Question

the hernandez family orders 3 large pizzas. they cut the pizzas so that each pizza has the same number of slices, giving them a total of 24 slices.
the wilson family also orders several large pizzas from the same pizza restaurant. they also cut the pizzas so that all their pizzas have the same number of slices. for the wilson family, the equation \\(y = 10x\\) represents the relationship, where \\(x\\) represents the number of pizzas and \\(y\\) represents the number of total slices.
which statements best describe the pizzas bought by the hernandez and wilson families? select two options.

for the hernandez family, the equation \\(y = 3x\\) represents the relationship between the number of slices and number of pizzas, where \\(x\\) represents the number of pizzas and \\(y\\) represents the number of total slices.
for each family, a graph of the relationship between number of pizzas, \\(x\\), and number of slices, \\(y\\), goes through the point \\((0, 0)\\) and is a straight line.
the hernandez familys slices are smaller than the wilson familys slices.
if the wilson family ordered 3 large pizzas from the same restaurant as the hernandez family, they would have more slices than the hernandez family after each family cut their pizzas into slices.
the constant of proportionality is 10 for the wilson family, but cannot be determined for the hernandez family.

Explanation:

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<reasoning>

Find unit rate for Hernandez family

Using the Constant of Proportionality knowledge point
\[
k = \frac{\text{total slices}}{\text{total pizzas}} = \frac{24}{3} = 8\text{ slices per pizza}
\]

Write Hernandez family equation

Using the Proportional Relationship Equation knowledge point
\[
y = 8x
\]

Analyze graph properties

Using the Graphing Proportional Relationships knowledge point
\[

$$\begin{aligned} &\text{Hernandez: } y = 8x \implies (0,0) \text{ is on the line, straight line}\\ &\text{Wilson: } y = 10x \implies (0,0) \text{ is on the line, straight line} \end{aligned}$$

\]

Compare slice counts for 3 pizzas

Using the Constant of Proportionality knowledge point
\[

$$\begin{aligned} &\text{Hernandez: } 24\text{ slices}\\ &\text{Wilson: } y = 10(3) = 30\text{ slices}\\ &30 > 24 \end{aligned}$$

\]

Evaluate given options

Using the Proportional Relationship Equation knowledge point

• Option 1: Incorrect, equation is \(y = 8x\).
• Option 2: Correct, both are proportional relationships.
• Option 3: Incorrect, slice size cannot be determined from slice count.
• Option 4: Correct, 30 slices is more than 24 slices.
• Option 5: Incorrect, constant is 8 for Hernandez.

</reasoning>

<answer>
<mcq-option>For the Hernandez family, the equation \(y = 3x\) represents the relationship between the number of slices and number of pizzas, where \(x\) represents the number of pizzas and \(y\) represents the number of total slices.</mcq-option>
<mcq-correct>For each family, a graph of the relationship between number of pizzas, \(x\), and number of slices, \(y\), goes through the point \((0, 0)\) and is a straight line.</mcq-correct>
<mcq-option>The Hernandez family's slices are smaller than the Wilson family's slices.</mcq-option>
<mcq-correct>If the Wilson family ordered 3 large pizzas from the same restaurant as the Hernandez family, they would have more slices than the Hernandez family after each family cut their pizzas into slices.</mcq-correct>
<mcq-option>The constant of proportionality is 10 for the Wilson family, but cannot be determined for the Hernandez family.</mcq-option>
</answer>

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Answer:

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<reasoning>

Find unit rate for Hernandez family

Using the Constant of Proportionality knowledge point
\[
k = \frac{\text{total slices}}{\text{total pizzas}} = \frac{24}{3} = 8\text{ slices per pizza}
\]

Write Hernandez family equation

Using the Proportional Relationship Equation knowledge point
\[
y = 8x
\]

Analyze graph properties

Using the Graphing Proportional Relationships knowledge point
\[

$$\begin{aligned} &\text{Hernandez: } y = 8x \implies (0,0) \text{ is on the line, straight line}\\ &\text{Wilson: } y = 10x \implies (0,0) \text{ is on the line, straight line} \end{aligned}$$

\]

Compare slice counts for 3 pizzas

Using the Constant of Proportionality knowledge point
\[

$$\begin{aligned} &\text{Hernandez: } 24\text{ slices}\\ &\text{Wilson: } y = 10(3) = 30\text{ slices}\\ &30 > 24 \end{aligned}$$

\]

Evaluate given options

Using the Proportional Relationship Equation knowledge point

• Option 1: Incorrect, equation is \(y = 8x\).
• Option 2: Correct, both are proportional relationships.
• Option 3: Incorrect, slice size cannot be determined from slice count.
• Option 4: Correct, 30 slices is more than 24 slices.
• Option 5: Incorrect, constant is 8 for Hernandez.

</reasoning>

<answer>
<mcq-option>For the Hernandez family, the equation \(y = 3x\) represents the relationship between the number of slices and number of pizzas, where \(x\) represents the number of pizzas and \(y\) represents the number of total slices.</mcq-option>
<mcq-correct>For each family, a graph of the relationship between number of pizzas, \(x\), and number of slices, \(y\), goes through the point \((0, 0)\) and is a straight line.</mcq-correct>
<mcq-option>The Hernandez family's slices are smaller than the Wilson family's slices.</mcq-option>
<mcq-correct>If the Wilson family ordered 3 large pizzas from the same restaurant as the Hernandez family, they would have more slices than the Hernandez family after each family cut their pizzas into slices.</mcq-correct>
<mcq-option>The constant of proportionality is 10 for the Wilson family, but cannot be determined for the Hernandez family.</mcq-option>
</answer>

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