Question

At the beginning of a political science class at a large university, the students were asked which term, liberal or conservative, best described their political views. They were asked the same question at the end of the course, to see what effect the class discussions had on their views. Of those that characterized themselves as “liberal” initially,$30\%$ held conservative views at the end. Of those who were conservative initially,$40\%$ moved to the liberal camp. It turned out that there were just as many students with conservative views at the end as there had been liberal students at the beginning. Out of the$260$ students in the class, how many held liberal and conservative views at the beginning of the course and at the end? (No students joined or dropped the class between the surveys, and they all participated in both surveys.)

Short answer

At the beginning of the course, $120$students held liberal views and $140$students held conservative views and at the end of the course,$140$students held liberal views and localid="1668167170915" $120$students held conservative views.

Step-by-step solution

Represent the number of students having liberal and conservative views in terms of a system of equations. 

The number of students having liberal and conservative views can be found using the augmented matrix.

The total number of students is, $L+C=260 ......\left(1\right)$

The students with conservative views at the end equal the students with liberal views at the beginning.

The number of students with liberal view equals,

$\begin{array}{rcl}0.3L+0.6C& =& L\\ & \Rightarrow & 0.6C=0.7L\\ & \Rightarrow & 0.7L-0.6C=0 ......\left(2\right)\end{array}$

Consider the equations (1) and (2). The system of equations is,

$\left|\begin{array}{c}L+C=260\\ 0.7L-0.6C=0\end{array}\right|$

Represent the above equations in terms of a matrix.

$\left|\begin{array}{ccc}1& 1& 260\\ 0.7& -0.6& 0\end{array}\right|$

Perform the row operations on the matrix.

Consider the obtained matrix and reduce it to the row echelon form.

$rref\left|\begin{array}{ccc}1& 1& 260\\ 0.7& -0.6& 0\end{array}\right|\to \left|\begin{array}{ccc}1& 0& 120\\ 0& 1& 140\end{array}\right|$

The values are, $L=120,C=140$.

Hence, at the beginning of the course, $120$students held liberal views and students held conservative views and $140$at the end of the course, $140$students held liberal views and $120$students held conservative views.