Chapter 3, Problem 10STP

(a)

To determine

To write: System of equations to model the situation.

Answer to Problem 10STP

System of equations,

For total number:

  $q+d=14$

For total value:

  $0.25q+0.10d=2.75$

Explanation of Solution

Given information:

Total number of quarters and dimes is 14.

Total value of all the coins is $2.75.

Let

qbe the number of quarters.

And

d be the number of dimes.

We know that

Total number of quarters and dimes is 14.

Thus,

For total number

  $q+d=14$

We also know

Total value of all the coins is $2.75.

Since we do not know the number of quarters and dimes,

Thus,

For total value of coins:

  $0.25q+0.10d=2.75$

(b)

To determine

To write: Matrix equation for solving d and q .

Answer to Problem 10STP

Matrix equation:

  $\left[\begin{array}{cc}1& 1\\ 0.25& 0.10\end{array}\right]\cdot \left[\begin{array}{c}q\\ d\end{array}\right]=\left[\begin{array}{c}14\\ 2.75\end{array}\right]$

Explanation of Solution

Given information:

Total number of quarters and dimes is 14.

Total value of all the coins is $2.75.

From Part (a) result,

We have

System of equations,

For total number:

  $q+d=14$

For total value:

  $0.25q+0.10d=2.75$

Through the use of the system of equation,

The matrix equation becomes

  $\left[\begin{array}{cc}1& 1\\ 0.25& 0.10\end{array}\right]\cdot \left[\begin{array}{c}q\\ d\end{array}\right]=\left[\begin{array}{c}14\\ 2.75\end{array}\right]$

(c)

To determine

Number of dimes and quarters Valeria has.

Answer to Problem 10STP

Valeria has 9 quarters and 5 dimes.

Explanation of Solution

Given information:

Total number of quarters and dimes is 14.

Total value of all the coins is $2.75.

We have

The matrix:

  $\left[\begin{array}{cc}1& 1\\ 0.25& 0.10\end{array}\right]$

To know if the inverse exists,

Calculate the determinant first:

  $\text{Det}=1\left(0.10\right)-1\left(0.25\right)=0.10-0.25=-0.15$

Now,

We need to proceed to the inverse of matrix because the determinant is not equal to 0.

Thus,

  $\begin{array}{l}{\left[\begin{array}{cc}1& 1\\ 0.25& 0.10\end{array}\right]}^{-1}=-\frac{1}{0.15}\left[\begin{array}{cc}0.10& -1\\ -0.25& 1\end{array}\right]\\ =\left[\begin{array}{cc}-\frac{1}{0.15}\left(0.10\right)& -\frac{1}{0.15}\left(-1\right)\\ -\frac{1}{0.15}\left(-0.25\right)& -\frac{1}{0.15}\left(1\right)\end{array}\right]\\ =\left[\begin{array}{cc}-\frac{0.10}{0.15}& \frac{1}{0.15}\\ \frac{0.25}{0.15}& -\frac{1}{0.15}\end{array}\right]\end{array}$

Then

Multiply each of the matrix equation obtained in Part (b) with inverse matrix:

  $\left[\begin{array}{cc}-\frac{0.10}{0.15}& \frac{1}{0.15}\\ \frac{0.25}{0.15}& -\frac{1}{0.15}\end{array}\right]\cdot \left[\begin{array}{cc}1& 1\\ 0.25& 0.10\end{array}\right]\cdot \left[\begin{array}{c}q\\ d\end{array}\right]=\left[\begin{array}{cc}-\frac{0.10}{0.15}& \frac{1}{0.15}\\ \frac{0.25}{0.15}& -\frac{1}{0.15}\end{array}\right]\cdot \left[\begin{array}{c}14\\ 2.75\end{array}\right]$

Now,

Inverse matrix gets eliminated:

  $\left[\begin{array}{c}q\\ d\end{array}\right]=\left[\begin{array}{cc}-\frac{0.10}{0.15}& \frac{1}{0.15}\\ \frac{0.25}{0.15}& -\frac{1}{0.15}\end{array}\right]\cdot \left[\begin{array}{c}14\\ 2.75\end{array}\right]$

Perform the multiplication to find the values of q and d :

  $\begin{array}{l}\left[\begin{array}{c}q\\ d\end{array}\right]=\left[\begin{array}{c}-\frac{0.10}{0.15}\left(14\right)+\frac{1}{0.15}\left(2.75\right)\\ \frac{0.25}{0.15}\left(14\right)+\left(-\frac{1}{0.15}\right)\left(2.75\right)\end{array}\right]\\ =\left[\begin{array}{c}-\frac{1.4}{0.15}+\frac{2.75}{0.15}\\ \frac{3.5}{0.15}+\left(-\frac{2.75}{0.15}\right)\end{array}\right]\\ =\left[\begin{array}{c}\frac{1.35}{0.15}\\ \frac{0.75}{0.15}\end{array}\right]\\ =\left[\begin{array}{c}9\\ 5\end{array}\right]\end{array}$

Therefore,

Valeria has 9 quarters and 5 dimes.