- Division (\(5 \div 8\)): The solution to \(8x = 5\) is \(\frac{5}{8}\) (Option C).
- Part of a whole: Jane ate \(\frac{5}{8}\) of the pizza (Option A).
- Ratio: The ratio of boys to girls is \(5\) to \(8\) (Option B).
- The final mappings are: \(\boxed{a \to C, b \to A, c \to B}\).
Explanation
- Analyzing the Interpretations of the Fraction \(\frac{5}{8}\)
The fraction \(\frac{5}{8}\) can represent different mathematical concepts depending on the context in which it is used. To solve this, we analyze each requested interpretation and map it to the provided options:
AAs a division problem: The fraction \(\frac{a}{b}\) is equivalent to \(a \div b\). Thus, \(\frac{5}{8}\) represents \(5 \div 8\). Setting up the equation \(8x = 5\), we find \(x = \frac{5}{8}\), which directly corresponds to option C.
BAs part of a whole: This interpretation views the fraction as taking \(5\) equal pieces out of a total of \(8\) equal pieces that make up a whole object. Option A, "Jane ate \(\frac{5}{8}\) of the pizza," illustrates this perfectly, as the pizza is the whole, divided into \(8\) slices, and Jane consumes \(5\) of them.
CAs a ratio: A ratio compares two quantities, indicating how many times one quantity relates to another. The expression \(\frac{5}{8}\) can describe a relationship of 5 units to 8 units. Option B, "The ratio of boys to girls is 5 to 8," explicitly states this comparison between two distinct groups.
Option D states \(5x = 8\), which leads to \(x = \frac{8}{5}\), not \(\frac{5}{8}\), and therefore does not represent the requested fraction.
- Matching Interpretations to Sentences
Based on our analysis, we have successfully matched the interpretations to the sentences:
- Interpretation (a) (division) corresponds to option C: The solution to \(8x = 5\) is \(\frac{5}{8}\).
- Interpretation (b) (part of a whole) corresponds to option A: Jane ate \(\frac{5}{8}\) of the pizza.
- Interpretation (c) (ratio) corresponds to option B: The ratio of boys to girls is 5 to 8.
Examples
Understanding different interpretations of fractions is essential for daily life. For instance, if you are baking a cake and need to split a recipe (part of a whole), calculate ingredients for a group (division), or mix liquids in a specific proportion (ratio), you are using these concepts constantly. Recognizing whether a fraction represents a division problem, a portion of an item, or a comparative relationship helps in choosing the right mathematical tools to solve everyday problems efficiently.