Chapter 7.3, Problem 95E

Solution

To calculate: The number of each coin Lucas has in his coin box.

The obtained result is 22 nickels, 35 dimes, and 17 quarters.

Given Information:

Lucas has 74 coins consisting of nickels, dimes, and quarters in his coin box. The total value of the coins is $8.85. If the number of nickels and quarters is four more than the number of dimes.

Calculation:

Consider the given information,

Suppose that x is representing the number of nickels ($0.05), and y is representing the number of dimes ($0.10) , and z is representing the number quarters ($0.25).

Now, write the equation, as Lucas has 74 coins consisting of nickels, dimes, and quarters. So,

  $x+y+z=74$   ...... (1)

The total value of the coins is $8.85.

  $0.05x+0.10y+0.25z=8.85$   ...... (2)

And, the number of nickels and quarters is 4 more than the number of dimes.

  $\begin{array}{c}x+z=y+4\\ x-y+z=4\end{array}$   ...... (3)

Now, write the system of the matrix as $AX=B$ .

  $A=\left[\begin{array}{ccc}1& 1& 1\\ 0.05& 0.10& 0.25\\ 1& -1& 1\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\text{ and }B=\left[\begin{array}{c}74\\ 8.85\\ 4\end{array}\right]$

The equation can be solved as,

  $\begin{array}{c}{A}^{-1}AX=A^{-1}B\\ X=A^{-1}B\end{array}$

Solve for  X by multiplying both sides by  $A^{-1}$ . That is,  $X=A^{-1}B$. Using a graphing calculator (as mention in the book), find the solution of the system as shown below:

  

The obtained solutions are defined as,

  $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}22\\ 35\\ 17\end{array}\right]$

Hence, the obtained result is 22 nickels, 35 dimes, and 17 quarters.