Q. 135 - Intermediate Algebra

Question

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

Lucinda had a pocketful of dimes and quarters with a value of $$6.20$. The number of dimes is eighteen more than three times the number of quarters. How many dimes and how many quarters does Lucinda have?

Step-by-Step Solution

Answer

Lucinda has $42$ number of dimes and $8$ number of quarters.

Step 1. Given Information

The total value of quarters and dimes is $$6.20$.

Also, the number of dimes is eighteen more than three times the number of quarters. 

Step 2. Identify and name what we are looking for

We need to find the number of quarters and dimes Lucinda has.

Let $x$ represents the number of quarters and $y$ represents the number of dimes.

Step 3. Form the equations

The value of each quarter is $$0.25$ and the value of each dime is $$0.10$. The value of $x$ quarters and $y$ dimes is $$6.20$. So an equation can be written as

$0.25x+0.10y=6.20 ...\left(1\right)$

The number of dimes is eighteen more than three times the number of quarters, so

$y=3x+18 ...\left(2\right)$

Step 4. Solve using substitution

Using the second equation, substitute $3x+18$ for $y$ in the first equation and solve for $x$

$\begin{array}{rcl}0.25x+0.10y& =& 6.20\\ 0.25x+0.10(3x+18)& =& 6.20\\ 0.25x+0.30x+1.8& =& 6.20\\ 0.55x+1.8-1.8& =& 6.20-1.8\\ \frac{0.55x}{0.55}& =& \frac{4.4}{0.55}\\ x& =& 8\end{array}$

Step 5. Solve for y

In the second equation, substitute $8$ for $x$ and find the value of $y$

$$\begin{gathered} y=3y+18 \\ y=3\times 8+18 \\ y=24+18 \\ y=42 \end{gathered}$$

So number of quarters is $8$ and number of dimes is $42$

Step 6. Check the solution.

Substitute $8$ for $x$ and $42$ for $y$ in the first equation formed.

$\begin{array}{rcl}0.25x+0.10y& =& 6.20\\ 0.25·8+0.10·42& =& 6.20\\ 2+4.20& =& 6.20\\ 6.20& =& 6.20\end{array}$

It is a true statement.

Again, substitute the values in the second equation formed.

$\begin{array}{rcl}y& =& 3x+18\\ 42& =& 3·8+18\\ 42& =& 24+18\\ 42& =& 42\end{array}$

This is also a true statement.

So the point satisfies both the equations.