Chapter 3.8, Problem 46STP

To determine

To find: The number of girls and boys voted for the theme.

Answer to Problem 46STP

380 boys and 325 girls (option C).

Explanation of Solution

Given information:

For the topic "A Night under the Stars" girls voted 40% and 60% for " The Time of My life". Whereas, for "The Time of My life" boys voted 75% and 25% for " A Night under the Stars".The total number of votes received for the topic "A Night under the Stars" is 225 and 480 votes for "The Time of My Life".

Calculation:

The following equation represents the case:

  $\begin{array}{l}0.4x+0.25y=225\\ 0.6x+0.75y=480\\ 0.6x+0.10y=480\end{array}$

Represent equation in matrix form:

  $\left[\begin{array}{cc}0.4& 0.25\\ 0.6& 0.75\end{array}\right]\left[\begin{array}{l}x\hfill \\ y\hfill \end{array}\right]=\left[\begin{array}{l}225\hfill \\ 480\hfill \end{array}\right]$

The solution is given by the following formula:

  $X=A^{-1}B$

First, determinant of coefficient matrix is calculated below.

  $\begin{array}{l}\left|\begin{array}{l}0.40.25\\ 0.60.75\end{array}\right|\\ =0.4\times 0.75-0.6\times 0.25\\ =0.3-0.15\\ =0.15\end{array}$

Second, inverse of the coefficient matrix is calculated below:

  $\begin{array}{cc}{A}^{-1}=& \frac{1}{\left|A\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]\\ =& \frac{1}{0.15}\left[\begin{array}{cc}0.75& -0.25\\ -0.6& 0.4\end{array}\right]\\ & =\left[\begin{array}{cc}\frac{0.75}{0.15}& -\frac{0.25}{0.15}\\ -\frac{0.6}{0.15}& \frac{0.4}{0.15}\end{array}\right]\\ & =\left[\begin{array}{cc}5& -\frac{5}{3}\\ -4& \frac{8}{3}\end{array}\right]\end{array}$

Substitute value of $A^{-1}$ in formula:

  $\begin{array}{l}X=\left[\begin{array}{cc}5& -\frac{5}{3}\\ -4& \frac{8}{3}\end{array}\right]\left[\begin{array}{l}225\hfill \\ 480\hfill \end{array}\right]\\ =\left[\begin{array}{c}5\times 225-\frac{5}{3}480\\ -4\times 225+\frac{8}{3}480\end{array}\right]\\ =\left[\begin{array}{c}1125-800\\ -900+1280\end{array}\right]\\ =\left[\begin{array}{c}1125-800\\ -900+1280\end{array}\right]\end{array}$

Hence, 380 boys and 325 girls voted. Hence the answer is C.