- For Part (a), note that fractions with the same positive numerator decrease as the denominator increases, resulting in the order \(\frac{7}{10} > \frac{7}{13} > \frac{7}{15}\).
- For Part (b), compare the decimal values or use common denominators to find that the decreasing order is \(-\frac{3}{7} > -\frac{11}{16} > -\frac{31}{35}\).
- Match the result for part (a) to the provided options, which gives the correct selection as Option A.
- Final answer: \(\boxed{A}\)
Explanation
- Arranging Part (a) in Decreasing Order
To arrange the fractions in Part (a), \(\frac{7}{10}, \frac{7}{13}, \frac{7}{15}\), we observe that they all have the same positive numerator, \(7\). When fractions have the same positive numerator, the value of the fraction decreases as the denominator increases. Comparing the denominators, we have \(10 < 13 < 15\). Thus, the order of these fractions from greatest to smallest is \(\frac{7}{10} > \frac{7}{13} > \frac{7}{15}\).
- Arranging Part (b) in Decreasing Order
To arrange the negative fractions in Part (b), \(-\frac{31}{35}, -\frac{11}{16}, -\frac{3}{7}\), it is helpful to compare their magnitudes. First, we find decimal approximations:
\(-\frac{31}{35} \approx -0.8857\)
\(-\frac{11}{16} = -0.6875\)
\(-\frac{3}{7} \approx -0.4286\)
Since \(-0.8857 < -0.6875 < -0.4286\), the numbers in increasing order are \(-\frac{31}{35}, -\frac{11}{16}, -\frac{3}{7}\). Consequently, the decreasing order is \(-\frac{3}{7}, -\frac{11}{16}, -\frac{31}{35}\).
- Conclusion and Selecting the Option
Comparing our result for Part (a), which is \(\frac{7}{10}, \frac{7}{13}, \frac{7}{15}\), with the provided options:
Option A: \(\frac{7}{10}, \frac{7}{13}, \frac{7}{15}\)
Option B: \(\frac{7}{15}, \frac{7}{13}, \frac{7}{10}\)
Option C: \(\frac{7}{10}, \frac{7}{15}, \frac{7}{13}\)
Option D: \(\frac{7}{13}, \frac{7}{10}, \frac{7}{15}\)
The correct sequence is Option A.
Examples
Understanding how to order fractions is essential in daily life, such as when comparing discount percentages or portion sizes. For instance, if you are looking at different sales offers, knowing that \(\frac{1}{2}\) off is a larger discount than \(\frac{1}{4}\) off helps you make better financial decisions. Similarly, if you are splitting resources, like land or food, being able to identify which fraction is larger ensures you understand how much of a whole each person receives. Accurate comparison of these values is a foundational skill for budgeting, cooking, and resource management.