Question

explaining the error in a proportion

a student uses the ratio of 4 oranges to 6 fluid ounces to find the number of oranges needed to make 24 fluid ounces of juice. the student writes this proportion:

\\\frac{4}{6} = \frac{24}{16}\\

explain the error in the students work.

Explanation:

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<reasoning>

Identify the given ratio

Using the Algebraic Error Analysis knowledge point
\[
\text{Given ratio} = \frac{4\text{ oranges}}{6\text{ fluid ounces}}
\]

Analyze the student's proportion

Using the Algebraic Error Analysis knowledge point
\[
\text{Student's proportion: } \frac{4}{6} = \frac{24}{16}
\]

Identify the error in setup

Using the Algebraic Error Analysis knowledge point
\[

$$\begin{aligned} &\text{Left side: } \frac{\text{oranges}}{\text{fluid ounces}} = \frac{4}{6}\\ &\text{Right side: } \frac{\text{fluid ounces}}{\text{oranges}} = \frac{24}{16} \end{aligned}$$

\]

Formulate the correct proportion

Using the Algebraic Error Analysis knowledge point
\[
\frac{4\text{ oranges}}{6\text{ fluid ounces}} = \frac{x\text{ oranges}}{24\text{ fluid ounces}} \implies \frac{4}{6} = \frac{x}{24}
\]
</reasoning>

<answer>
The student set up the proportion incorrectly by not aligning the corresponding units in the numerators and denominators.

On the left side, the ratio is written as \(\frac{\text{oranges}}{\text{fluid ounces}}\) (\(\frac{4}{6}\)). On the right side, the student inverted the relationship and wrote it as \(\frac{\text{fluid ounces}}{\text{oranges}}\) (\(\frac{24}{16}\)).

To correct the error, the units must match across both ratios. The correct proportion should be:
\[\frac{4}{6} = \frac{x}{24}\]
where \(x\) represents the unknown number of oranges.
</answer>

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Answer:

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<reasoning>

Identify the given ratio

Using the Algebraic Error Analysis knowledge point
\[
\text{Given ratio} = \frac{4\text{ oranges}}{6\text{ fluid ounces}}
\]

Analyze the student's proportion

Using the Algebraic Error Analysis knowledge point
\[
\text{Student's proportion: } \frac{4}{6} = \frac{24}{16}
\]

Identify the error in setup

Using the Algebraic Error Analysis knowledge point
\[

$$\begin{aligned} &\text{Left side: } \frac{\text{oranges}}{\text{fluid ounces}} = \frac{4}{6}\\ &\text{Right side: } \frac{\text{fluid ounces}}{\text{oranges}} = \frac{24}{16} \end{aligned}$$

\]

Formulate the correct proportion

Using the Algebraic Error Analysis knowledge point
\[
\frac{4\text{ oranges}}{6\text{ fluid ounces}} = \frac{x\text{ oranges}}{24\text{ fluid ounces}} \implies \frac{4}{6} = \frac{x}{24}
\]
</reasoning>

<answer>
The student set up the proportion incorrectly by not aligning the corresponding units in the numerators and denominators.

On the left side, the ratio is written as \(\frac{\text{oranges}}{\text{fluid ounces}}\) (\(\frac{4}{6}\)). On the right side, the student inverted the relationship and wrote it as \(\frac{\text{fluid ounces}}{\text{oranges}}\) (\(\frac{24}{16}\)).

To correct the error, the units must match across both ratios. The correct proportion should be:
\[\frac{4}{6} = \frac{x}{24}\]
where \(x\) represents the unknown number of oranges.
</answer>

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