Chapter 7.3, Problem 96E

Solution

To calculate: The number of bills of each carrying by Diana.

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

Given Information:

Diana has saved $177 to take with her on the family vacation. She has 51 bills consisting of $1, $5, and $10 bills. If the number of $5 bills is three times the number of $10 bills.

Calculation:

Consider the given information,

Suppose that x is representing the number of bills of $1, and y is representing the number of bills of $5, and z is representing the number bills of $10 bills.

Now, write the equation, as Diana has saved $177. So,

  $x+5y+10z=177$   ...... (1)

The total number of bills are 51.

  $x+y+z=51$   ...... (2)

And, the number of $5 bills is 3 times the number of $10 bills.

  $y-3z=0$   ...... (3)

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

  $A=\left[\begin{array}{ccc}1& 5& 10\\ 1& 1& 1\\ 0& 1& -3\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\text{ and }B=\left[\begin{array}{c}177\\ 51\\ 0\end{array}\right]$

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,

  $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}27\\ 18\\ 6\end{array}\right]$

Hence, the number of bills with amounts are 27 with $1 bills, 18 with $5 bills, and 6 with $10 bills.