= \frac{5}{3}B - B + 3B - B + B = \frac{11}{3}B = 110 \end{aligned} \\ B = 30\\ \textsf{The number of students offering both}\\ \textsf{Physics and Chemistry is}\,\, 30.$