Q. 137 - Intermediate Algebra

Question

In the following exercise, translate to a system of equations and solve. 

A cashier has $54$ bills, all of which are \($10$ or \)$20$ bills. The total value of the money is $$910$. How many of each type of bill does the cashier have?

Step-by-Step Solution

Answer

The number of $$10$ bills cashier has is $17$ and the number of $$20$ bills cashier has is $37$.

Step 1. Given Information

There are total $54$ bills of $$10$ and $$20$.

The total value of the money is $\$910$.

Step 2. Identify and name what we are looking for

We need to find the number of $$10$ and $$20$ bills.

Let $x$ represents the number of $$10$ bills

and $y$ represents the number of $$20$ bills.

Step 3. Form the equations

The total number of bills is $54$, so the equation is

$x+y=54 ...\left(1\right)$

Now, the total amount is $$910$ where $x$ is the number of $$10$ bills and $y$ is the number of $$20$ bills. So the equation is

$10x+20y=910 ...\left(2\right)$

Step 4. Solve using substitution

Solve the first equation for $y$

$\begin{array}{rcl}x+y& =& 54\\ x+y-x& =& 54-x\\ y& =& 54-x ...\left(3\right)\end{array}$

Using the third equation substitute $54-x$ for $y$ in the second equation and solve for $x$

$\begin{array}{rcl}10x+20y& =& 910\\ 10x+20(54-x)& =& 910\\ 10x+1080-20x& =& 910\\ 10x-20x+1080-1080& =& 910-1080\\ -10x& =& -170\\ x& =& 17\end{array}$

Step 5. Find the value of y

Substitute $17$ for $x$ in the third equation

$$\begin{gathered} y=54-x \\ y=54-17 \\ y=37 \end{gathered}$$

Thus the number of $$10$ bills is $17$ and the number of $$20$ bills is $37$.

Step 6. Check the solution

Substitute $17$ for $x$ and $37$ for $y$ in the first equation formed.

$\begin{array}{rcl}x+y& =& 54\\ 17+37& =& 54\\ 54& =& 54\end{array}$

It is a true statement.

Again, substitute the values in the second equation formed.

$\begin{array}{rcl}10x+20y& =& 910\\ 10·17+20·37& =& 910\\ 170+740& =& 910\\ 910& =& 910\end{array}$

This is also a true statement.

So the point satisfies both the equations.