The company inventory consists of 51 bags of Colombian and 33 bags of Brazilian beans. Each bag holds 80 pounds of beans, so in total the company has 4080 pounds of Colombian and 2640 pounds of Brazilian beans.
The company wants to use up its entire inventory, a total of 6720 pounds of beans.
Let [tex]r[/tex] and [tex]m[/tex] denote the amount (in pounds) of the robust and mild blends, respectively, that the company should end up producing.
To use the entire inventory, we must have
[tex]r+m=6720[/tex]
Each pound of the robust blend uses 12 ounces (3/4 = 0.75 pound) of Colombian beans, and each pound of the mild blend uses 6 ounces (3/8 = 0.375 pound) of Colombian beans, so that
[tex]0.75r+0.375m=4080[/tex]
while each pound of the robust blend uses 4 ounces (1/4 = 0.25 pound) of Brazilian beans, and each pound of the mild blend uses 10 ounces (5/8 = 0.625 pound) of Brazilian beans, so that
[tex]0.25r+0.625m=2640[/tex]
Multiply both equations by 8 to get rid of the rational coefficients:
[tex]\begin{cases}6r+3m=32640\\2r+5m=21120\end{cases}[/tex]
Subtract 3(second equation) from (first equation) to eliminate [tex]r[/tex]:
[tex](6r+3m)-3(2r+5m)=32640-3\cdot21120[/tex]
[tex]-12m=-30720\implies\boxed{m=2560}[/tex]
Then
[tex]r+2560=6720\implies\boxed{r=4160}[/tex]
So the company needs to produce 4160 pounds of the robust blend and 2560 pounds of the mild blend.
