The arrangement of rational numbers $$\frac{-7}{10}, \frac{5}{-8}, \frac{2}{-3}$$ in ascending order is

### Concept & Strategy To arrange negative fractions in ascending order, standardize the negative signs to the numerator, calculate their decimal values (or make denominators equal), and order them from the largest magnitude negative number to the smallest. ### Step-by-Step Solution * First, rewrite the fractions in a standard format: $$\frac{-7}{10}, \frac{-5}{8}, \frac{-2}{3}$$ * Next, convert them into approximate decimal values for easy comparison: * $$\frac{-7}{10} = -0.700$$ * $$\frac{-5}{8} = -0.625$$ * $$\frac{-2}{3} \approx -0.666$$ * Remember that for negative numbers, a larger absolute value means a smaller actual value. * Comparing the decimals: $$-0.700 < -0.666 < -0.625$$ * Replacing them with their original fractional forms: $$\frac{-7}{10} < \frac{2}{-3} < \frac{5}{-8}$$ ### Exam Strategy & Shortcut **Common Denominator (LCM):** If you dislike decimals, find the LCM of $$10, 8$$, and $$3$$, which is $$120$$. * $$\frac{-7 \times 12}{120} = \frac{-84}{120}$$ * $$\frac{-5 \times 15}{120} = \frac{-75}{120}$$ * $$\frac{-2 \times 40}{120} = \frac{-80}{120}$$ Comparing numerators directly: $$-84 < -80 < -75$$. The correct order instantly reveals itself as $$\frac{-7}{10}, \frac{2}{-3}, \frac{5}{-8}$$. ### Common Pitfall Treating negative fractions like positive ones. Students often order them as $$0.625, 0.666, 0.700$$ and accidentally arrange them in descending order instead of ascending. Always double-check the negative sign rules. ### Final Answer **Therefore, the correct answer is option (d).**