To arrange the rational numbers $-\frac{7}{10}, \frac{5}{-8}, \frac{2}{-3}$ in ascending order, we first express them with positive denominators and then find a common denominator.

Standardizing Rational Numbers

Rewrite the given fractions to have positive denominators:

• $-\frac{7}{10}$ remains $-\frac{7}{10}$
• $\frac{5}{-8}$ becomes $-\frac{5}{8}$
• $\frac{2}{-3}$ becomes $-\frac{2}{3}$

The numbers to order are now: $-\frac{7}{10}, -\frac{5}{8}, -\frac{2}{3}$.

Finding Common Denominator

The denominators are 10, 8, and 3. We find the Least Common Multiple (LCM) of these numbers.

• Prime factorization of 10 = $2 \times 5$
• Prime factorization of 8 = $2^3$
• Prime factorization of 3 = $3$
• LCM(10, 8, 3) = $2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120$.

The common denominator is 120.

Converting to Equivalent Fractions

Convert each fraction to an equivalent fraction with a denominator of 120:

• $-\frac{7}{10} = -\frac{7 \times 12}{10 \times 12} = -\frac{84}{120}$
• $-\frac{5}{8} = -\frac{5 \times 15}{8 \times 15} = -\frac{75}{120}$
• $-\frac{2}{3} = -\frac{2 \times 40}{3 \times 40} = -\frac{80}{120}$

Comparing and Ordering

We need to arrange the fractions $-\frac{84}{120}, -\frac{75}{120}, -\frac{80}{120}$ in ascending order. Since all numbers are negative, the number with the largest numerator (in absolute value) is the smallest.

Comparing the numerators: -84, -75, -80.

The ascending order of the numerators is -84, -80, -75.

Therefore, the ascending order of the fractions is:

$-\frac{84}{120} < -\frac{80}{120} < -\frac{75}{120}$

Final Ascending Order

Convert the ordered fractions back to their original form:

• $-\frac{84}{120}$ corresponds to $-\frac{7}{10}$
• $-\frac{80}{120}$ corresponds to $-\frac{2}{3}$ (or $\frac{2}{-3}$)
• $-\frac{75}{120}$ corresponds to $-\frac{5}{8}$ (or $\frac{5}{-8}$)

The rational numbers in ascending order are $-\frac{7}{10}, \frac{2}{-3}, \frac{5}{-8}$.