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:

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

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Formulate the correct proportion

Using the Algebraic Error Analysis knowledge point

$$ \frac{4\text{ oranges}}{6\text{ fluid ounces}} = \frac{x\text{ oranges}}{24\text{ fluid ounces}} \implies \frac{4}{6} = \frac{x}{24} $$

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.