Question

3 a piggy bank contains quarters and dimes. there are 6 more quarters than dimes. the total value of the coins is $4.30. which system of equations can be used to determine the number of quarters, \\(q\\), and the number of dimes, \\(d\\), in the bank?

f \\(\

$$\begin{aligned} 0.10q + 0.25d &= 6 \\\\ q - d &= 4.30 \\end{aligned}$$

\\)

g \\(\

$$\begin{aligned} 0.10q + 0.25d &= 4.30 \\\\ d - q &= 6 \\end{aligned}$$

\\)

h \\(\

$$\begin{aligned} 0.25q + 0.10d &= 4.30 \\\\ q - d &= 6 \\end{aligned}$$

\\)

j \\(\

$$\begin{aligned} 0.25q + 0.10d &= 6 \\\\ d - q &= 4.30 \\end{aligned}$$

\\)

Explanation:

Define variables and values

Using the Systems of Linear Equations knowledge point
Let \(q\) be the number of quarters and \(d\) be the number of dimes.
Quarters are worth \(\$0.25\) each, and dimes are worth \(\$0.10\) each.

Model the total value equation

Using the Systems of Linear Equations knowledge point

$$ 0.25q + 0.10d = 4.30 $$

Model the coin quantity relationship

Using the Systems of Linear Equations knowledge point

$$ q = d + 6 \implies q - d = 6 $$

Formulate the system of equations

Using the Systems of Linear Equations knowledge point

$$
LATEXBLOCK0
$$

Analyze incorrect option F

Using the Systems of Linear Equations knowledge point
Option F swaps the coefficients and constants:

• It assigns \(\$0.10\) to quarters and \(\$0.25\) to dimes, setting their sum to \(6\) instead of the total value.
• It sets the difference in coin counts \(q - d\) equal to the monetary value \(4.30\).

Answer:

• F \(

$$\begin{aligned} 0.10q + 0.25d &= 6 \\ q - d &= 4.30 \end{aligned}$$

\)

• G \(

$$\begin{aligned} 0.10q + 0.25d &= 4.30 \\ d - q &= 6 \end{aligned}$$

\)

• **H \(

$$\begin{aligned} 0.25q + 0.10d &= 4.30 \\ q - d &= 6 \end{aligned}$$

\) (Correct answer)**

• J \(

$$\begin{aligned} 0.25q + 0.10d &= 6 \\ d - q &= 4.30 \end{aligned}$$

\)