Question

roni wants to write an equation to represent a proportional relationship that has a constant of proportionality equal to \\(\frac{7}{25}\\). she writes the equation \\(y = x + \frac{7}{25}\\). what error is roni making?

she should have written \\(y = -x + \frac{7}{25}\\) so that \\(x\\) and \\(y\\) have a constant sum.
she should have written \\(xy = \frac{7}{25}\\) so that \\(x\\) and \\(y\\) have a constant product.
she should have written \\(y = \frac{7}{25}x\\) so that \\(x\\) and \\(y\\) have a constant quotient.
she should have written \\(y = \frac{7}{25}\\) so that \\(y\\) has a constant value.

Explanation:

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<reasoning>

Identify the given values

The constant of proportionality is given as:
\[k = \frac{7}{25}\]
Roni's incorrect equation is:
\[y = x + \frac{7}{25}\]

Recall proportional relationship form

Using the Proportional Relationship Equation knowledge point, a proportional relationship between two variables \(x\) and \(y\) is represented by the equation:
\[y = kx\]
where \(k\) is the constant of proportionality.

Analyze the constant ratio

Using the Constant of Proportionality knowledge point, the constant \(k\) represents the constant quotient of the two variables:
\[k = \frac{y}{x}\]
This means \(x\) and \(y\) must have a constant quotient, not a constant sum or product.

Identify Roni's error

Roni added the constant of proportionality instead of multiplying it by \(x\). She should have written:
\[y = \frac{7}{25}x\]
This matches the third option.
</reasoning>

<answer>
<mcq-option>She should have written \(y = -x + \frac{7}{25}\) so that x and y have a constant sum.</mcq-option>
<mcq-option>She should have written \(xy = \frac{7}{25}\) so that x and y have a constant product.</mcq-option>
<mcq-correct>She should have written \(y = \frac{7}{25}x\) so that x and y have a constant quotient.</mcq-correct>
<mcq-option>She should have written \(y = \frac{7}{25}\) so that y has a constant value.</mcq-option>
</answer>

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"question_type": "Multiple Choice",
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Answer:

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<reasoning>

Identify the given values

The constant of proportionality is given as:
\[k = \frac{7}{25}\]
Roni's incorrect equation is:
\[y = x + \frac{7}{25}\]

Recall proportional relationship form

Using the Proportional Relationship Equation knowledge point, a proportional relationship between two variables \(x\) and \(y\) is represented by the equation:
\[y = kx\]
where \(k\) is the constant of proportionality.

Analyze the constant ratio

Using the Constant of Proportionality knowledge point, the constant \(k\) represents the constant quotient of the two variables:
\[k = \frac{y}{x}\]
This means \(x\) and \(y\) must have a constant quotient, not a constant sum or product.

Identify Roni's error

Roni added the constant of proportionality instead of multiplying it by \(x\). She should have written:
\[y = \frac{7}{25}x\]
This matches the third option.
</reasoning>

<answer>
<mcq-option>She should have written \(y = -x + \frac{7}{25}\) so that x and y have a constant sum.</mcq-option>
<mcq-option>She should have written \(xy = \frac{7}{25}\) so that x and y have a constant product.</mcq-option>
<mcq-correct>She should have written \(y = \frac{7}{25}x\) so that x and y have a constant quotient.</mcq-correct>
<mcq-option>She should have written \(y = \frac{7}{25}\) so that y has a constant value.</mcq-option>
</answer>

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