- Convert boundary fractions to decimal form: \(\frac{1}{2} = 0.5\) and \(\frac{4}{5} = 0.8\).
- Compare the given options (\(\frac{1}{3}, \frac{2}{4}, \frac{2}{3}, \frac{5}{6}\)) against the interval \((0.5, 0.8)\).
- Identify that \(\frac{2}{3} \approx 0.667\) is the only value strictly between \(0.5\) and \(0.8\).
- The correct fraction is \(\boxed{\frac{2}{3}}\).
Explanation
- Analyzing the Boundaries and Inequality
The problem asks us to identify which fraction from the set \(\{\frac{1}{3}, \frac{2}{4}, \frac{2}{3}, \frac{5}{6}\}\) satisfies the inequality \(\frac{1}{2} < x < \frac{4}{5}\). First, we convert the boundaries into decimal form to make comparison easier:
\[ \frac{1}{2} = 0.5 \quad \text{and} \quad \frac{4}{5} = 0.8. \]
The inequality we need to satisfy is \(0.5 < x < 0.8\).
- Evaluating the Options
Now, we convert each of the given options into decimal form or compare them directly to the range \((0.5, 0.8)\):
- Option A: \(\frac{1}{3} \approx 0.333\). Since \(0.333 < 0.5\), this value does not satisfy the condition.
- Option B: \(\frac{2}{4} = \frac{1}{2} = 0.5\). The condition requires a value strictly greater than \(0.5\), so \(0.5 < 0.5\) is false.
- Option C: \(\frac{2}{3} \approx 0.667\). Since \(0.5 < 0.667 < 0.8\), this value satisfies the condition.
- Option D: \(\frac{5}{6} \approx 0.833\). Since \(0.833 > 0.8\), this value is outside the specified range.
- Conclusion
Based on our comparison, the only fraction that satisfies the inequality \(\frac{1}{2} < x < \frac{4}{5}\) is \(\frac{2}{3}\). Thus, the correct option is C.
Examples
In real life, inequalities are used whenever you have a range of acceptable values. For instance, if you are mixing ingredients for a recipe and a specific component must be added in an amount between 0.5 and 0.8 cups to maintain the correct consistency, you can use these comparisons to ensure your measurement is within the correct bounds. Understanding how to compare fractions helps in adjusting ratios for cooking, mixing chemicals, or even scaling financial investments.