To solve the problem of finding eight rational numbers between the given pairs of fractions, here’s how it can be approached:

(i) Between -1/3 and 2/7

1. Convert these fractions to have a common denominator (e.g., 21):

   • $-\frac{1}{3} = -\frac{7}{21}$
   • $\frac{2}{7} = \frac{6}{21}$

2. Rational numbers between $-\frac{7}{21}$ and $\frac{6}{21}$: Examples: $-\frac{6}{21}, -\frac{5}{21}, -\frac{4}{21}, 0, \frac{1}{21}, \frac{2}{21}, \frac{3}{21}, \frac{5}{21}$.

(ii) Between 2/3 and 4/3

1. Convert these fractions to have a common denominator (e.g., 3):

   • $\frac{2}{3}$ and $\frac{4}{3}$ already have common denominators.

2. Rational numbers between $\frac{2}{3}$ and $\frac{4}{3}$: Examples: $\frac{7}{9}, \frac{8}{9}, \frac{5}{6}, 1, \frac{11}{9}, \frac{4}{3}$.

(iii) Between 3/-4 and -1/2

1. Convert these fractions to have a common denominator (e.g., 8):

   • $\frac{3}{-4} = -\frac{6}{8}$
   • $-\frac{1}{2} = -\frac{4}{8}$

2. Rational numbers between $-\frac{6}{8}$ and $-\frac{4}{8}$: Examples: $-\frac{5}{8}, -\frac{9}{16}, -\frac{10}{16}, -\frac{7}{16}, -\frac{3}{8}$.

(iv) Between -1/5 and 1/5

1. Convert to common denominators:

   • $-\frac{1}{5}$ and $\frac{1}{5}$ already have common denominators.

2. Rational numbers between $-\frac{1}{5}$ and $\frac{1}{5}$: Examples: $-\frac{3}{25}, -\frac{2}{25}, 0, \frac{1}{25}, \frac{2}{25}$.

(v) Between -2 and -1

1. Rational numbers between $-2$ and $-1$: Examples: $-\frac{9}{5}, -\frac{8}{5}, -\frac{3}{2}, -\frac{5}{4}, -\frac{4}{3}$.

(vi) Between -2 and 0

1. Rational numbers between $-2$ and $0$: Examples: $-\frac{7}{4}, -\frac{6}{4}, -\frac{5}{4}, -1, -\frac{3}{2}, -\frac{4}{5}, -\frac{1}{2}$.

Would you like a more detailed explanation or breakdown for any specific part? Let me know!

Related Questions:

1. How do you determine common denominators for fractions?
2. What defines a rational number?
3. Can irrational numbers exist between these pairs?
4. How do we graph these rational numbers on a number line?
5. What strategies can we use for similar problems?

Tip: Always convert fractions to a common denominator for clarity when finding rational numbers between them.