Chapter 3.8, Problem 26PPS

To determine

To find: The number of millilitres of blue and red food colour to be mixed for making the amount of purple food colouring.

Answer to Problem 26PPS

  $\left[\begin{array}{c}48.24\\ 23.24\end{array}\right]$ Millilitres.

Explanation of Solution

Given information:

Peggy needs 25 millilitres of a $44\text{\%}$ concentration food colouring. The store has a 25 % red and 50 % blue concentration of food colouring.

Calculation:

Consider the case of Peggy requiring coloured frosting for ice. Peggy requires $25\text{mm}$ of coloured frosting with $44\text{\%}$ concentration. The store has $25\text{\%}$ red and $50\text{\%}$ blue food colouring.

Let the quantity of red colouring be $x$

Let the quantity of blue colouring be $y$

Hence,

  $\begin{array}{l}x+y=25\\ x+y=20\\ 0.25x+0.5y=0.44\end{array}$

Represent the equations in matrix form as follows,

  $\left[\begin{array}{cc}1& 1\\ 0.25& 0.5\end{array}\right]\left[\begin{array}{l}x\hfill \\ y\hfill \end{array}\right]=\left[\begin{array}{c}25\\ 0.44\end{array}\right].$

Determinant of the coefficient matrix is,

  $\begin{array}{l}\left|A\right|=\left|\begin{array}{cc}1& 1\\ 0.25& 0.5\end{array}\right|\\ =1\times 0.5-1\times 0.25\\ =0.25\end{array}$

Inverse of the matrix is,

  $\begin{array}{l}{A}^{-1}=\frac{1}{\left|A\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ =\frac{1}{0.25}\left[\begin{array}{cc}0.5& -1\\ -0.25& 1\end{array}\right]\\ =\left[\begin{array}{cc}\frac{0.5}{0.25}& -\frac{1}{0.25}\\ -\frac{0.25}{0.25}& \frac{1}{0.25}\end{array}\right]\\ =\left[\begin{array}{cc}2& -4\\ -1& 4\end{array}\right]\end{array}$.

The solution is given by,

  $\begin{array}{l}X=A^{-1}B\\ X=\left[\begin{array}{cc}2& -4\\ -1& 4\end{array}\right]\left[\begin{array}{c}25\\ 0.44\end{array}\right]\\ =\left[\begin{array}{c}2\times 25+(-4)0.44\\ -1\times 25+4\times 0.44\end{array}\right]\\ =\left[\begin{array}{c}48.24\\ 23.24\end{array}\right]\end{array}$

Hence, the answer is $\left[\begin{array}{c}48.24\\ 23.24\end{array}\right]$ millilitres.