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:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Systems of Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"Systems of Linear Equations",
"Linear Function Modeling"
]
}
</pre_analysis>

<reasoning>

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
\[

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

\]

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 \(4.30\).

</reasoning>

<answer>
<mcq-option>F \(

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

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

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

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

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

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

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

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

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Systems of Linear Equations"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"tutor",
"mastery_adaptive_brevity"
],
"needs_drawing": false,
"concepts_used": [
"Systems of Linear Equations"
],
"new_concepts": [],
"current_concepts": [
"Systems of Linear Equations",
"Linear Function Modeling"
]
}
</pre_analysis>

<reasoning>

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
\[

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

\]

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 \(4.30\).

</reasoning>

<answer>
<mcq-option>F \(

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

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

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

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

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

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

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

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

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Algebra",
"Systems of Linear Equations"
]
}
</post_analysis>