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:

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<reasoning>

Define the variables and coin values

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

Model the total value equation

Using the Linear Function Modeling and Systems of Linear Equations knowledge points
\[
0.25q + 0.10d = 4.30
\]

Model the coin quantity relationship

Using the Linear Function Modeling and Systems of Linear Equations knowledge points
\[
q = d + 6 \implies q - d = 6
\]

Combine into a system of equations

Using the Systems of Linear Equations knowledge point
\[

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

\]

Analyze the incorrect option F

Using the Systems of Linear Equations knowledge point
Option F is incorrect because it swaps the values of the coins, assigning \$0.10 to quarters (LXI0) and \$0.25 to dimes (\(0.25d\)), and incorrectly sets the total value of the coins to 6 instead of 4.30.
</reasoning>

<answer>
<mcq-option>F \(

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

\)</mcq-option>
<mcq-option>G \(

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

\)</mcq-option>
<mcq-correct>H \(

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

\)</mcq-correct>
<mcq-option>J \(

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

\)</mcq-option>
</answer>

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Answer:

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<reasoning>

Define the variables and coin values

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

Model the total value equation

Using the Linear Function Modeling and Systems of Linear Equations knowledge points
\[
0.25q + 0.10d = 4.30
\]

Model the coin quantity relationship

Using the Linear Function Modeling and Systems of Linear Equations knowledge points
\[
q = d + 6 \implies q - d = 6
\]

Combine into a system of equations

Using the Systems of Linear Equations knowledge point
\[

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

\]

Analyze the incorrect option F

Using the Systems of Linear Equations knowledge point
Option F is incorrect because it swaps the values of the coins, assigning \$0.10 to quarters (LXI0) and \$0.25 to dimes (\(0.25d\)), and incorrectly sets the total value of the coins to 6 instead of 4.30.
</reasoning>

<answer>
<mcq-option>F \(

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

\)</mcq-option>
<mcq-option>G \(

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

\)</mcq-option>
<mcq-correct>H \(

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

\)</mcq-correct>
<mcq-option>J \(

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

\)</mcq-option>
</answer>

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