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Let's work through the question, which asks for **eight rational numbers** between the given pairs of fractions:

### (i) Between \(-\frac{1}{3}\) and \(\frac{2}{7}\)
We find fractions between these two values by finding their decimal equivalents:
\[
-\frac{1}{3} \approx -0.333, \quad \frac{2}{7} \approx 0.2857
\]
Thus, we need eight fractions in the interval \([-0.333, 0.2857]\). Examples:
\[
-\frac{5}{15}, -\frac{1}{5}, -\frac{1}{6}, -\frac{1}{10}, 0, \frac{1}{10}, \frac{1}{6}, \frac{1}{5}
\]

### (ii) Between \(\frac{2}{3}\) and \(\frac{4}{3}\)
Decimal equivalents:
\[
\frac{2}{3} \approx 0.6667, \quad \frac{4}{3} \approx 1.333
\]
Eight rational numbers between them:
\[
\frac{3}{4}, \frac{7}{10}, \frac{5}{6}, 1, \frac{7}{6}, \frac{5}{4}, \frac{3}{2}, \frac{11}{8}
\]

### (iii) Between \(-\frac{3}{4}\) and \(-\frac{1}{2}\)
Decimal equivalents:
\[
-\frac{3}{4} \approx -0.75, \quad -\frac{1}{2} \approx -0.5
\]
Eight rational numbers between them:
\[
-\frac{5}{8}, -\frac{11}{16}, -\frac{9}{16}, -\frac{13}{24}, -\frac{7}{12}, -\frac{2}{3}, -\frac{3}{5}, -\frac{14}{25}
\]

### (iv) Between \(-\frac{1}{5}\) and \(\frac{1}{5}\)
Decimal equivalents:
\[
-\frac{1}{5} \approx -0.2, \quad \frac{1}{5} \approx 0.2
\]
Eight rational numbers between them:
\[
-\frac{1}{10}, -\frac{1}{15}, -\frac{1}{25}, 0, \frac{1}{25}, \frac{1}{15}, \frac{1}{10}, \frac{1}{9}
\]

Let me know if you'd like further clarifications!

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Here are 5 related questions:
1. Can you explain how to find rational numbers between any two given fractions?
2. What is the relationship between decimal and fraction forms of numbers?
3. How can we verify if a number is rational?
4. Can you find irrational numbers between two given fractions instead of rational ones?
5. What is the general method for simplifying fractions?

**Tip:** When finding rational numbers, ensure the denominators are compatible for easier interpolation!

Find eight rational numbers between: (i) -1/3 and 2/7 (ii) 2/3 and 4/3 (iii) -3/4 and -1/2 (iv) -1/5 and 1/5

Learn how to find eight rational numbers between pairs of fractions, such as -1/3 and 2/7 or 2/3 and 4/3. This problem demonstrates rational number interpolation and fraction manipulation, perfect for Grade 8 students.

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