Chapter 2.5, Problem 48E

To determine

To find: the gallons of 10% and 25% alcohol solution required to make 12 gallons of 15% alcohol solution.

Answer to Problem 48E

8 gallons of 10% alcohol solution

4 gallons of 25% alcohol solution

Explanation of Solution

Given information:

Let x and y be the number of gallons of 10% alcohol solution and 25% alcohol solution respectively. The equations depicting the data are as follows:

  $\begin{array}{l}x+y=12\\ 0.10x+0.25y=0.15\left(x+y\right)\to \text{ 0}\text{.05}x-0.10y=0\end{array}$

Calculation:

Set up the matrix for the given equations as follows:

  $\begin{array}{l}\begin{array}{ccc}A& X& B\end{array}\\ \left[\begin{array}{cc}\begin{array}{l}1\\ 0.05\end{array}& \begin{array}{l}1\\ -0.10\end{array}\end{array}\right]\left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{l}12\\ 0\end{array}\right]\end{array}$

For $A=\left[\begin{array}{cc}\begin{array}{l}a\\ c\end{array}& \begin{array}{l}b\\ d\end{array}\end{array}\right]$ , the inverse formula is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}\begin{array}{l}d\\ -c\end{array}& \begin{array}{l}-b\\ a\end{array}\end{array}\right]$ . Put the values in the above formula and evaluate the inverse for the given matrix as follows:

  $\begin{array}{c}{A}^{-1}=\frac{1}{-0.1-0.05}[\begin{array}{c}\begin{array}{l}-0.10\\ -0.05\end{array}\end{array}& \begin{array}{l}-1\\ 1\end{array}\end{array}]=-\frac{1}{0.15}\left[\begin{array}{cc}\begin{array}{l}-0.10\\ -0.05\end{array}& \begin{array}{l}-1\\ 1\end{array}\end{array}\right]\\ =\left[\begin{array}{cc}\begin{array}{l}2/3\\ 1/3\end{array}& \begin{array}{l}20/3\\ -20/3\end{array}\end{array}\right]$

Solve $A^{-1}B=X$ as follows:

  $\begin{array}{c}\left[\begin{array}{cc}\begin{array}{l}2/3\\ 1/3\end{array}& \begin{array}{l}20/3\\ -20/3\end{array}\end{array}\right]\cdot \left[\begin{array}{l}12\\ 0\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\\ \left[\begin{array}{l}8\\ 4\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\end{array}$

Thus, the value of x and y is 8 gallon of 10% alcohol solution and 4 gallon of 25% alcohol solution respectively.