Chapter 7.4, Problem 92E

To determine

To calculate: the number of pounds of glossy, semi-gloss and matte paper in a $100$ -pound package.

Answer to Problem 92E

the number of pounds of paper having glossy grade: $50$ pound

the number of pounds of paper having gloss grade: $35$ pound

the number of pounds of paper having matte grade: $15$ pound

Explanation of Solution

Given information:

a wholesale company sells a $100$ -pound package of computer paper that consists of three grades, glossy, semi-gloss and matte. Glossy costs $\$5.50$ per pound, gloss costs $\$4.25$ per pound and matte costs $\$3.75$ per pound.one half of the $100$ -pound package consists of the two less expensive grades.

Calculation Let the number of pounds of paper having glossy grade: $x$

the number of pounds of paper having gloss grade: $y$

the number of pounds of paper having matte grade: $z$

per pound glossy grade paper costs: $\$5.50$

  $x$ pound glossy grade paper costs: $5.50x$

per pound gloss grade paper costs: $\$4.25$

  $y$ pound gloss grade paper costs: $4.25y$

per pound matte grade paper costs: $\$3.75$

  $z$ pound matte grade paper costs: $3.75z$

Total package: 100

  $x+y+z=100$

Total cost of the $100$ -pound package: $\$480$

  $5.50x+4.25y+3.75z=480$

one half of the $100$ -pound package consists of the two less expensive grades.

  $\begin{array}{l}y+z=\frac{100}{2}\\ y+z=50\end{array}$

Equation 1: $x+y+z=100$

Equation 2: $5.50x+4.25y+3.75z=480$

Equation 3: $y+z=50$

Now solving this system

Augmented matrix: $\left(\begin{array}{ccccc}1& 1& 1& ⋮& 100\\ 5.50& 4.25& 3.75& ⋮& 480\\ 0& 1& 1& ⋮& 50\end{array}\right)$

Reducing augmented matrix in Echelon form

  $\begin{array}{l}\left(\begin{array}{ccccc}1& 1& 1& ⋮& 100\\ 5.50& 4.25& 3.75& ⋮& 480\\ 0& 1& 1& ⋮& 50\end{array}\right)\\ R_2\to R_2-5.50R_1\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 100\\ 0& -1.25& -1.75& ⋮& -70\\ 0& 1& 1& ⋮& 50\end{array}\right)\\ R_2\leftrightarrow R_3\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 100\\ 0& 1& 1& ⋮& 50\\ 0& -1.25& -1.75& ⋮& -70\end{array}\right)\\ R_3\to R_3+1.25R_1\\ \left(\begin{array}{ccccc}1& 1& 1& ⋮& 100\\ 0& 1& 1& ⋮& 50\\ 0& 0& -0.5& ⋮& -7.5\end{array}\right)\end{array}$

Now, system of linear equation corresponding to reduced echelon matrix

  $\begin{array}{l}x+y+z=100\\ y+z=50\\ -0.5z=-7.5\\ z=15\\ y=35\\ x=50\end{array}$

Therefore,

the number of pounds of paper having glossy grade: $50$ pound

the number of pounds of paper having gloss grade: $35$ pound

the number of pounds of paper having matte grade: $15$ pound