Question

A video store specializes in three types of movies: children's, American West, and horror. It's known that:
60% of children's movies plus 50% of westerns account for 30% of all movies.
20% of children's movies plus 60% of westerns plus 60% of horror movies represent half of all movies.

Answer 1

X, y, z denote children's movies, western movies, and horror movies, respectively. The related system of equations is formed and solved.

$\left\{\begin{matrix} \qquad\qquad\ \dfrac{60}{100}x + \dfrac{50}{100}y =\dfrac{30}{100}(x+y+z)\\ \dfrac{20}{100}x+ \dfrac{60}{100}y+ \dfrac{60}{100}z= \dfrac{1}{2}(x+y+z) \\ \ \ \ x+100=y \end{matrix} \right.$

Simplifying the system of equations, we have

$\left\{\begin{matrix} 3x + 2y -3z=0\\ -3x+ y+ z= 0 \\ \ \ \ x+100=y \end{matrix} \right.\ \ \Longrightarrow \ \left\{\begin{matrix} 5x-3z=-200\\ -2x+ z= -100 \end{matrix} \right.$

$\ \ \ 5x-3z=-200\\\underline{-6x+3z=-300} \ \Rightarrow \ x=500\\\ \ -x+0= \ 500$

                                       $y&=500+100=600\\ z&=3(500)-600=900$

The video store has 500 children's movies, 600 westerns and 900 horror movies.