Question

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

Peter has been saving his loose change for several days. When he counted his quarters and dimes, he found they had a total value $13.10. The number of quarters was fifteen more than three times the number of dimes. How many quarters and how many dimes did Peter have?

Short answer

The number of quarters are 48 and number of dimes are 11.

Step-by-step solution

Step 1. Given Information 

The given data is that Peter has been saving his loose change for several days. When he counted his quarters and dimes, he found they had a total value $13.10. The number of quarters was fifteen more than three times the number of dimes.

Step 2. Calculation  

Let x be the number of quarters and ybe the number of dimes.

Total number of quarters and dimes is $\$13.10$. Quarter has one-fourth of the value and one dime has one tenth value that can be shown as $\frac{1}{4}x+\frac{1}{10}y=13.10--\left(1\right)$

The number of quarters was fifteen more than three times the number of dimes that is$x=3y+15--\left(2\right)$

Substitute the value of x from equation (2) in equation (1) to find the value of y.

$$\begin{gathered} \frac{1}{4}(3y+15)+\frac{1}{10}y=13.10 \\ 0.75y+3.75+0.1y=13.10 \\ y=\frac{13.10-3.75}{0.85} \\ y=11 \end{gathered}$$

Substitute the value of y in equation (2) to find the value of x.

$$\begin{gathered} x=3\left(11\right)+15 \\ x=33+15 \\ x=48 \end{gathered}$$